![Preface
This is a compact introduction to some of the principal topics of mathemat-
ical logic. In the belief that beginners should be exposed to the easiest and
most natural proofs, I have used free-swinging set-theoretic methods. The
significance of a demand for constructive proofs can be evaluated only after a
certain amount of experience with mathematical logic has been obtained. If
we are to be expelled from ‘‘Cantor’s paradise’’ (as nonconstructive set
theory was called by Hilbert), at least we should know what we would be
missing. Readers who wish to learn about constructive mathematics and
intuitionism can consult the books by van Atten (2003), van Atten, Boldini,
Bourdeau, and Heinzmann (2008), Brouwer (1976), George and Velleman
(2001), Heyting (1956), Mancosu (1997), and Troelstra and van Dalen (1988).
The more important changes in this new edition are the following:
1. There is an entirely new exposition, Section 3.7, of some basic ideas
and results about nonstandard models of number theory.
2. A second appendix, Appendix B, has been added as an introduction
to modal propositional logic. The most important application in this
area has to do with provability logic (see Boolos [1993]), but it is also
of great interest elsewhere in logic.
3. The Bibliography has been greatly enlarged, although it is still not
able to do full justice to the enormous expansion of research and
scholarship in logic and related areas.
4. The exercises and the Answers to Selected Exercises have been
expanded and necessary corrections have been made.
The material in this book can be covered in two semesters, but Chapters 1–3
are quite adequate for a one-semester course (Sections 1.5, 1.6, 3.7, and
2.10–2.16 can be omitted in an abbreviated course). I have adopted the
convention of prefixing a ‘‘D’’ to any section or exercise that will probably
be difficult for a beginner, and an ‘‘A’’ to any section or exercise that
presupposes familiarity with a topic that has not been carefully explained
in the text. Bibliographic references are given to the best source of informa-
tion, which is not always the earliest; hence these references give no indi-
cation as to priority.
I believe that the essential parts of the book can be read with ease by
anyone with some experience in abstract mathematical thinking. There is,
however, no specific prerequisite.
xiii](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-18-320.jpg)
![containing all sets. Now, consider the set A of all those sets X such that
X is not a member of X. Clearly, by definition, A is a member of A if
and only if A is not a member of A. So, if A is a member of A, then A is
also not a member of A; and if A is not a member of A, then A is a
member of A. In any case, A is a member of A and A is not a member
of A [See Link, (2004)].
2. Cantor’s paradox (1899). This paradox involves the theory of cardinal
numbers and may be skipped by those readers having no previous
acquaintance with that theory. The cardinal number Y of a set Y is a
measure of the size of the set; Y ¼ Z if and only if Y is equinumerous
with Z (that is, there is a one–one correspondence between Y and Z).
We define Y 4 Z to mean that Y is equinumerous with a subset of Z;
by Y < Z we mean Y 4 Z and Y 6¼ Z. Cantor proved that, if p(Y) is
the set of all subsets of Y, then Y < p(Y). Let V be the universal set—
that is, the set of all sets. Now, p(V) is a subset of V; so it follows
easily that p(V) 4 V. On the other hand, by Cantor’s theorem,
V < p(V). Bernstein’s theorem asserts that, if Y 4 Z and Z 4 Y,
then Y ¼ Z. Hence, V ¼ p(V), contradicting V < p(V).
3. Burali-Forti’s paradox (1897). This paradox is the analogue in the
theory of ordinal numbers of Cantor’s paradox and requires famil-
iarity with ordinal number theory. Given any ordinal number, there
is a still larger ordinal number. But the ordinal number determined
by the set of all ordinal numbers is the largest ordinal number.
4. The liar paradox. A man says, ‘‘I am lying’’. If he is lying, then what he
says is true and so he is not lying. If he is not lying, then what he says is
true, and so he is lying. In any case, he is lying and he is not lying.*
5. Richard’s paradox (1905). Some phrases of the English language
denote real numbers; for example, ‘‘the ratio between the circumfer-
ence and diameter of a circle’’ denotes the number p. All the phrases
of the English language can be enumerated in a standard way: order
all phrases that have k letters lexicographically (as in a dictionary)
and then place all phrases with k letters before all phrases with a
larger number of letters. Hence, all phrases of the English language
that denote real numbers can be enumerated merely by omitting all
other phrases in the given standard enumeration. Call the nth real
* The Cretan ‘‘paradox,’’ known in antiquity, is similar to the liar paradox. The Cretan philoso-
pher Epimenides said, ‘‘All Cretans are liars.’’ If what he said is true, then, since Epimenides is
a Cretan, it must be false. Hence, what he said is false. Thus, there must be some Cretan who is
not a liar. This is not logically impossible; so we do not have a genuine paradox. However, the
fact that the utterance by Epimenides of that false sentence could imply the existence of some
Cretan who is not a liar is rather unsettling.
xvi Introduction](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-21-320.jpg)
![number in this enumeration the nth Richard number. Consider the
phrase: ‘‘the real number whose nth decimal place is 1 if the nth
decimal place of the nth Richard number is not 1, and whose nth
decimal place is 2 if the nth decimal place of the nth Richard number
is 1.’’ This phrase defines a Richard number—say, the kth Richard
number; but, by its definition, it differs from the kth Richard number
in the kth decimal place.
6. Berry’s paradox (1906). There are only a finite number of symbols
(letters, punctuation signs, etc.) in the English language. Hence, there
are only a finite number of English expressions that contain fewer
than 200 occurrences of symbols (allowing repetitions). There are,
therefore, only a finite number of positive integers that are denoted
by an English expression containing fewer than 200 occurrences of
symbols. Let k be the least positive integer that is not denoted by an
English expression containing fewer than 200 occurrences of symbols. The
italicized English phrase contains fewer than 200 occurrences of
symbols and denotes the integer k.
7. Grelling’s paradox (1908). An adjective is called autological if the prop-
erty denoted by the adjective holds for the adjective itself. An adjec-
tive is called heterological if the property denoted by the adjective
does not apply to the adjective itself. For example, ‘‘polysyllabic’’
and ‘‘English’’ are autological, whereas ‘‘monosyllabic’’ and
‘‘French’’ are heterological. Consider the adjective ‘‘heterological.’’
If ‘‘heterological’’ is heterological, then it is not heterological. If
‘‘heterological’’ is not heterological, then it is heterological. In either
case, ‘‘heterological’’ is both heterological and not heterological.
8. Löb’s paradox (1955). Let A be any sentence. Let B be the sentence: ‘‘If
this sentence is true, then A.’’ So, B asserts: ‘‘If B is true, then A.’’
Now consider the following argument: Assume B is true; then, by B,
since B is true, A holds. This argument shows that, if B is true, then A.
But this is exactly what B asserts. Hence, B is true. Therefore, by B,
since B is true, A is true. Thus, every sentence is true. (This paradox
may be more accurately attributed to Curry [1942].)
All of these paradoxes are genuine in the sense that they contain no obvious
logical flaws. The logical paradoxes (1–3) involve only notions from the theory
of sets, whereas the semantic paradoxes (4–8) also make use of concepts like
‘‘denote,’’ ‘‘true,’’ and ‘‘adjective,’’ which need not occur within our standard
mathematical language. For this reason, the logical paradoxes are a much
greater threat to a mathematician’s peace of mind than the semantic paradoxes.
Analysis of the paradoxes has led to various proposals for avoiding them.
All of these proposals are restrictive in one way or another of the ‘‘naive’’
concepts that enter into the derivation of the paradoxes. Russell noted the
self-reference present in all the paradoxes and suggested that every object
Introduction xvii](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-22-320.jpg)
![b1 ¼ c1, b2 ¼ c2, . . . , bk ¼ ck. Thus, hb1, b2i ¼ hb2, b1i if and only if b1 ¼ b2.
Ordered 2-tuples are called ordered pairs. The ordered 1-tuple hbi is taken to be
b itself. If X is a set and k is a positive integer, we denote by Xk
the set of all
ordered k-tuples hb1, . . . , bki of elements b1, . . . , bk of X. In particular, X1
is X
itself.IfYandZaresets,thenbyY Zwedenotethesetofallorderedpairshy, zi
such that y 2 Y and z 2 Z. Y Z is called the Cartesian product of Y and Z.
An n-place relation (or a relation with n arguments) on a set X is a subset of
Xn
—that is, a set of ordered n-tuples of elements of X. For example, the
3-place relation of betweenness for points on a line is the set of all 3-tuples
hx, y, zi such that the point x lies between the points y and z. A 2-place
relation is called a binary relation; for example, the binary relation of father-
hood on the set of human beings is the set of all ordered pairs hx, yi such that
x and y are human beings and x is the father of y. A 1-place relation on X is a
subset of X and is called a property on X.
Given a binary relation R on a set X, the domain of R is defined to be the set
of all y such that hy, zi 2 R for some z; the range of R is the set of all z such that
hy, zi 2 R for some y; and the field of R is the union of the domain and range
of R. The inverse relation R1
of R is the set of all ordered pairs hy, zi such that
hz, yi 2 R. For example, the domain of the relation on the set v of nonne-
gative integers* is v, its range is v {0}, and the inverse of is . Notation:
Very often xRy is written instead of hx, yi 2 R. Thus, in the example just
given, we usually write x y instead of hx, yi 2 .
A binary relation R is said to be reflexive if xRx for all x in the field of R; R is
symmetric if xRy implies yRx; and R is transitive if xRy and yRz imply xRz.
Examples: The relation on the set of integers is reflexive and transitive but
not symmetric. The relation ‘‘having at least one parent in common’’ on the
set of human beings is reflexive and symmetric, but not transitive.
A binary relation that is reflexive, symmetric, and transitive is called an
equivalence relation. Examples of equivalence relations are: (1) the identity
relation IX on a set X, consisting of all pairs hx, xi, where x 2 X; (2) given a
fixed positive integer n, the relation x y (mod n), which holds when x and y
are integers and x y is divisible by n; (3) the congruence relation on the set
of triangles in a plane; (4) the similarity relation on the set of triangles in a
plane. Given an equivalence relation R whose field is X, and given any y 2 X,
define [y] as the set of all z in X such that yRz. Then [y] is called the
R-equivalence class of y. Clearly, [u] ¼ [v] if and only if uRv. Moreover, if
[u] 6¼ [v], then [u] [v] ¼ ;; that is, different R-equivalence classes have no
elements in common. Hence, the set X is completely partitioned into the
R-equivalence classes. In example (1) above, the equivalence classes are just
the unit sets {x}, where x 2 X. In example (2), there are n equivalence classes,
the kth equivalence class (k ¼ 0, 1, . . . , n 1) being the set of all integers that
leave the remainder k upon division by n.
* v will also be referred to as the set of natural numbers.
xx Introduction](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-25-320.jpg)
![show that, if X ffi Y1 Y and Y ffi X1 X, then X ffi Y (see Bernstein’s the-
orem in Chapter 4). If X ffi Y, one says that X and Y have the same cardinal
number, and if X is equinumerous with a subset of Y but Y is not equinumer-
ous with a subset of X, one says that the cardinal number of X is smaller than
the cardinal number of Y.*
A set X is denumerable if it is equinumerous with the set of positive integers.
A denumerable set is said to have cardinal number Q0, and any set equinu-
merous with the set of all subsets of a denumerable set is said to have the
cardinal number 2Q0
(or to have the power of the continuum). A set X is finite if
it is empty or if it is equinumerous with the set {1, 2, . . . , n} of all positive
integers that are less than or equal to some positive integer n. A set that is not
finite is said to be infinite. A set is countable if it is either finite or denumerable.
Clearly, any subset of a denumerable set is countable. A denumerable sequence
is a function s whose domain is the set of positive integers; one usually writes
sn instead of s(n). A finite sequence is a function whose domain is the empty set
or {1, 2, . . . , n} for some positive integer n.
Let P x, y1, . . . , yk
ð Þ be some relation on the set of nonnegative integers.
In particular, P may involve only the variable x and thus be a property.
If P 0, y1, . . . , yk
ð Þ holds, and, if, for every n, P n, y1, . . . , yk
ð Þ implies
P n þ 1, y1, . . . , yk
ð Þ, then P x, y1, . . . , yk
ð Þ is true for all nonnegative integers x
(principle of mathematical induction). In applying this principle, one usually
proves that, for every n, P n, y1, . . . , yk
ð Þ implies P n þ 1, y1, . . . , yk
ð Þ by assum-
ing P n, y1, . . . , yk
ð Þ and then deducing P n þ 1, y1, . . . , yk
ð Þ; in the course of this
deduction, P n, y1, . . . , yk
ð Þ is called the inductive hypothesis. If the relation P
actually involves variables y1, . . . , yk other than x, then the proof is said to
proceed by induction on x. A similar induction principle holds for the set
of integers greater than some fixed integer j. An example is: to prove
by mathematical induction that the sum of the first n odd integers
1 þ 3 þ 5 þ þ (2n 1) is n2
, first show that 1 ¼ 12
(that is, P(1)), and
then, that if 1 þ 3 þ 5 þ þ (2n 1) ¼ n2
, then 1 þ 3 þ 5 þ þ (2n 1) þ
(2n þ 1) ¼ (n þ 1)2
(that is, if P(n) then P(n þ 1)). From the principle of math-
ematical induction one can prove the principle of complete induction: If, for
every nonnegative integer x the assumption that P u, y1, . . . , yk
ð Þ is true for all
u x implies that P x, y1, . . . , yk
ð Þ holds, then, for all nonnegative integers
x, P x, y1, . . . , yk
ð Þ is true, (Exercise: Show by complete induction that every
integer greater than 1 is divisible by a prime number.)
A partial order is a binary relation R such that R is transitive and, for every x
in the field of R, xRx is false. If R is a partial order, then the relation R0
that is
the union of R and the set of all ordered pairs hx, xi, where x is in the field of
R, we shall call a reflexive partial order; in the literature, ‘‘partial order’’ is used
for either partial order or reflexive partial order. Notice that (xRy and yRx) is
* One can attempt to define the cardinal number of a set X as the collection [X] of all sets
equinumerous with X. However, in certain axiomatic set theories, [X] does not exist, whereas in
others [X] exists but is not a set.
xxii Introduction](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-27-320.jpg)
![Proof
Notice that b _ c is logically equivalent to :(:b ^ :c). Hence, by the
second part of Proposition 1.4, any statement form in ^, _, and : is logically
equivalent to a statement form in only ^ and : [obtained by replacing all
expressions b _ c by :(:b ^ :c)]. The other parts of the corollary are
similar consequences of the following tautologies:
b ^ c , :(:b _ :c)
b _ c , (:b ) c)
b ^ c , :(b ) :c)
We have just seen that there are certain pairs of connectives — for example,
^ and : — in terms of which all truth functions are definable. It turns out that
there is a single connective, # (joint denial), that will do the same job. Its truth
table is
A B A # B
T T F
F T F
T F F
F F T
A # B is true when and only when neither A nor B is true. Clearly,
:A , (A # A) and (A ^ B) , ((A # A) # (B # B)) are tautologies. Hence,
the adequacy of # for the construction of all truth functions follows from
Corollary 1.6.
Another connective, j (alternative denial), is also adequate for this purpose.
Its truth table is
A B AjB
T T F
F T T
T F T
F F T
AjB is true when and only when not both A and B are true. The adequacy of j
follows from the tautologies :A , (AjA) and (A _ B) , ((AjA)j(BjB)).
PROPOSITION 1.7
The only binary connectives that alone are adequate for the construction of
all truth functions are # and j.
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![(B _ C) ^ (A _ B), (B _ :C) ^ (A _ D), A ^ (B _ A) ^ (:B _ A), A _ :B,
A ^ B, A. Note that our terminology considers a literal to be a (degenerate)
conjunction and a (degenerate) disjunction.
(a) The proof of Proposition 1.5 shows that every statement form b is
logically equivalent to one in disjunctive normal form. By applying
this result to :b, prove that b is also logically equivalent to a form
in conjunctive normal form.
(b) Find logically equivalent dnfs and cnfs for :(A ) B) _ (:A ^ C) and
A , ((B ^ :A) _ C). [Hint: Instead of relying on Proposition 1.5, it is
usually easier to use Exercise 1.27(b) and (c).]
(c) A dnf (cnf) is called full if no disjunct (conjunct) contains two
occurrences of literals with the same letter and if a letter that occurs
in one disjunct (conjunct) also occurs in all the others. For example,
(A ^ :A ^ B) _ (A ^ B), (B ^ B ^ C) _ (B ^ C) and (B ^ C) _ B are
not full, whereas (A ^ B ^ :C) _ (A ^ B ^ C) _ (A ^ :B ^ :C) and
(A ^ :B) _ (B ^ A) are full dnfs.
(i) Find full dnfs and cnfs logically equivalent to (A ^ B) _ :A and
:(A ) B) _ (:A ^ C).
(ii) Prove that every noncontradictory (nontautologous) statement
form b is logically equivalent to a full dnf (cnf) c, and, if c
contains exactly n letters, then b is a tautology (is contradictory)
if and only if c has 2n
disjuncts (conjuncts).
(d) For each of the following, find a logically equivalent dnf (cnf), and
then find a logically equivalent full dnf (cnf):
(i) (A _ B) ^ (:B _ C)
(ii) :A _ (B ) :C)
(iii) (A ^ :B) _ (A ^ C)
(iv) (A _ B) , :C
(e) Construct statement forms in : and ^ (respectively, in : and _ or in
: and )) logically equivalent to the statement forms in (d).
1.43 A statement form is said to be satisfiable if it is true for some assignment
of truth values to its statement letters. The problem of determining the
satisfiability of an arbitrary cnf plays an important role in the theory of
computational complexity; it is an example of a so-called np-complete
problem (see Garey and Johnson, 1978).
(a) Show that b is satisfiable if and only if :b is not a tautology.
(b) Determine whether the following are satisfiable:
(i) (A _ B) ^ (:A _ B _ C) ^ (:A _ :B _ :C)
(ii) ((A ) B) _ C) , (:B ^ (A _ C))
(c) Given a disjunction d of four or more literals: L1 _ L2 _ . . . _ Ln, let
C1, . . . , Cn2 be statement letters that do not occur in d, and con-
struct the cnf e:
(L1 _ L2 _ C1) ^ (:C1 _ L3 _ C2) ^ (:C2 _ L4 _ C3) ^ . . .
^ (:Cn3 _ Ln1 _ Cn2) ^ (:Cn2 _ Ln _ :C1)
22 Introduction to Mathematical Logic](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-51-320.jpg)
![contradiction is (A _ B) ^ B ^ (C _ D) ^ :B. Prove that if b is unsatis-
fiable, then res(b) is a blatant contradiction. [Hint: Use induction on
the number n of letters that occur in b. In the induction step, use (c).]
(e) Prove that b is unsatisfiable if and only if res(b) is a blatant contra-
diction.
1.45 Let b and d be statement forms such that b ) d is a tautology.
(a) If b and d have no statement letters in common, show that either b
is contradictory or d is a tautology.
(b) (Craig’s interpolation theorem) If b and d have the statement letters
B1, . . . , Bn in common, prove that there is a statement form c having
B1, . . . , Bn as its only statement letters such that b ) c and c ) d
are tautologies.
(c) Solve the special case of (b) in which b is (B1 ) A) ^ (A ) B2) and
d is (B1 ^ C) ) (B2 ^ C).
1.46 (a) A certain country is inhabited only by truth-tellers (people who
always tell the truth) and liars (people who always lie). Moreover,
the inhabitants will respond only to yes or no questions. A tourist
comes to a fork in a road where one branch leads to the capital and
the other does not. There is no sign indicating which branch to take,
but there is a native standing at the fork. What yes or no question
should the tourist ask in order to determine which branch to take?
[Hint: Let A stand for ‘‘You are a truth-teller’’ and let B stand for
‘‘The left-hand branch leads to the capital.’’ Construct, by means of
a suitable truth table, a statement form involving A and B such that
the native’s answer to the question as to whether this statement
form is true will be yes when and only when B is true.]
(b) In a certain country, there are three kinds of people: workers (who
always tell the truth), businessmen (who always lie), and students
(who sometimes tell the truth and sometimes lie). At a fork in the
road, one branch leads to the capital. A worker, a businessman and
a student are standing at the side of the road but are not identifiable
in any obvious way. By asking two yes or no questions, find out
which fork leads to the capital (Each question may be addressed to
any of the three.)
More puzzles of this kind may be found in Smullyan (1978, Chapter 3;
1985, Chapters 2, 4–8).
1.4 An Axiom System for the Propositional Calculus
Truth tables enable us to answer many of the significant questions concern-
ing the truth-functional connectives, such as whether a given statement form
is a tautology, is contradictory, or neither, and whether it logically implies or
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![were placed, side by side, that fifty could go up abreast. The blue
Voltigeur flag, now full of holes, was planted on the parapet. A tide
of brave Americans overflowed the fort. Resistance was vain. A little
before half-past nine Bravo gave up his diamond-hilted sword, and
the tricolor, that had been waving placidly amidst the uproar, came
down with a jerk.[17]
Fire was opened then upon the Mexicans at the gateway below, and
fearless Captain Roberts of Casey’s storming party, at the head of all
the troops on the causeway and supported by General Smith’s
brigade, carried the gateway batteries. Many from Quitman’s and
Smith’s commands rushed to the summit, dealing with flying
enemies as they went. Scott himself came up—the hero of Chippewa
and Lundy’s Lane. The men pressed round him. He told them how
glad he was, and how proud of them; and how proud their country,
their wives, their sisters and their sweethearts would be; and it
seemed as if such cheering had never been heard, anywhere in the
world, before.[17]
Exultant but weary, the soldiers now looked about them as they took
breath. From this eyrie the whole wonderful Valley of Mexico could
be surveyed. All round the west the great wall of rugged mountains
closed it in, and two vast, snowy peaks guarded its portal on the
east. As if reluctantly the mountains gradually subsided into verdant
hills and a wide plain, enamelled in a thousand soft hues. The broad,
smooth lakes gleamed like molten silver. The gold of ripening grain,
penciled lines of pale-green maguey, cottages radiant in the sun like
the sails of distant ships, country-houses and villas half hiding in
foliage, and many straight, converging avenues, lined with trees,
delighted the eye. In the midst, clear-cut as a medallion, lay the city
of Mexico, the capital, its roofs and towers black with people; and
there, just yonder, stood the Halls of the Montezumas, the Jerusalem
of these ardent young crusaders. Unfortunately breastworks,
redoubts, cannon and a Mexican army were still to be reckoned with.
Santa Anna had probably lost not more than 1800 killed, wounded
and missing this day, and apparently Scott’s loss had been about one
fourth as great.[18]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-56-320.jpg)
![QUITMAN’S
OPERATIONS
The Citadel in 1840.
But the Americans quickly prepared to
advance—first of all, Quitman. Naturally a
certain discretion had been given to the
commanding generals, and he intended to
make the most of it. Looking from the hill along the Belén causeway,
he saw a wide avenue divided through the middle by a stone
aqueduct some eight feet wide and fifteen feet high, resting on
heavy arches and pillars of masonry. Owing to fine weather the road
was unusually firm. A small number of troops, fleeing in the utmost
confusion, could be seen upon it, but at only one point fortifications.
Borrowing all of Pillow’s troops except the Fifteenth Infantry, which
remained to hold Chapultepec and guard the prisoners, he quietly
gave orders that his men should assemble near the main gateway.
At once the inspiring words began to circulate, “Quitman’s division to
the city!” and as soon as possible the Rifles, in their crimson sashes,
were leading the march forward. About a mile on, a two-gun battery,
with a field redan at its right on the marsh, blocked the way. For an
hour or so Drum used a small gun upon it. Then the Rifles, after
creeping along the aqueduct from arch to arch, took it by assault,
and the march continued toward the fortifications at the garita.[19]
As at the other garitas, no gates existed here,
but a ditch and a parapet blocked one half of
the causeway and a zigzag redoubt the other.
Just at the north was the stone house intended
for guards and customs officials, beyond which
lay the wide Paseo (Promenade). South, on the
Piedad road, were artillery and infantry that
could fire through the arches. Inside the garita,
buildings extending toward the east offered
shelter, and in open ground a little more
toward the north and about 300 yards distant,
the extensive edifice called the citadel,
protected with a wall and a wet ditch, constituted a serious obstacle.
[19]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-57-320.jpg)
![Santa Anna, after acting like a madman when Chapultepec fell, came
to this garita. General Terrés, a brave old Spaniard, commanded
here with about 180 infantry and some artillerymen. Santa Anna
gave him three guns of medium power, and stationed General
Ramírez in the Paseo, Brevet General Argüelles on the opposite side,
and General Perdigón Garay and Colonel Barrios in the rear with
substantial reserves.[19]
On approaching this formidable position, Quitman encountered a
withering storm of bullets, grape and solid shot from both sides and
the front, and suffered rather severely. But Drum and Benjamin, iron
men, bringing up as soon as possible a long 18-pounder and a 24-
pound howitzer on the opposite sides of the aqueduct, dampened
the ardor of the Mexicans not a little, and splinters from the masonry
did havoc among the sheltered artillerymen at the garita. Some
troops already beaten at Chapultepec and at the intermediate
battery soon became demoralized. At about one o’clock rumors crept
in that Americans from the southern front were turning the position.
Ramírez, Garay, Argüelles and Barrios retired without the formality of
saying good-by; and Terrés, whose cannon ammunition had failed,
withdrew prudently to the citadel with two of the guns and about
seventy panicky men, the remnant of his garrison. The Rifles now
dashed over the parapet; and at exactly twenty minutes past one a
tall, slender man with short, bristling, grayish hair stood on it,
smoking a cigar and waving a red handkerchief tied to a rifle. It was
Quitman, self-possessed but exultant; and in a few moments the
Palmetto colors and the green banner of the Rifles, with its blazing
gold eagle, were flying at the portal of the city.[19]
The advance then continued for some little distance, and, as the
ammunition of our two heavy guns had been exhausted, the
captured Mexican 8-pounder was made to do good service. But
Santa Anna, who had thought the position safe and gone on to San
Cosme, soon arrived with ordnance and troops. The citadel was
reinforced, and infantry and cannon were placed at other points.
Quitman’s last artillery cartridges were used, and under the enemy’s
fire no more could be brought up. Solid shot cut down both Drum](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-58-320.jpg)
![THE SAN COSME
APPROACH
and Benjamin. Our infantry had to retire to the vicinity of the garita.
Attempts were then made to strengthen the position; but they did
not accomplish very much. Ammunition gave out entirely, and firing
ceased. The enemy grew bolder. Again and again they charged, and
though repulsed they did not appear to be discouraged. By this time
every member of Quitman’s staff, Beauregard, his engineer officer,
and all his artillery officers had been killed or wounded, and he
longed anxiously for night.[19]
Meanwhile, events had occurred on Scott’s other wing. Trousdale’s
command, supplemented with Jackson’s guns, pushed along the
road and aqueduct on the north side of the rectangle, and the latter
distinguished himself by fearlessly attacking a one-gun redoubt,
which, supported by infantry and by fire from the summit of the hill,
barred the way. To check Mexican reinforcements and threaten the
enemy—particularly the troops in Quitman’s front—Scott now had
Worth, Garland’s brigade, C. F. Smith’s battalion, Duncan’s battery,
the rest of Magruder’s battery and Sumner’s dragoons pursue the
same route. The one-gun redoubt was flanked and occupied; and
Worth’s forces arrived at the northeast corner of the rectangle in
time to annoy the retreat of Rangel and other departing Mexicans.
[20]
Here began the broad, straight Verónica
causeway—closely similar to that of Belén—
which extended almost north for nearly two
miles (3530 yards) to the English cemetery,
and there joined the San Cosme highway at approximately a right
angle. Understanding the difficulties of the Belén approach, Scott
intended to make only a feint in that quarter, and let his left wing
break into the city. He therefore sent the brigades of Clarke and
Cadwalader and also Huger with siege guns to Worth. To organize
the attacking column, replenish the ammunition, make other needed
preparations, and sweep away the resistance encountered at several
minor fortifications, especially near the cemetery, required time; but
at about four o’clock Worth found himself on the straight highway
about half a mile from the San Cosme garita.[20]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-59-320.jpg)
![This entrance to the city had been included in the general scheme of
defence, but on account of its remoteness from pressing danger few
workmen had been employed here; and when Chapultepec fell, it lay
entirely open except for a small parapet without a ditch extending
partly across the highway some 250 yards to the west. General
Peña, however, coming this way from Chapultepec, stopped at the
parapet, and Rangel placed at the garita such troops as he could
assemble. Santa Anna, who displayed on this occasion reckless valor
and an almost fiendish activity, sent three available cannon and
brought additional troops. The roofs of buildings in the vicinity were
occupied. A redoubt with embrasures was hastily erected at the
garita, the near arches of the aqueduct were stopped up with sand-
bags, and some guns in the Paseo were prepared to coöperate.[20]
On attempting to advance, therefore, Worth found the highway
swept with bullets, canister, grape and shells. Garland, however, was
ordered to creep forward under the protection of the arches, and
endeavor to reach the south flank of the garita, and Clarke to
burrow through the continuous line of buildings on the other side,
and strike the northern flank. Lieutenant U. S. Grant, who was
reported as acquitting himself at this time “most nobly,” waded some
ditches with a party of men and a mountain howitzer, and planted
the gun on the roof of a church at the right; and Lieutenant Raphael
Semmes of the navy performed a similar exploit on the left. Artillery
fire compelled Peña, who—reinforced by Santa Anna with two
companies of the Eleventh Infantry—was fighting gallantly, to leave
the parapet; and Hunt, of Duncan’s battery, though he lost more
than half his men in dashing 150 yards at full speed, landed a gun at
that point, where he could load in safety and then fire from the one
embrasure.[20]
By five o’clock these preparations were complete. On the other hand
Rangel had been severely wounded, and his principal gun, a 24-
pound howitzer, had become unserviceable. Suddenly, to his utter
astonishment, Americans appeared on the top of a three-story house
that commanded the interior of his redoubt, and with a single volley
disposed of almost every gunner and artillery mule. Then some of](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-60-320.jpg)
![them hurried down to the front door of the house, burst it open, and
rushed into the redoubt, where they met Americans just arrived by a
flanking movement from the other side of the highway. In a panic
the Mexicans fled, literally sweeping away Santa Anna and a body of
troops, who had come at all speed from Belén to support the
position. Many of them scattered, but with no little difficulty others
were conducted to the citadel. By six o’clock Worth entered Mexico.
Near the garita his forces were safely housed, and by way of “good-
night” and good advice, Huger dropped a few shells in the vicinity of
the palace.[20]
The end, however, was not yet in view. Santa Anna had some 5000
infantry and fifteen cannon at the citadel, with probably about 7000
more troops not far away, and the Americans, besides having lost
many in the day’s fighting, were now fearfully divided. Not only
Worth but Quitman, who planted three heavy guns in battery during
the night, intended to advance in the morning, and apparently a day
of carnage was to ensue.[20]
But Santa Anna probably began to feel the reaction that always
followed his great efforts. Funds and provisions were scanty. The
army was demoralized, and the mass of the people felt
disheartened. Within the town there were no fortifications, and it
looked as if another battle under these conditions might scatter the
troops, and involve the loss of nearly all the war material. Besides,
leading persons in the city had always been strenuously anxious to
prevent bombardment and assault; and the President was urged
now, as four months previously, to spare it. Early in the evening,
therefore, he briefly discussed the situation with Olaguíbel, the
minister of war and three generals. The Governor was for acting
deliberately; but Santa Anna, declaring that honor had been satisfied
and the city could not be defended successfully, ordered immediate
evacuation; and by one o’clock the troops retired in a somewhat
orderly fashion to Guadalupe Hidalgo. About three hours later a
commission of the city council (ayuntamiento) offered terms of
capitulation at the American headquarters in Tacubaya. These were](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-61-320.jpg)
![THE AMERICANS
CAPTURE MEXICO
of course rejected, for the town lay at our mercy; but Scott gave
informally the usual assurances of good treatment.[21]
So when the first thin streak of dawn
glimmered forth behind the gray volcanoes,
and our cannon at Belén garita were on the
point of opening fire, a white flag and an
invitation to enter the capital reached Quitman. First making sure
there was no deception, he advanced; and after stopping about half
an hour at the citadel he moved forward under a splendid sun to the
grand plaza, which fronted the palace and the cathedral, with
Smith’s brigade, the Marines, the New York volunteers and Steptoe’s
battery. As a triumphal procession the command looked rather
strange. Quitman and Smith marched at its head on foot—the
former with only one shoe; and behind them came troops decorated
with mud, the red stains of battle and rough bandages, carrying
arms at quite haphazard angles. Not less astonishing looked the city,
for sidewalks, windows, balconies and housetops were crowded with
people. Except for the silence, the countless white handkerchiefs and
the foreign flags, it might have been thought a holiday. Before the
palace, which filled the east side of the plaza, the troops formed in
line of battle. Officers took their places at the front, and when
Captain Roberts hoisted a battle-scarred American flag on the staff
of the palace at seven o’clock, arms were presented and the officers
saluted.[22]
Soon loud cheering was heard. A few squares away the commander-
in-chief, escorted by cavalry with drawn swords, had reached
Worth’s command, which had stopped at six o’clock by orders
opposite the high ash trees of the Alameda. A clatter of galloping
hoofs followed; and in another moment, amidst the involuntary
applause of the Mexicans, General Scott, dressed in full uniform and
mounted on a tall, heavy bay charger, dashed with his staff and
Harney’s dragoons into the grand plaza—his noble figure, gold
epaulets and snowy plumes, resplendent under the brilliant sun, fitly
typifying the invisible glory of his unkempt and limping army.
Uncovering, he rode slowly along the line of battle to the music of](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-62-320.jpg)
![our national airs; the troops, presenting arms again, cheered and
hurrahed till it seemed as if the earthquake-proof cathedral must be
shaking, and the cavalry escort waved high their flashing blades.[22]
In stentorian tones the commander-in-chief appointed Quitman
governor of the city; and then, dismounting in the courtyard, he
clanked up the broad stairway of the palace, to indite
congratulations on the “many glorious victories” of his army.
Presently cross-belted American Marines were calmly patrolling the
Halls of the Montezumas as if they owned them, while the rest of
the troops gazed with profound exultation at the long pinkish façade
and the endless balconies of the upper story, and the people gazed
silently at the troops. “They are all and each of them heroes,”
commented a foreigner present, and others in the world thought the
same.](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-63-320.jpg)
![“Light up your homes, O fathers,
For those young hero bands,
Whose march is still through vanquished towns
And over conquered lands,
Whose valor, wild, impetuous
In all its fiery glow,
Pours onward like a lava-tide,
And sweeps away the foe!”[22]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-64-320.jpg)
![XXIX
FINAL MILITARY OPERATIONS
January, 1847–April, 1848
At the north, after the Buena Vista campaign and the embarrassments growing out
of it came to an end, Taylor probably wished, in what an officer calledto advance as
far as San Luis Potosí, and retained troops urgently needed by Scott; but by the
middle of June, 1847, he doubtless realized that effective operations on so long a
line, especially through hostile and much of the way through barren territory, were
impracticable, and advised that Scott’s column alone should act on the offensive. A
month later orders of a corresponding tenor were issued at Washington, and then
some 3000 surplus troops of the northern army proceeded toward the capital,
though too late, of course, to assist in the decisive struggle.[1]
Valencia, during his brief stay at San Luis Potosí in the early summer of 1847, not
only requested permission to move against Saltillo, but planned that General
Filisola, aided by a brigade under Avalos, then lying at Matehuala, by Reyes, the
comandante general of Zacatecas, and by Urrea—who still commanded theand
could easily pass across the Sierra Madre from Tula—should threaten, if not attack,
Saltillo and Monterey, and at least keep the Americans on the defensive. Some
disquieting movements of these troops resulted; but Valencia was soon called to
Mexico, and various difficulties, chiefly a lack of means resulting from the American
occupation, proved fatal to this ambitious enterprise, besides hindering the Mexican
preparations to receive Taylor at San Luis Potosí.[1]
During the winter of 1846–47 and to some extent later, the garrison of Tampico was
menaced by plans for an uprising, to be assisted by outside forces, and sometimes
it was feared that a move to capture the city would be launched from Tula in the
hope of embarrassing Scott’s communications; but the Americans, though not
strong in numbers there, were vigilant and well protected by fortifications. Besides,
the authorities of Tamaulipas, now living on fairly good terms with the invaders,
had little wish to take part in active hostilities. They quarrelled bitterly with Urrea,
who naturally attempted to draw supplies and money from the region, and in
November, 1847, with a view to bringing about harmony, that officer was removed.
Scott’s victories and especially the fall of Mexico had no little effect in this quarter;
the prospect of serious operations entirely disappeared; and early in November,
1847, General Taylor, who had reached the conclusion some time before that his
country wanted him for President, and had laid aside his old brown coat in favor of
checked shirt sleeves, set out for home on a leave of absence, which actually](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-65-320.jpg)
![FIGHTING IN THE CAPITAL
continued until the close of the war. Wool took his place; but nothing occurred in
this region except guerilla affairs, of which a due account will be given presently.[1]
In the northwest, meantime, Price, who commanded in New Mexico and was
disturbed by rumors of danger from the south, decided on his own responsibility,
ignoring instructions to do otherwise, that he must assume the aggressive. Early in
March, 1848, the city of Chihuahua was therefore reoccupied; and on the sixteenth
of that month, after a little brisk fighting, the town of Rosales, about sixty miles to
the southeast, which Angel Trias held with some 800 men, chiefly National Guards,
was captured by assault with a trifling loss. But this campaign had no general effect
on the war—indeed, the treaty of peace had already been signed—and Price was
ordered by Marcy to retire.[1]
In Scott’s department the final military operations
began very promptly. Immediately after the Americans
took possession of the grand plaza at Mexico on the
morning of September 14, a multitude of blanketed léperos crowded closely upon
them. Already these miscreants had tasted the disorder they loved, for the palace
had been left unguarded, and they had sacked it; and now they showed signs of
turbulence. The plaza was cleared, however, and no further trouble seemed likely.
But when our troops began to march away to their quarters, a shot was heard. A
bullet probably intended for Worth struck Garland, and almost instantly firing from
street corners, windows and the tops of houses became general, though not
systematic. Thousands of convicts from the jail supported the populace, and in one
way or another not a few of the better class coöperated. By Tornel’s order paving
stones had been taken to many of the azoteas with a view to resisting the invader
step by step, and these, like every other sort of weapon, were now used.[2]
Though surprised, the Americans promptly accepted the challenge. Skirmishers
drove back the mobs. Grape and canister swept the streets. As a rule, every house
from which a shot flew became a target for our heavy cannon, which seemed to
shake the very foundations of the city, and when breached was immediately
sacked; and sharpshooters worked effectively on towers and roofs. Scott
threatened even sterner measures; and the city authorities not only put up notices,
embodying his threats and imploring the people to desist from a vain and
imprudent contest, but interceded personally with them in the streets. By about
noon the Americans held all the points of vantage, and as evening approached, the
firing died away. A fearful night ensued. It was dark and cold. No lights relieved the
gloom. Wild mobs ran shouting through the streets, and the hoof-beats of American
patrols resounded from square to square.[2]
Santa Anna, finding it impossible on the morning of the fourteenth to subsist his
army at Guadalupe, had ordered the infantry and heavy guns to Querétaro under
General Herrera, and proceeded with four small pieces and the cavalry to San
Cristóbal, a point about fourteen miles northeast of the capital. After seeing the](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-66-320.jpg)
![people of Mexico view with indifference his efforts of the previous day, he expected
nothing of them; but on learning of the outbreak he marched back to Guadalupe,
and at a late hour sent into Mexico a small force of cavalry and infantry to
investigate and assist. This met Duncan’s battery and retreated; but Santa Anna,
assured that on the next day there would be a rising en masse, erected a
breastwork at the Peralvillo garita on the north side of the town, and waited.[2]
As soon as day broke, gloomy and wet, the shooting was in fact resumed, at least
in the northern quarters. But he soon perceived that no general movement was
taking place, and again marched away. This disheartened the people still more; the
efforts of the authorities influenced them greatly; and by the end of the afternoon,
realizing that much was to be suffered and nothing gained, they generally
abandoned hope. During the next day or two scattering shots could be heard, but
real fighting was over. Extravagant hopes of destroying the small American army
were still entertained by lightheaded men.wrote one of these, and attempts were
made by military officers to organize a real conspiracy; but lack of courage, means,
confidence and mutual trust—as well as the watchfulness of the Americans—made
success impossible. Scott repeatedly warned his troops to be vigilant and orderly, to
keep together, and to refrain from drinking. As the danger grew less menacing,
however, they became less careful, and for probably a month assassinations were
frequent. From first to last several hundred Americans perished in the hostilities,
and no doubt far more of the enemy. But by the middle of October the city was
tranquil.[2]
The concluding field operations in Scott’s department resembled for the most part
the fighting just described, for they had to do chiefly with guerillas. That style of
warfare suited the national character. It had figured prominently in the Spanish
struggle against Napoleon and in the Mexican war of independence; and when the
public began to see clearly that battles could not stop the Americans, it was
invoked—even though by universal military practice in Europe those who robbed
and fought at will, while pretending to be inoffensive, were considered brigands
and assassins—as the one hope.[3]
Thoughtful persons like J. F. Ramírez and General Mora pointed out serious
dangers: the impossibility of discipline, the relaxation of morale, the destruction of
all standards, and the certainty that a spirit of violence and rapine would grow by
what it fed upon; and they recognized the improbability that such methods could
prevail against the strength, equipment, compactness and skill of the Americans.
But the obvious advantages of the guerilla system, which it required far less
intelligence to perceive and appreciate, counted powerfully on the other side. How
much the Spanish themselves had suffered from their irregulars during the
hostilities against Napoleon was not understood, and patriotic pride in the war of
independence had tended to draw a veil over its horrors.[3]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-67-320.jpg)
![THE GUERILLAS OF THE
NORTHEAST
The dagger, said the official newspaper, was the favorite weapon of the people.
Unarmed men could burn wagons and intercept communications, it was pointed
out. Even women and children could help. A thorough knowledge of the country, its
mountains and its by-paths, would evidently constitute an enormous advantage.
Light corps of the abstemious rancheros, embarrassed with no baggage, could
travel quickly day and night, concentrate in large numbers against an American
detachment, strike, vanish, and then, when least expected, reappear, making the
most of all neglects, all mistakes, nullifying superior strength by avoiding it, and
nullifying discipline by fighting in a style that had no need of discipline. Situated
even more favorably than Spain for such warfare, the Mexicans were to outdo her
example.[3]
This is what will save us, proclaimed in effect the legislature of México
state.exclaimed the congress of Vera Cruz. Santa Anna endorsed the plan. Salas
organized thewhich were to make” “in every manner imaginable”; and in April,
1847, the government, pinning its faith to the system, set it on foot in earnest.
Scott, thewas to be routed after all.[3]
In the north February, 1847, was the golden month of
the irregulars, for the approach of the Mexican army
under Santa Anna encouraged the rancheros to lay
aside the habits of peace. Canales boasted of 161
Americans killed that month, and Urrea with his combined force of regulars and
guerillas, besides engaging in other operations, captured a train of wagons at Agua
Negra, and horribly slaughtered a large number of guards and teamsters. To
avenge this butchery a party of Rangers, teamsters and other civilians murdered
twenty-four men in a village not far distant. Upon this Canales declared what he
called martial law, announcing that every American, armed and unarmed, and every
Mexican living peaceably would be shot; and many were led by fear or a lust for
plunder to take up arms.[4]
The American trains in particular seemed likely to be easy prey. As they commonly
stretched out for some two miles and were guarded only—for so the character of
the road usually dictated—at the ends, the Mexicans, trained to charge at full speed
through an ordinary thicket, could readily attack them from ambush at about the
middle point, create a stampede, and do a great deal of mischief. Infantry could not
pursue the guerillas with success, and the number of our mounted men was always
comparatively small, for every Mexican ranchero had at least one smart pony. In
September, 1847, a band even attacked Mier. Governor Aguirre of Coahuila exerted
himself particularly to organize forces of this character, and not only alcaldes but
priests aided the cause.[4]
The American leaders, however, pursuing a course that was now conciliatory, now
severe, and in many instances technically unjust, succeeded in coping with a
system that was itself unjust. Taylor levied on the people of Nuevo Léon a tax of](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-68-320.jpg)
![GUERILLAS IN VERA CRUZ
STATE
$96,000, the estimated value of the goods destroyed at Agua Negra, but suspended
it indefinitely, when representative authorities proved the substantial innocence of
the population and begged for mercy. Cavalry patrols and detachments pursuing
culprits fairly wore out their horses. Villages, if even suspected of harboring
thewere burned. Contributions were imposed wherever connivance appeared
probable. By April, 1847, Canales was in despair.[5]
Then Wool determined to stamp out the evil, and announced in July that any
guerillas caught by him would be executed. In December, 1847, he issued his
famous Order 11, which not only made the Mexican authorities and their towns
responsible for all damages done, but required them to hunt down theAguirre
attempted to retaliate, but in vain. The Americans had power enough to carry out
threats, whereas he had not; and he admitted his failure. Besides, the mass of the
population were indolent in mind as well as body, and looked upon submission as
preferable to danger. In February, 1848, finding the guilty rancheros were anxious
to give up the business, Wool enabled them to resume peaceful occupations by
declaring an amnesty, and in the following May he stated that the country had
never before been so free from highway robbery.[5]
In the south, Vera Cruz, a state of mountains, gorges,
thickets and forests threaded with blind paths, was
the chief home of the guerilla, and it looked as if
Scott’s line of communication might be virtually
destroyed. Not only many hardy, hot-blooded and unscrupulous natives, but a great
many desperadoes hailing from Cuba were ready to enlist. After the fall of Vera
Cruz, and still more after the battle of Cerro Gordo, a large number of regular
officers, to say nothing of privates, could scarcely find bread, and some men, like
the ex-divinity student, ex-Carlist, Jarauta—whose small, close beard, fierce black
eyes, braided jacket, graceful cloak and gold-laced sombrero gave him a romantic
air—had acquired in Spain a taste for this adventurous, reckless life; but a vastly
greater number were prosaic felons, liberated from prison under a pledge to rob
and murder. Nominally J. C. Rebolledo, a fine looking man of rather humane
instincts, was the chief in this district, but the 800 or so persons belonging to many
small bands, while occasionally acting more or less in concert, were mainly
independent. The decree under which all goods coming from points occupied by the
Americans were lawful booty opened possibilities of large gains, and Rebolledo’s
capture of ten loaded wagons in April, 1847, set the people aflame with cupidity.[6]
Brevet Colonel McIntosh and his inexperienced officers, who left Vera Cruz for the
interior—it will be recalled—about the first of June, 1847, with a well-advertised
convoy including a large amount of specie, dependent on wild mustangs under raw,
half-mutinous drivers largely ignorant of English, received the full benefit of this
ambitious feeling. Near Tolomé and at Paso de Ovejas he lost men, wagons and
pack-mules; and at the national bridge there was a genuine skirmish, in which a
number of Americans were killed or wounded. Out of about 130 wagons twenty-](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-69-320.jpg)
![four had to be abandoned in the low country; and a little way above Jalapa, though
strongly reinforced, the troops had to fight again. General Pierce, who left Vera
Cruz about six weeks later than McIntosh, had similar experiences. Early in August
Major Lally set out from the coast with a few more than 1000 soldiers, two 6-
pounders and sixty-four wagons. He lost no merchandise, but his four fights cost
him nearly 100 men killed, wounded and missing; and Captain Wells, who followed
Lally with some 200 recruits and additional ammunition, lost forty and had to
retreat. These and other affairs proved that irregulars, favored by the geography of
the region, were capable of doing substantial harm.[7]
But in Vera Cruz, as in every other quarter where they operated, though perhaps
nowhere else in so marked a degree, the lack of morale, which enabled the guerilla
system to exist, proved the cause of its failure. Poor arms, poor ammunition, poor
marksmanship, and the want of artillery might have been remedied, or at least
might have been offset by the counter-balancing advantages; but this defect was
fatal. The Mexican guerillas were very different from what the guerillas of Spain had
been. They fought like savages without the excuse of savages, for they knew
better. Infuriated by their treacheries and cruelties, the Americans were persistent
and unsparing in severity. Patrols who seemed never to sleep hunted out their nests
in the mountains. On the march, flanking parties would force their way through the
woods five miles or more from the road to catch them between two fires. The torch
was applied with much liberality on suspicion, and sometimes on general principles,
to huts and villages; and in the end a black swath of devastation, leagues in width,
marked the route.[8]
Scott ordered that in every case of outrage the nearest alcalde, if he failed to
deliver up the guilty, should be fined at least $300 for a murder or the value of the
stolen property for a robbery, and that any robber or murderer and any person
belonging to a known party of such miscreants might, when caught, be summarily
tried by three officers, and either flogged or executed. This plan, however, did not
quite satisfy those on the ground—especially the Texas troops. Captain Walker, on
his cream-colored horse, and Colonel Hays, in his blue roundabout, black trousers
and black leather cap, impressed themselves on the Mexican imagination as the
agents of diabolical wrath; and in general it was a tale of merciless atrocities
followed by merciless reprisals.[8]
At the same time this lack of morale deprived the guerillas of Mexican support. By
taking bribes for letting merchandise pass up to the interior and sometimes even
guarding it, they violated the laws on which their existence rested. Mostly they
were brave only where they felt safe. When laden with booty they would scatter to
their homes, no matter how important the business in hand. Rivalries and even
hostilities between parties operating in the same district arose. Cooperation could
seldom be reckoned upon, and hardly any would face the climate far above Jalapa.
Soon learning that it was more wholesome to waylay Mexicans than Americans,
they plundered their fellow-countrymen without ceremony; and they would rob](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-70-320.jpg)
![GUERILLAS OF THE CENTRE
even old women or young children of their needful clothing. Sheafs of complaints
against them piled up in the state and national archives. People organized to fight
them, and sometimes appealed to the Americans against the very men who were to
have been their champions.said an American officer.[9]
In the states of Puebla, México and Oaxaca also
guerillas were organized, and in Puebla all these
parties could find an opportunity. General Rea, a pupil
of Morelos and the Mexican revolution, had the discredit of the chief command,
though Bravo, who stood at the summit of the social scale, was mainly responsible
for their iniquities, since during his brief term as comandante general of Puebla he
issued a great number of patents to unfit leaders. What Rea did particularly in this
regard was to combine individuals and small groups, and place them under some
kind of supervision. He loved to answer critics by saying that his guerillas were in
the field because honorable men were not; and that, had not the government
condoned their crimes, they would have served the Americans as counter-guerillas.
After a time his officers adopted a set of rules which aimed to regulate operations,
but even this measure seems to have accomplished little. The guerillas robbed the
people, seized funds belonging to the state, and pillaged even churches. Some
gangs were large enough to attack haciendas. One party called themselves the
“Lancers of the Poisoned Spear.”[10]
Soon after Scott left Puebla for Mexico early in August, 1847, these banditti and
every individual ruffian of that vicinity hurried to the city. Mexicans and even foreign
residents were robbed and outraged, and about the first of September, in the hope
of more booty, the Americans also were attacked. Two thousand soldiers were
needed for a garrison, and Colonel Childs, the civil and military governor, actually
had 2193; but 1800 of these were in hospitals. His effectives consisted of about
fifty cavalry, 100 artillery, 250 of the First Pennsylvania volunteers, and a small spy
company of Mexicans.[11]
Headquarters, Lieutenant Colonel Black of Pennsylvania and the main body of
troops occupied thea large rectangular building on the eastern side of the town,
which had a plaza of its own opening toward a public promenade called the Tivoli.
To this position five howitzers were allotted, and within a hundred and fifty yards of
it all the sick were placed. Half a mile or so from the town on a hill stood Loreto
fort, a stone affair equipped with two 12-pound field guns and a 10-inch mortar,
where Major Gwynn of the Sixth Infantry commanded; and not far distant, on a
higher point of the same hill, was Guadalupe church, now protected with mountain
howitzers, a ditch and an earthen wall, under Captain Morehead of the
Pennsylvania regiment. But the chief element of the defence was the large, robust,
finely-featured Childs, a skilful and veteran officer, cold in manner, clear in
judgment, and inflexible in courage. September 13 thebegan in earnest, and from
that day on there was a continual small-arm attack, particularly at night, upon San
José, which replied with a musket and howitzer fire that kept the assailants at a](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-71-320.jpg)
![THEOF PUEBLA
respectful distance. What was more serious than guerilla shooting, all supplies were
now cut off. Such was the state of things at the second city of Mexico when Santa
Anna retired from the first.[11]
Santa Anna’s real intention was probably to seek an
asylum in Guatemala. But many of his friends urged
that he could make himself dictator as the sole hope
of the country, and it was clear that, if he should recover Puebla and cut off Scott,
he would still be able to boast of a triumph. His cavalry, though greatly reduced by
desertion, included some 2000 men backed with four light guns. Alvarez, who was
ordered to Puebla, still had about 600 foot and horse. Rea, Santa Anna understood,
commanded 600 irregulars; 2500 National Guards lay near him with two field
pieces, it was reported; and the Pueblans were described as eager to fight. Six
thousand men and six guns appeared quite enough to dispose ofas Mexicans
described the garrison; and he therefore presented himself at Puebla on September
21. Two days later Alvarez arrived there. But between these two events Childs
appeared at a second-story balcony, as a soldier expressed it, and announced that
Scott had taken the capital. Evidently, therefore, the Mexican President was not
greatly to be feared.[11]
After looking about, Santa Anna concluded that it would not be easy to capture the
American positions by assault, and appealed to the minister of war—wherever that
official might be—for 1000 infantry, a 16-pounder, a 12-pounder, ammunition and
supplies. He now had ten cannon, but all of them were light; owing to desertion his
force included only some 4000 men; and the citizens had no arms, he reported.
Probably, too, the annoyances and outrages inflicted upon them by him and his
troops, and his appointing the guerilla chief military commandant of the city
dampened whatever ardor they had possessed.[12]
On September 23 and 24 unsuccessful attempts were made at Guadalupe, and the
next day Santa Anna summoned Childs, describing his army as 8000 strong, and
graciously announcing thatthe Americans might retirewith the honors of war. Childs
replied as was proper, and then, riding to the posts, gave notice amid cheers that
no surrender need be expected. To add the touch of humor that soldiers love, an
American flag was manufactured out of an old Mexican uniform, and raised aloft;
and the garrison settled down to severe duty, stern discipline, short rations and
incessant watchfulness at all hours. The Mexicans tried to approach San José by
throwing up successive breastworks at night in the streets leading that way, but
shot, shell and rockets from Loreto kept them back. September 30 Santa Anna
learned that no ammunition could be provided for the heavy cannon demanded of
the minister, and resolved apparently to make a bold effort. With two 6-pounders
he fired all day on the weak, plaza face of San José. But Childs, anticipating such a
manoeuvre, had brought a 12-pounder from Loreto the night before; and this,
protected with bags of tobacco, made an assault impracticable.[12]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-72-320.jpg)
![THE FIGHT AT HUAMANTLA
A new factor now entered the military situation. About the middle of September
GeneralLane, one of Taylor’s chief officers at Buena Vista, arrived at Vera Cruz from
the Rio Grande, and on the nineteenth his brigade set out for the interior. Aware of
the situation at Puebla but not aware what was to be encountered on the route, the
General had not made adequate preparations, and on meeting guerillas at the
national bridge he was obliged to send back for ammunition and supplies. By
October 1, however, he managed to leave Jalapa.[13]
It was a hard march that ensued. Torrents of rain deluged the troops. Sometimes
the road lay deep under water. For dinner they had a thin slice of beef, a couple
ofand some coffee; for supper, after darkness fell upon them with tropical
abruptness, the same without the beef; and perhaps mud for a couch. But Lane, a
hearty westerner with a stout frame and unbounded vigor, led on unshrinkingly in
his black hat and old blue overcoat, and the rest followed him eagerly. October 5,
after incorporating additional troops at Perote, he left that place with a force of
about 3300 and seven guns, and marched on across hot plains, where water sold
for five dollars a drink, and men died of sheer fatigue.[13]
Santa Anna, informed by spies that 1000 Americans were approaching, and
anxious, not only to prevent them from joining Childs, but still more to win the
glory of routing them, had set out from Puebla four days earlier with perhaps 3500
men, leaving Rea to continue the fighting. Desertion played havoc with his
command, especially when the strength of Lane’s force was ascertained; but, after
sending back a large part of the faithful in order to keep control of them, he took
possession of El Pinal, where the national highway passed between a precipitous
mountain and a ravine, with about 1000 cavalry and six guns, and made
preparations to ambush Lane’s rear. This done, he moved to Huamantla, a sizable
town eight miles distant, and waited.[13]
Early on October 9 the drums and bugles awoke
Lane’s troops at the hacienda of San Antonio Tamaris,
approximately ten miles from Huamantla and twelve
from El Pinal; and the men, leaping from the damp grass and buckling their muddy
belts, found the white walls of the hacienda, the church towers of neighboring
villages, the dark woods on the hillsides, and the distant, snowy peaks all aglow
under a splendid sun. Never, perhaps, did soldiers feel more like having an
adventure. Santa Anna had just marched from Huamantla to conceal his force at El
Pinal, leaving behind him with no scouts or outposts his six guns, a very small
guard for them and a party of irregulars; but a spy reported to the Americans that
he was at Huamantla, and Lane moved off to attack him. First rode four mounted
companies, and at their head a rather short, slender, spare, slouchy man, with
reddish hair, a small reddish beard, mild blue eyes and a quiet, kindly manner,
whom nobody would have picked out as a fearless, indomitable fighter, the scourge
of the guerillas, but in fact he was Captain Walker; and then marched Lane with
five guns and some 1800 men.[14]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-73-320.jpg)
![When about three miles from their destination, Walker and his 200 cavalry, seeing a
party of Mexican horse approach the town, dashed ahead. Entering Huamantla they
formed in fours, and then with a yell, a flash of sabres and a thunder of hoofs they
swept through to the plaza. The Mexicans had time to get four of the guns away,
but the others were captured, and most of the American troopers, concluding their
work had been finished, scattered to drink, loot or hunt for cannon and
ammunition. But now Santa Anna, who had observed Lane’s movement from a
church tower near El Pinal, appeared with his full command. They were a beautiful
sight—galloping horses, red and green uniforms, brilliant pennons and a billowy sea
of flashing lance points; but they were enemies, and the Americans accepted their
challenge.[14]
“Take it cool, my boys, but run like the devil!” cried Lane. Every nerve was taxed.
Blood gushed from nostrils. The Mexicans, lashing their steeds into foam, reached
the goal first, however, and the American troopers found themselves attacked on all
sides. Walker was shot from a house, and soon expired; but he lived long enough
to give a final order: “Don't surrender boys; the infantry will soon be here.” And so
they were—“with a shout and a bound,” said one of them. The tide was quickly
turned, and giving up the town, as the soldiers loved to call Santa Anna, passed the
night some distance away. So ended the Tale of Huamantla or The Biter Bitten,
which received no little applause at the time.[14]
While these events were taking place, the garrison of Puebla continued to be
hemmed in, starved and harassed. Their casualties numbered in all only fifty-two,
but they felt severely the effect of so long a strain. Though a number of sorties
were made, and their persecutors had to retire from several annoying positions, the
Americans were not strong enough to do more. Their day of deliverance was
approaching, however. October 10 Lane moved forward, dogged and somewhat
annoyed by Santa Anna. Two days later his men saw the spires of Puebla,
dominated by the sombre towers of the cathedral, and set off by white volcanoes
veiled with clouds. And now and again the numberless bells of the city, great and
small, pealed forth harmonious tones of many colors, that seemed to blend and
interweave in rich and varied tapestries of sound, hung out in the mediaeval style
to honor their triumphal approach.[15]
At about one o’clock, announced by the bells of Guadalupe, they entered the
suburbs—not a few of them at a run. One column then advanced by the main
street, while another flanked the town by the left. For two hours there was
considerable firing from houses, though Rea’s guerillas had begun to leave their
posts the night before; but at length Lane extended histo Childs, with a sunny smile
on his rather hard features, and the garrison joyously welcomed their deliverers. In
the main plaza a bugler playedand all sang the chorus:
“The star-spangled banner, Oh, long may it wave
O’er the land of the free and the home of the brave!”[15]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-74-320.jpg)
![POLITICAL CHAOS IN
MEXICO
Now followed the punishment of Rea. Some twenty-five miles from Puebla toward
the southwest lay beautiful Atlixco, a defensible point that not only reconciled the
climates of the temperate and the cold zones of Mexico, and controlled a region
fertile in grains, flocks and herds, but, while fairly remote from the Americans, gave
convenient access to important roads. Here, in the midst of flowers, fruits and
snowy mountains, the government of Puebla had found a refuge, when the
Americans under Worth approached the state capital; and to this asylum Rea now
withdrew. De facto, at least, the guerilla chief was the most important person on
the ground. The authorities did not relish his prominence; they detested his men
and his methods; and on October 18, tired of spending money fruitlessly on the
National Guards for Rea to command, they dissolved the corps. But many of the
irregulars proposed to make the best of what appeared to be a good situation, in
which they could live on plunder, if not paid.[16]
October 18 Lane, who apparently never slept nor expected his followers to sleep,
ordered them to be ready in the morning for an expedition. Many of the soldiers
were barefoot, but they borrowed shoes; and at about nine o’clock, cheered by the
fife and the drum, some 1500 men set out round the base of Popocatepetl under a
hot sun. At about four in the afternoon, after making twenty miles or so, they came
in sight of the enemy, and a running fight began. Blistered feet and parching
tongues were now forgotten. The Mexicans, holding some good position and
protected by chaparral, could make a stand against cavalry, but when the infantry
came up they always fled. Shortly after sunset Lane reached Atlixco, which stood
on the slope of a lofty hill. As it was unsafe to risk a street fight in an unknown
town at night, he ordered the artillery to open. The moon was full. Marks were
easily selected. By their burning fuses the shells could be traced until they fell
amidst the shadows; and then a burst of red fire, the crash of roofs and walls, and
the cries of the people told the rest.[17]
After about an hour of cannonading, the troops advanced into the town—which
surrendered at once—and there slept as best they could. Rea, with two guns and
the disordered remnants of his force, retreated to Izucar de Matamoros, about
thirty miles farther down the valley; but from that point he was routed a month
later. These and other exploits of Lane’s discouraged as well as dispersed the chief
guerilla forces of the plateau, and in February, 1848, Rea asked permission of the
Mexican authorities to leave the country.[17]
Neither in these affairs nor in any other military
operations did Santa Anna figure at this time, and
there was a good reason for his inactivity. Officially he
no longer existed. As General Scott had feared, our
entering the capital had resulted in the destruction of the Mexican government.
September 16 Santa Anna resigned, explaining that it was advisable to preserve the
chief magistracy from the hazards of war, and fix it near the centre of wealth and
population, whereas he proposed to continue the hostilities wherever that should](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-75-320.jpg)
![SANTA ANNA ELIMINATED
be possible. The same proclamation or decree assigned the executive power to a
triumvirate: the president of the supreme court, General Herrera and General
Alcorta, and Santa Anna then ceased actually to exercise any civil authority.[18]
But as Congress was not in session to accept his resignation, some doubted
whether it became effective; the presidency of the supreme court was vacant on
account of the incumbent’s death; the appointment of Herrera and Alcorta needed
to be made, or at least confirmed, by the council of government, a body no longer
acting; and it was denied broadly that Santa Anna had the power to issue such a
decree. Peña y Peña, to be sure, was regarded as a member of the court, and, if he
was, he could claim by right of seniority to act as the chief justice; but the legality
of his membership was questioned, and the presidency of that body was really an
elective office. Peña was old, feeble and even timid; his ill-success as Herrera’s
minister of relations doubtless weighed heavily upon him; and he was now living,
almost as a recluse, in the country. Indeed there was really no organic law even, for
the amended constitution of 1824, though formally adopted, had not come into
effect. In short, chaos reigned, and the states were officially “resuming” their
individual sovereignty.[18]
But a number of good and able men, particularly Cuevas and Couto, determined to
ward off ruin, and awakened others. Peña, drawn from his retirement, consented
for patriotic reasons to override all the technical difficulties; and on September 22
he announced formally that, in order to give the nation a head, he would act as the
Executive until an interim President could somehow be chosen. At the small city of
Toluca, capital of the state of México, just outside the Valley, this fiction of a
government pitched its tent; and perhaps it gained some feeling of security from
the vast bastioned, battlemented ridge between it and the Americans, from snowy
Mt. Miguel towering above the city, and from the peacefully shining lagoons of the
intervening meadows. What was more important, Herrera, Olaguíbel and many
others of the best men rallied to the support of Peña, the representatives of neutral
governments recognized him, and the states began to concur. Early in October,
however, he removed to Querétaro, a safer yet central place, and with Luis de la
Rosa as sole minister addressed himself to his task.[19]
The programme that he announced was honorable
and straightforward. My tenure of office will be
extremely brief, he said in effect, for Congress will be
assembled as soon as possible; I will usurp no powers, but will not be turned from
the path of duty by insurrections; the closest economy will be practised, the
necessary taxes laid fairly, and all interests respected; union and harmony will be
the watchwords, and the national rights will be maintained. His most urgent
problem, of course, was to deal with Santa Anna, who not only held the chief
military command, but insisted that he could resume the Presidential authority by
simply withdrawing his resignation; and in this matter the government showed a
decision that earned it no little prestige. All Santa Anna’s protests against political](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-76-320.jpg)
![effacement were disregarded, and on October 7 he was instructed both to give up
his troops and to submit, as did other unsuccessful commanders, to a military trial.
[20]
At about the time this order overtook him, the Huamantla affair occurred. From a
military point of view he was now prostrate. He saw it himself, and knew that the
country would see it. Evidently his countless political enemies would make the most
of his complete failure, and he was doubtless aware that his military reports had
offended many officers. His chief executive merits—decision and activity—had led
only to a useless expenditure of life and money, it was now pointed out, and his
ostensible patriotism was attributed to passion and obstinacy. Even his confidence
in himself broke down. Unable to understand why failure had attended all his
efforts, he fell into a sombre depression, and without a struggle he placed his
troops at the orders of General Reyes, who joined him on October 11 with about
1000 men. His part in the war was over; and in the following January, realizing that
nothing could be gained through intrigue or conspiracy and fearing the Americans
would make him a prisoner, he asked for permission to leave the country. Both his
own government and our authorities consented. And after giving a dinner at El
Encero to the American officers of that vicinity, who had treated him with
distinguished consideration, he sailed once more, about the first of April, from what
he regarded as an ungrateful country.[21]
Santa Anna being now eliminated, the government had to face its military
difficulties without his assistance. In general the problem was to make bricks with
neither straw nor clay. Almost every good cannon had been taken by the
Americans, and the muskets had nearly all been captured, thrown away or sold.
Ammunition was almost wholly wanting. The engineering material had been lost or
destroyed. Vast sums of money were needed to provide fortifications as well as
replace all this equipment, and the government could hardly obtain enough, day by
day, to cover its minimum expenses. Even officers had to sell their shoes for bread.
[22]
As for an army, Santa Anna and Alvarez together had some 2000 troops the first
week of October, Reyes had about 1000, about 3000 from Mexico City concentrated
at Querétaro under Herrera, about 1000 from Jalisco were on their way to the same
point, and small detachments existed at various other places. But nearly all of these
men were utterly demoralized. “Almost useless,” they were officially termed; and
the army as a whole felt the crushing weight of general contempt. Herrera, the
commander-in-chief, became so disgusted over the uncontrolled excesses of the
troops that he resigned. Rincón declined on the ground of ill-health to serve. Arista,
when summoned to Querétaro, declared he would not command a soldier until
exonerated for his conduct on the Rio Grande. No officers of high distinction,
indeed, were available except the aged, torpid and infirm Bustamante and “the old
woman,” Filisola, as Bancroft described him.[22]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-77-320.jpg)
![THE AMERICAN POLICY
Attempts were made to lay plans of campaign, but an expert summed up one of
them by saying it appeared excellent—only it was based upon things as they should
have been, not as they were; and all the others had the same defect. Schemes
were devised to reform, reorganize and build up the forces, and quotas amounting
to 16,000 were assigned to the states; but México, which had been expected to
furnish nearly a quarter of these men, promptly answered that she could not, and
other states did not even reply. In fact, the regular forces decreased instead of
multiplying, for sometimes a general could not feed his troops, and frequently,
when soldiers were let out of the barracks on service, they vanished; and the
people, instead of helping to support the Mexican troops, even dreaded to see
them approach, for their coming was liable to draw an American attack, and more
than liable to mean extortion, outrage and robbery. Nowhere on the military horizon
could a glimmer of light be seen.[22]
Over against this pitiful government stood the United
States—wealth against poverty, strength against
weakness; and the antithesis was complete, for while
the Mexicans could only plan, that was the hardest thing for us to do. The idea of
retiring to a defensive line still persisted. Taylor himself adhered to it. But in
addition to the other overwhelming objections to this project, it seemed improbable
that a majority in Congress could agree where to draw the line. Even Calhoun,
though qualified to make a better argument for an untenable proposition than any
other man in the country, was unable to present this policy in such a manner as to
satisfy either the friends or the opponents of the war. Some advised holding, in
addition to the territory thus to be cut off, the chief ports of Mexico; and some
advocated retaining the capital also, and the line to Vera Cruz. Others favored the
occupation of still more cities; and many were for subjugating and holding the
entire country.[23]
To this last plan, however, even had it been practicable to levy all the costs upon
Mexico, there were tremendous objections. It would have involved keeping under
arms 80,000 or possibly 100,000 young men, seriously needed at home for the
most part, in order to be sure of having effectives enough at the front. The troops
in Mexico would have become corrupted both physically and morally; and the
commanders would have acquired the ideas and vices of proconsuls. It seemed to
be almost an insoluble problem. No final decision was made. But the government
determined to occupy the capital, hold the line to Vera Cruz, retain the chief ports,
and extend our holdings according to circumstances.[23]
To Scott, however, the lack of a definitive plan signified little. Not one reinforcement
entered the capital until after the first of November, and even at the end of that
month he was barely able to garrison Mexico and Chapultepec. December 4 his
army included only about 8000 privates, of whom a quarter were sick. During the
next three weeks Generals Patterson, Butler and Cushing, Colonel Hays, Lieutenant](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-78-320.jpg)
![Colonel Johnston and Major Lally, each with troops, arrived; and the forces then
numbered about 11,000 effectives and 3000 sick.[24]
Scott therefore announced, with no doubt a strategic purpose as well as a rhetorical
flourish, that our army was “about to spread itself over and to occupy the Republic
of Mexico.” What he really intended was to take possession successively of the
principal mining regions—those of Zacatecas and San Luis Potosí—and the capitals
of such important states as lay within easy reach. Even for the former purpose,
however, two columns of some 5000 effectives each were needed, and the men as
well as clothing for them could not be provided. The only immediate operations,
therefore, aside from the establishment of new posts on the road to Vera Cruz,
were the peaceful occupation of Pachuca, a mining town about fifty miles northeast
of Mexico, Toluca, about thirty-eight miles distant in the opposite direction, and
Cuernavaca, the key to the Acapulco region, a little farther away toward the
southwest.[24]
February 6, 1848, an expedition of more consequence marched, by Scott’s orders,
from Vera Cruz. Most of the guerillas who infested the road to the interior lived and
found a market at or near Córdoba, a city about sixty-five miles to the southwest,
and Orizaba, sixteen or eighteen miles beyond it in the same direction; and
Bankhead was instructed to occupy those towns. Very different from Lane’s rough
trips on the plateau was this march. Near Córdoba flourished such genuine tropical
wonders as the bread tree, the butter tree, the milk tree, and a kind of palm called
“the traveller’s friend,” which covered the wayfarer’s head with a tent, and
quenched his thirst with abundant sap. Going on, one found enormous masses of
vegetation—thick, matted, boundlessly prolific—moulded into astonishing yet
harmonious forms by the bays and promontories of the rapidly mounting foothills;
terraces of luxuriant foliage piled on sheer cliffs, castles on the terraces, and
cathedrals on the castles; verdure, verdure everywhere, dripping, flowing, spurting,
tumbling in every hue and shade of green, with a dark, velvety mist in the gorges
that became clear sapphire when the sun touched it, and here and there a cascade
letting fall its crystal thread from a mossy crag.[25]
Then came the rich Orizaba valley, hemmed in with jungles, and winding off
between sombre, precipitous mountains until lost in the dreamy distance; and
above it the sparkling snows of the vast peak sent down a torrent of gray glacier
water, that leaped into mid-air, and then, gathering itself below, wound on through
splendid, odorous trees full of parrots, canaries and mocking-birds, hurried past
fragrant orange groves and still more fragrant blossoms, poured through the arches
of a noble old bridge, and buried itself in the woods. But the Americans did not
forget their orders. Both cities were occupied without resistance, and both were
garrisoned; and the guerillas now found their proceedings considerably hampered.
[25]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-79-320.jpg)
![THE REMOVAL OF SCOTT
The final military operations of Scott seemed thus
rather tame, as was natural; but Polk executed one
that could be termed startling, if not exactly brilliant.
His principal assistants were Pillow, Worth and Duncan; and in different ways each
had excellent qualifications for the work. Pillow was not “The Lie Incarnate,” as Trist
believed, nor even “a perfect ass,” as many thought; but vanity, ambition, lack of
probity, and a gift for dark and cunning methods characterized him. His instincts
and talents, indeed, were those of the criminal lawyer who minds nothing about his
case except the verdict. When the President’s brother shot a man down in the
street at Nashville, Pillow got him off. With reference to his work in helping bring
about Polk’s nomination at Baltimore he wrote, “The fatal blow was given, but it
was not seen nor known what produced such a result—nor where the blow came
from.” “I feel as boyant as the air,” he said in December, 1846, when great
dissatisfaction with Taylor prevailed at Washington, because I know “that I have
done the work.... I have paid him in full” for his treatment of me. And one could
seldom get a finger on Pillow’s back, when he was not wriggling actively toward
some object of selfish desire.[26]
Without a particle of real military ability or success to his credit, he now stood
second in our army, and hence logically enough saw no reason why he might not,
by some devious path, arrive at the first position and even at the Presidency.
“Modesty,” said Burke, “does not long survive innocence.” To plant such a person,
with urgent recommendations, at open, big-hearted Scott’s right hand, to win his
confidence, to spy upon, criticise and undermine him, and inevitably to scheme for
his place, was indecent; but Polk did it.[26]
Very unlike Pillow was the courtly and fascinating Worth; but his mind was intense,
narrow and self-centred. After the battle of Monterey he exclaimed, “I am satisfied
with myself. The most vindictive foes crouch at my feet, and my friends choke with
joy and delight.” And there is one sin of which even angels are capable, we have
been told. All his military recognition he owed to Scott, but probably the debt
weighed heavily on his proud and restive nature; and, while apparently
reciprocating the genuine affection of his chief, he had inwardly rejected Scott’s
principles and methods nearly thirty years before the Mexican war. Regarding his
friend, fellow New Yorker and brother Democrat, Marcy, an adroit politician, he felt
very differently. “I would not give an ounce” of his wisdom, he wrote in June, 1846,
for all Scott’s glory; Scott “is determined to sink and draw his friends down with
him.”[27]
At Vera Cruz the commander-in-chief, relying on their long intimacy, told Worth
frankly that he believed the administration intended to ruin him, and the
subordinate officer evidently determined not to be drawn down. At the same place
a brother officer suggested to Worth a higher position than was even the highest in
the army. The suggestion appears to have struck root. All military men believed the
next President would be one of them, and what commander had acquired a more](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-80-320.jpg)
![THE REMOVAL OF SCOTT
brilliant reputation? The New York Sun recommended him for the place; and the
idea of his candidacy was favorably received by many. This prospect naturally
turned him still more against his old friend, for either Scott or Taylor seemed almost
certain to be the Whig nominee. Through a series of clashes, for which little—if any
—justification can be seen, and in spite of Scott’s efforts to conciliate him, Worth
proceeded then to gain emancipation from his burden of gratitude, and place
himself in open antagonism to his former patron.[28]
Duncan’s motives were different again. He was
intimate with Worth; and Pillow, who offered to marry
the Colonel to a rich and handsome widow, doubtless
promised him the post of inspector general. At any rate he urged Polk to make the
appointment, hinting at other reasons than mere qualifications, and it was made;
and we know that Duncan gave himself much trouble to assist Pillow as a partisan
supporter. The power of such a combination, headed by the President himself, to
gather adherents from the many ambitious officers hardly needs to be pointed out;
and finally there were, of course, jealous and envious men. “Since we cannot attain
to greatness, let us revenge ourselves by railing at it,” said Montaigne for the
benefit of such persons; and many of the officers knew that greatness was beyond
their powers. None of them could monologue as Scott did; none could look in a
cocked hat as he looked; none had won the Mexican war; and, moreover, he was
the sole general-in-chief.[29]
The result was a powerful movement against the prestige and authority properly
belonging to Scott. Pillow’s reports on the battles of Contreras and Chapultepec
tended to represent the General as a nonentity; and Worth not only did somewhat
the same, but referred to the Commander in terms of ridicule and contempt. A
letter, doubtless written directly or indirectly by Pillow over the signature of
“Leonidas,” extolling Pillow shamelessly and belittling Scott, was trickily worked into
the New Orleans Delta of September 10, 1847; and another letter, containing a
passage intended to show that Worth and Duncan had saved Scott from choosing
the wrong approach to the capital, appeared in the United States, then in a
Tampico newspaper and finally at Mexico City. Both letters were grossly improper,
especially since the army lay in the enemy’s country; and Scott found it necessary
to act. As he well said, “The general-in-chief who once submits to an outrage from
a junior, must lay his account to suffer the like from all the vicious under him,” and
“even the great mass of the spirited, intelligent, and well affected, among his
brothers in arms, would soon reduce such commander to utter imbecility, by
holding him in just scorn and contempt” for his recreancy to himself and the
country.[30]
On November 12, therefore, he issued his General Orders 349, which aimed to
stigmatize these offences in such a way as to prevent a recurrence of them.
Duncan then assumed in a plainly defiant manner the paternity of the Tampico
letter, although in fact the offensive passage had not been written by him. His](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-81-320.jpg)
![primary object in doing this was evidently to give Worth a handle, and the handle
was promptly seized. One thing led to another; and in the end formal charges were
brought by the commander-in-chief against Pillow, Worth and Duncan, and by the
two generals against him; appeals—insulting to Scott—were made by Pillow and
Worth to the government; and the technical “arrest” of the three officers followed.
[31]
The government then stepped in. Scott had no doubt given it offence during the
campaign, for his letters had plainly enough revealed a conviction that Polk had
broken faith with him, and purposely thrown difficulties and annoyances across his
path; but the circumstances had appeared to warrant his complaints, and Marcy
had at least “got even” by administering liberal censures in reply. The balance in
fact—aside, perhaps, from a mere acerbity of language—was against the
administration. Besides, having served the country well and saved the government
from disaster, Scott was entitled to some indulgence for irritation caused by the
peculiarly trying circumstances that surrounded him. He was a large man, had done
a large work and merited large treatment. But there was nothing large about the
administration. The confines of mediocrity hemmed it in. Pillow and Duncan were
therefore by its orders relieved of arrest; Worth was not only released, but assigned
to duty according to his highest brevet rank; and “in view of the present state of
things in the army,” chiefly or entirely caused by Polk’s agent and Marcy’s friend,
Scott was deposed. He had performed his task, said Robert E. Lee, and now was
“turned out as an old horse to die.” April 22, 1848, amidst the lamentations, cheers
and blessings of the army as a whole—trembling himself with emotion—he took his
leave, and Major General Butler, who was a Democrat and looked well on a horse,
bore sway at headquarters.[32]](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-82-320.jpg)
![XXX
THE NAVAL OPERATIONS OF THE WAR
1845–1848
In January, 1846, the United States had available for naval hostilities one ship-of-
the-line, seven frigates and razees, fifteen sloops-of-war, six brigs, one schooner
and three steamers—that is to say, thirty-three war craft. As ships-of-the-line
carried more than seventy guns, frigates about forty-four to fifty, sloops twenty,
brigs ten and other vessels in proportion, this fleet had 1155 cannon. Two of the
vessels, under Commodore James Biddle, were on the coast of Asia; several
occupied the Brazil station; and five cruised in African waters to check the trade in
slaves. The Pacific squadron, commanded by Commodore John D. Sloat, comprised
on July 1, 1846, the frigate Savannah, the sloops Portsmouth, Levant, Warren and
Cyane, the schooner Shark and the storeship Erie, to which the frigate Congress,
the razee Independence and the sloops Dale, Saratoga and Preble were added later
in the year, while the Levant went home; and substantially all the rest of the fleet,
known as the Home Squadron, attended to the West Indies and Gulf service, under
Commodore David Conner.[1]
The appropriation for the year ending with June, 1846, was a little less than ten
millions, but only about six and a half millions were expended. The war bill of May
13 permitted the completion of all vessels then building and the purchase of others;
and by November, 1847, after suffering a number of losses, the navy had in
commission five ships-of-the-line, one razee, four frigates, thirteen sloops, six brigs,
eleven schooners, four bomb-vessels, twelve steamers and six storeships.[1]
The peace establishment created by Congress in 1844 provided for 7500 petty
officers, seamen, landsmen and boys, and in August, 1846, this number was raised
to 10,000 for the period of the war; but owing to the remarkable activity of the
merchant marine and the consequently high wages, men could not easily be
obtained. During the most important year—November, 1846, to November, 1847—
not over 8000 were in the fleet at any one time. The whole number of seamen
employed in the course of the war did not exceed 7000; and hence plans to
strengthen our forces in the Gulf and the Pacific had to be curtailed. The service,
too, did not enjoy unqualified popularity. In the sailor’s decalogue appeared this
commandment:](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-83-320.jpg)
![PRIVATEERING
“Six days shalt thou work
And do all thou art able,
On the seventh thou shalt holystone
The deck and scrape the cable”;
and the cannon had to be rubbed with fragrant “sea pitch” from the bottom of the
ocean until they shone like Japanese lacquer. Discipline, therefore, not reinforced by
the enthusiasm and the necessities of war, fell considerably below its reputation,
and the crews were eager to be free when their time expired. The officers, even,
had become lax after thirty years of peace, and in too many instances their
standards of conduct had given way.[2]
In the administration of the navy, also, the effects of a long peace could be seen.
The control of matters had fallen, though not by accident, into the hands of shrewd
officers deeply interested in themselves and their friends. Supernumeraries
abounded. Those who drew the most pay often rendered the least service. The pet
ambition was for a safe, quiet and easy position. Shore billets were too numerous.
No field officer of the Marines had cruised since his promotion, and one of them
had been in the service more than a generation without going to sea. Secretary
Bancroft, eager for distinction, undertook to eliminate the abuses, but only
succeeded in eliminating himself. He had taught Greek, and was ridiculed by the
naval men as undertaking to play the pedagogue over them. Having no dominating
force of character nor even a commanding presence, he could not stand against the
governing clique. The requirements of the war, which might have assisted an abler
administrator to win the day, only increased his difficulties. The Senate refused to
confirm some of his appointees; and early in September, 1846, he became our
minister to England.[3]
J. Y. Mason, who succeeded him, was a fat, easy, agreeable man, quite innocent of
the desire to achieve reforms. Nobody disliked him, but nobody felt obliged to obey
him; and as late as the twentieth of February, 1847, suddenly discovering that Scott
had mentioned certain designs of the army against Vera Cruz, he awoke to the fact
that his department had failed to give the anticipated assistance. Just what could
be expected of the navy under all these prejudicial conditions was, therefore, in
some minds, a little uncertain.[3]
One of the most serious duties imposed upon it was to
guard against privateering, for not only our commerce
but the supplies required by our troops depended
upon free lanes. About the middle of 1845 the government issued orders that any
activity of such a kind on the part of Mexico should be considered the signal for
war; and as a deterrent it was announced by the newspapers, though incorrectly,
that privateersmen were to be regarded as pirates. Crews not predominantly
composed of Mexicans, it was often asserted, could legally be “strung up to the](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-84-320.jpg)
![yard-arm,” since we were understood to have treaties that sanctioned this principle
with most countries.[4]
After the war actually began, a great deal of danger was apprehended. Desperate
characters were believed to be waiting at New Orleans, and “piratical gangs” in the
ports of Cuba, where Almonte seemed to be at work. News arrived early in August,
1846, that privateering regulations had been issued by Mexico, and suspicious craft
soon appeared off Key West. In December the Mexican minister of war openly
avowed that great hopes of injuring the United States in this manner were
entertained. Blank certificates and commissions reached Washington; information
regarding efforts to set vessels at work in various quarters arrived there; and finally
the Carmelita of Bangor, Maine, was captured near Gibraltar by a felucca named El
Unico, fitted out at Oran, Algeria, and run by Spanish desperadoes.[4]
Mexico had not in reality, after studying the subject with deep interest, much
expectation of accomplishing any large results by issuing letters of marque, and the
regulations of July, 1846, were intended principally or wholly to annoy this country;
but in September and October she took the matter up rather seriously. A new law
provided that any foreigner entering her naval service might become a Mexican at
once, and blank naturalization papers as well as thousands of privateering
commissions, duly signed but not filled out, were carried by agents to the West
Indies, Great Britain, France and Spain. Almonte did his best at Havana. J. N.
Pareda, appointed Mexican chargé d’affaires at Madrid, appears to have circulated
the documents actively in the Peninsular ports; and another privateer, a Spanish
steamer named La Rosita, put out from Oran.[5]
On the other hand, the representatives of the United States insisted upon our
treaties and the obligations of neutrality. Polk’s annual Message of December, 1846,
denounced the Mexican plan as inviting “all the freebooters upon earth,” who felt
like paying for the privilege, to cruise against American commerce, announced that
our own courts would say whether such papers could protect them from the pirate’s
doom, recommended that Congress provide at once for the trial of Spanish subjects
caught in such business, and suggested American privateers—intended mainly to
recapture vessels taken under Mexican letters. An American force hastened to the
Mediterranean, and our squadrons were expected to seize all the rovers putting
out, as well as intercept all prizes on their way to the enemy’s ports. These
precautions looked rather discouraging to enterprising desperadoes.[6]
In England there was a feeling, as will appear later, that Mexico should be allowed
the utmost license against us, and the Mexican minister at London received many
applications for letters; but Great Britain did not really wish her supplies of cotton
to be endangered, and all the seas to be filled with corsairs preying upon the trade
of the world; and in October, 1845, her minister to Mexico was instructed to
prevent that country, if he could, from issuing letters of marque indiscriminately.
Bankhead protested also, as did the Spanish minister, against important features of](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-85-320.jpg)
![THE BLOCKADE
the regulations. Palmerston himself, though he acted in a languid fashion, and gave
notice at Washington that British subjects, found on Mexican privateers, could not
be treated as pirates, announced that his government would faithfully do its duty.[7]
France was prompt and active in responding to our demands. Spain, placed under
stringent obligations by the treaty of October, 1795, promised full compliance with
its requirements, captured El Unico, punished its crew, pursued La Rosita, and
ordered O’Donnell, the captain general of Cuba, to act as her obligations required;
but she accepted Pareda, the colporteur of what was piracy under her agreement,
as consul of Mexico; and the captain general, while he convinced the American
representative of his good-will and in fact would not permit an open violation of the
treaty, suggested to the Mexicans ways—fortunately impracticable—of evading his
own rules. But the risks of privateering under so many embarrassments and the
virtual impossibility of converting a prize into cash, prevented all attempts except
the feeble ones already mentioned. In this field, consequently, our navy, though
incessantly watchful, could acquire no laurels.[7]
Another aspect of the situation concerned it more
seriously. On the day Congress passed the war bill
(May 13, 1846) orders were issued to blockade the
ports of Mexico. Several definite aims prompted this action. Primarily, of course, it
was desired to prevent supplies of all kinds from reaching the enemy, and to
deprive them of the almost indispensable revenues obtained in peace by taxing
imports; but there were also hopes that loss of business would induce Great Britain
and France, which had a profitable trade in that quarter, to urge upon Mexico the
acceptance of our terms. The blockade was therefore to be enforced vigorously. At
the same time neutrals were to be treated with all reasonable indulgence.
Theoretically only their war vessels had the right of entering closed ports, but
practically the intention was to broaden that narrow door considerably. Toward
itself, however, the United States determined to be strictly faithful in observing its
declared principles. Merely those ports where the order could become effective
were in view. The announcement of blockade was to be made as public as it could
be; and in particular the government required that a full warning should be given to
neutral ships.[8]
For the work thus imposed upon him Conner had ample time to prepare. As early
as August, 1845, he was directed to blockade the Gulf ports in case of war; early in
1846 he knew of Mexico’s attitude regarding Slidell; before the end of March his
vessels occupied convenient positions; and promptly on the outbreak of hostilities a
blockade was announced at the chief harbors. By July, with some assistance from
the revenue service, it extended from the Rio Grande to the Goatzacoalcos.[9]
But the difficulty of making it continuously effective proved to be extraordinary.
There were not vessels enough of the proper kind; occasionally a more or less
complete concentration became necessary; and accidents of many sorts occurred.](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-86-320.jpg)
![Uncharted shoals and rocks, currents of unknown direction and force, the frequent
haze, and the darkening of the lighthouses made extreme caution imperative. The
suddenness and violence of the storms almost surpassed belief. At Vera Cruz the
Somers was blown over and sunk before Semmes, her able commander, could take
steps to avert the disaster. Even at the anchorages one would suddenly hear on a
calm afternoon the clarion orders of the speaking trumpet; the ship would quiver
and reverberate as the cable of the heaviest anchor ran swiftly out; in a moment
the storm would burst; and for days it might be a question almost hourly of going
ashore. At such times all sailing vessels on patrol duty had to make instantly for the
open sea, and before they could return to their stations a lurking blockade runner
could perhaps enter the port. Owing to such difficulties Alvarado and Frontera, for
instance, could not be watched continuously.[9]
Embarrassments also of a minor yet serious character had to be encountered. Our
vessels, unlike those of England, were designed exclusively for war, and long
confinement impaired the efficiency of the men. The government supplies of
eatables needed to be eked out from New Orleans huckster boats and European
merchant ships. Water could not be obtained readily from a hostile shore. At the
Antigua River, in July, 1846, the boats going up with casks were fired upon, and
such affairs had to be expected. Vessels were despatched long distances
occasionally to obtain fresh provisions, but even then scurvy of a most serious
nature broke out in the summer of 1846, disabling some of the largest and most
efficient ships for several months. The Raritan had more than 200 cases. Nearly all
on the Potomac suffered. The Falmouth had to go as far north as Boston to throw it
off. Swampy shores and kelp rotting under the torrid sun produced myriads of
poisonous as well as otherwise annoying insects. During a brief stay in the river off
Tampico nearly all the officers and men contracted ague, and the yellow fever
scourged a number of the vessels. More than two thirds of those on the Saratoga
had the latter disease. In August, 1847, the Mississippi left her station with some
200 men suffering from it.[10]
Being strangers and enemies, the Americans labored under peculiar disadvantages.
The people gladly assisted blockade runners in every possible way. Spanish captains
in particular, having friends on shore and pilots thoroughly familiar with the coast,
could not be prevented from reaching harbor at night or in thick weather by way of
the shoals. Sometimes it looked, for one or another of these many reasons, as if
our officers were careless or incompetent. Army observers, not well informed
regarding the conditions, felt disposed now and then to pronounce the blockade a
humbug, and naturally some foreigners did so. This opinion had neither truth nor
probability in its favor. But naturally, in view of all the circumstances, it proved more
satisfactory to occupy the ports, and open them to commerce on the basis of a
reasonable contributory tariff.[11]
Besides cruising to watch for privateers and hovering off the chief harbors to
maintain a blockade, our fleet was expected to share in the general offensive. For](https://image.slidesharecdn.com/2342186-250524121351-dcf1fb39/85/Introduction-To-Mathematical-Logic-5th-Edition-Mendelson-Elliott-87-320.jpg)
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- 6. INTRODUCTION TO MATHEMATICAL LOGIC F I F T H E D I T I O N
- 7. DISCRETE MATHEMATICS ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. Juergen Bierbrauer, Introduction to Coding Theory Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Richard A. Brualdi and Dragos̆ Cvetković, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Darel W. Hardy, Fred Richman, and Carol L. Walker, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words Leslie Hogben, Handbook of Linear Algebra Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces
- 8. Continued Titles Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB® , Second Edition Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search C. C. Lindner and C. A. Rodger, Design Theory, Second Edition Hang T. Lau, A Java Library of Graph Algorithms and Optimization Elliott Mendelson, Introduction to Mathematical Logic, Fifth Edition Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications, Second Edition Richard A. Mollin, An Introduction to Cryptography, Second Edition Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers Dingyi Pei, Authentication Codes and Combinatorial Designs Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design W. D. Wallis, Introduction to Combinatorial Designs, Second Edition Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition
- 10. DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Elliott Mendelson Queens College Department of Mathematics Flushing, New York, U.S.A. INTRODUCTION TO MATHEMATICAL LOGIC F I F T H E D I T I O N
- 11. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-876-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Mendelson, Elliott. Introduction to mathematical logic / Elliott Mendelson. -- 5th ed. p. cm. -- (Discrete mathematics and its applications ; 48) Includes bibliographical references and index. ISBN-13: 978-1-58488-876-5 ISBN-10: 1-58488-876-8 1. Logic, Symbolic and mathematical. I. Title. QA9.M4 2009 511.3--dc22 2009002833 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 12. To Arlene
- 14. Contents Preface.................................................................................................................. xiii Introduction .......................................................................................................... xv 1. The Propositional Calculus............................................................................ 1 1.1 Propositional Connectives. Truth Tables.......................................... 1 1.2 Tautologies ............................................................................................ 5 1.3 Adequate Sets of Connectives.......................................................... 18 1.4 An Axiom System for the Propositional Calculus ........................ 24 1.5 Independence. Many-Valued Logics............................................... 34 1.6 Other Axiomatizations ...................................................................... 37 2. First-Order Logic and Model Theory......................................................... 41 2.1 Quantifiers........................................................................................... 41 2.1.1 Parentheses ............................................................................ 44 2.2 First-Order Languages and Their Interpretations. Satisfiability and Truth. Models.................................................... 48 2.3 First-Order Theories........................................................................... 61 2.3.1 Logical Axioms...................................................................... 62 2.3.2 Proper Axioms ...................................................................... 62 2.3.3 Rules of Inference ................................................................. 62 2.4 Properties of First-Order Theories................................................... 64 2.5 Additional Metatheorems and Derived Rules............................... 69 2.5.1 Particularization Rule A4 .................................................... 69 2.5.2 Existential Rule E4................................................................ 69 2.6 Rule C .................................................................................................. 73 2.7 Completeness Theorems ................................................................... 77 2.8 First-Order Theories with Equality.................................................. 88 2.9 Definitions of New Function Letters and Individual Constants ................................................................. 97 2.10 Prenex Normal Forms ..................................................................... 100 2.11 Isomorphism of Interpretations. Categoricity of Theories......... 105 2.12 Generalized First-Order Theories. Completeness and Decidability ............................................................................... 107 2.12.1 Mathematical Applications.............................................. 111 2.13 Elementary Equivalence. Elementary Extensions........................ 117 2.14 Ultrapowers. Nonstandard Analysis.......................................... 123 2.14.1 Reduced Direct Products ................................................. 125 2.14.2 Nonstandard Analysis ..................................................... 131 2.15 Semantic Trees.................................................................................. 135 2.16 Quantification Theory Allowing Empty Domains...................... 141 ix
- 15. 3. Formal Number Theory.............................................................................. 149 3.1 An Axiom System .............................................................................. 149 3.2 Number-Theoretic Functions and Relations .................................. 166 3.3 Primitive Recursive and Recursive Functions ............................... 171 3.4 Arithmetization. Gödel Numbers.................................................... 188 3.5 The Fixed-Point Theorem. Gödel’s Incompleteness Theorem................................................................... 202 3.6 Recursive Undecidability. Church’s Theorem ............................... 214 3.7 Nonstandard Models......................................................................... 224 4. Axiomatic Set Theory.................................................................................. 227 4.1 An Axiom System .............................................................................. 227 4.2 Ordinal Numbers ............................................................................... 242 4.3 Equinumerosity. Finite and Denumerable Sets ............................. 256 4.3.1 Finite Sets............................................................................... 261 4.4 Hartogs’ Theorem. Initial Ordinals. Ordinal Arithmetic............ 266 4.5 The Axiom of Choice. The Axiom of Regularity........................... 279 4.6 Other Axiomatizations of Set Theory.............................................. 290 4.6.1 Morse–Kelley (MK) .............................................................. 291 4.6.2 Zermelo–Fraenkel (ZF) ........................................................ 291 4.6.3 The Theory of Types (ST).................................................... 293 4.6.3.1 ST1 (Extensionality Axiom).................................. 294 4.6.3.2 ST2 (Comprehension Axiom Scheme) ...................................................... 294 4.6.3.3 ST3 (Axiom of Infinity)......................................... 295 4.6.4 Quine’s Theories NF and ML............................................. 297 4.6.4.1 NF1 (Extensionality).............................................. 297 4.6.4.2 NF2 (Comprehension)........................................... 298 4.6.5 Set Theory with Urelements ............................................... 300 5. Computability............................................................................................... 309 5.1 Algorithms. Turing Machines .......................................................... 309 5.2 Diagrams ............................................................................................. 315 5.3 Partial Recursive Functions. Unsolvable Problems....................... 322 5.4 The Kleene–Mostowski Hierarchy. Recursively Enumerable Sets ................................................................................. 338 5.5 Other Notions of Computability...................................................... 350 5.6 Decision Problems.............................................................................. 368 Appendix A: Second-Order Logic.................................................................. 375 Appendix B: First Steps in Modal Propositional Logic............................. 391 x Contents
- 16. Answers to Selected Exercises........................................................................ 403 Bibliography....................................................................................................... 433 Notation............................................................................................................... 447 Index .................................................................................................................... 453 Contents xi
- 18. Preface This is a compact introduction to some of the principal topics of mathemat- ical logic. In the belief that beginners should be exposed to the easiest and most natural proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. If we are to be expelled from ‘‘Cantor’s paradise’’ (as nonconstructive set theory was called by Hilbert), at least we should know what we would be missing. Readers who wish to learn about constructive mathematics and intuitionism can consult the books by van Atten (2003), van Atten, Boldini, Bourdeau, and Heinzmann (2008), Brouwer (1976), George and Velleman (2001), Heyting (1956), Mancosu (1997), and Troelstra and van Dalen (1988). The more important changes in this new edition are the following: 1. There is an entirely new exposition, Section 3.7, of some basic ideas and results about nonstandard models of number theory. 2. A second appendix, Appendix B, has been added as an introduction to modal propositional logic. The most important application in this area has to do with provability logic (see Boolos [1993]), but it is also of great interest elsewhere in logic. 3. The Bibliography has been greatly enlarged, although it is still not able to do full justice to the enormous expansion of research and scholarship in logic and related areas. 4. The exercises and the Answers to Selected Exercises have been expanded and necessary corrections have been made. The material in this book can be covered in two semesters, but Chapters 1–3 are quite adequate for a one-semester course (Sections 1.5, 1.6, 3.7, and 2.10–2.16 can be omitted in an abbreviated course). I have adopted the convention of prefixing a ‘‘D’’ to any section or exercise that will probably be difficult for a beginner, and an ‘‘A’’ to any section or exercise that presupposes familiarity with a topic that has not been carefully explained in the text. Bibliographic references are given to the best source of informa- tion, which is not always the earliest; hence these references give no indi- cation as to priority. I believe that the essential parts of the book can be read with ease by anyone with some experience in abstract mathematical thinking. There is, however, no specific prerequisite. xiii
- 19. This book owes an obvious debt to the standard works of Hilbert and Bernays (1934, 1939), Kleene (1952), Rosser (1953), and Church (1956). I am grateful to many people for their help, including those who helped with earlier editions, as well as Katalin Bimbo, Frank Cannonito, John Corcoran, George Hacken, Maidim Malkov, and Gordon McLean, Jr. I also want to thank my editor, Bob Stern, for his advice and patience. Elliott Mendelson xiv Preface
- 20. Introduction One of the popular definitions of logic is that it is the analysis of methods of reasoning. In studying these methods, logic is interested in the form rather than the content of the argument. For example, consider the two arguments: 1. All men are mortal. Socrates is a man. Hence, Socrates is mortal. 2. All cats like fish. Silvy is a cat. Hence, Silvy likes fish. Both have the same form: All A are B. S is an A. Hence, S is a B. The truth or falsity of the particular premisses and conclusions is of no concern to logi- cians. They want to know only whether the premisses imply the conclusion. The systematic formalization and cataloguing of valid methods of reasoning are a main task of logicians. If the work uses mathematical techniques or if it is primarily devoted to the study of mathematical reasoning, then it may be called mathematical logic. We can narrow the domain of mathematical logic if we define its principal aim to be a precise and adequate understanding of the notion of mathematical proof. Impeccable definitions have little value at the beginning of the study of a subject. The best way to find out what mathematical logic is about is to start doing it, and students are advised to begin reading the book even though (or especially if) they have qualms about the meaning and purpose of the subject. Although logic is basic to all other studies, its fundamental and apparently self-evident character discouraged any deep logical investigations until the late nineteenth century. Then, under the impetus of the discovery of non- Euclidean geometry and the desire to provide a rigorous foundation for calculus and higher analysis, interest in logic revived. This new interest, however, was still rather unenthusiastic until, around the turn of the century, the mathematical world was shocked by the discovery of the paradoxes— that is, arguments that lead to contradictions. The most important paradoxes are described here. 1. Russell’s paradox (1902). By a set, we mean any collection of objects— for example, the set of all even integers or the set of all saxophone players in Brooklyn. The objects that make up a set are called its members or elements. Sets may themselves be members of sets; for example, the set of all sets of integers has sets as its members. Most sets are not members of themselves; the set of cats, for example, is not a member of itself because the set of cats is not a cat. However, there may be sets that do belong to themselves—perhaps, for example, a set xv
- 21. containing all sets. Now, consider the set A of all those sets X such that X is not a member of X. Clearly, by definition, A is a member of A if and only if A is not a member of A. So, if A is a member of A, then A is also not a member of A; and if A is not a member of A, then A is a member of A. In any case, A is a member of A and A is not a member of A [See Link, (2004)]. 2. Cantor’s paradox (1899). This paradox involves the theory of cardinal numbers and may be skipped by those readers having no previous acquaintance with that theory. The cardinal number Y of a set Y is a measure of the size of the set; Y ¼ Z if and only if Y is equinumerous with Z (that is, there is a one–one correspondence between Y and Z). We define Y 4 Z to mean that Y is equinumerous with a subset of Z; by Y < Z we mean Y 4 Z and Y 6¼ Z. Cantor proved that, if p(Y) is the set of all subsets of Y, then Y < p(Y). Let V be the universal set— that is, the set of all sets. Now, p(V) is a subset of V; so it follows easily that p(V) 4 V. On the other hand, by Cantor’s theorem, V < p(V). Bernstein’s theorem asserts that, if Y 4 Z and Z 4 Y, then Y ¼ Z. Hence, V ¼ p(V), contradicting V < p(V). 3. Burali-Forti’s paradox (1897). This paradox is the analogue in the theory of ordinal numbers of Cantor’s paradox and requires famil- iarity with ordinal number theory. Given any ordinal number, there is a still larger ordinal number. But the ordinal number determined by the set of all ordinal numbers is the largest ordinal number. 4. The liar paradox. A man says, ‘‘I am lying’’. If he is lying, then what he says is true and so he is not lying. If he is not lying, then what he says is true, and so he is lying. In any case, he is lying and he is not lying.* 5. Richard’s paradox (1905). Some phrases of the English language denote real numbers; for example, ‘‘the ratio between the circumfer- ence and diameter of a circle’’ denotes the number p. All the phrases of the English language can be enumerated in a standard way: order all phrases that have k letters lexicographically (as in a dictionary) and then place all phrases with k letters before all phrases with a larger number of letters. Hence, all phrases of the English language that denote real numbers can be enumerated merely by omitting all other phrases in the given standard enumeration. Call the nth real * The Cretan ‘‘paradox,’’ known in antiquity, is similar to the liar paradox. The Cretan philoso- pher Epimenides said, ‘‘All Cretans are liars.’’ If what he said is true, then, since Epimenides is a Cretan, it must be false. Hence, what he said is false. Thus, there must be some Cretan who is not a liar. This is not logically impossible; so we do not have a genuine paradox. However, the fact that the utterance by Epimenides of that false sentence could imply the existence of some Cretan who is not a liar is rather unsettling. xvi Introduction
- 22. number in this enumeration the nth Richard number. Consider the phrase: ‘‘the real number whose nth decimal place is 1 if the nth decimal place of the nth Richard number is not 1, and whose nth decimal place is 2 if the nth decimal place of the nth Richard number is 1.’’ This phrase defines a Richard number—say, the kth Richard number; but, by its definition, it differs from the kth Richard number in the kth decimal place. 6. Berry’s paradox (1906). There are only a finite number of symbols (letters, punctuation signs, etc.) in the English language. Hence, there are only a finite number of English expressions that contain fewer than 200 occurrences of symbols (allowing repetitions). There are, therefore, only a finite number of positive integers that are denoted by an English expression containing fewer than 200 occurrences of symbols. Let k be the least positive integer that is not denoted by an English expression containing fewer than 200 occurrences of symbols. The italicized English phrase contains fewer than 200 occurrences of symbols and denotes the integer k. 7. Grelling’s paradox (1908). An adjective is called autological if the prop- erty denoted by the adjective holds for the adjective itself. An adjec- tive is called heterological if the property denoted by the adjective does not apply to the adjective itself. For example, ‘‘polysyllabic’’ and ‘‘English’’ are autological, whereas ‘‘monosyllabic’’ and ‘‘French’’ are heterological. Consider the adjective ‘‘heterological.’’ If ‘‘heterological’’ is heterological, then it is not heterological. If ‘‘heterological’’ is not heterological, then it is heterological. In either case, ‘‘heterological’’ is both heterological and not heterological. 8. Löb’s paradox (1955). Let A be any sentence. Let B be the sentence: ‘‘If this sentence is true, then A.’’ So, B asserts: ‘‘If B is true, then A.’’ Now consider the following argument: Assume B is true; then, by B, since B is true, A holds. This argument shows that, if B is true, then A. But this is exactly what B asserts. Hence, B is true. Therefore, by B, since B is true, A is true. Thus, every sentence is true. (This paradox may be more accurately attributed to Curry [1942].) All of these paradoxes are genuine in the sense that they contain no obvious logical flaws. The logical paradoxes (1–3) involve only notions from the theory of sets, whereas the semantic paradoxes (4–8) also make use of concepts like ‘‘denote,’’ ‘‘true,’’ and ‘‘adjective,’’ which need not occur within our standard mathematical language. For this reason, the logical paradoxes are a much greater threat to a mathematician’s peace of mind than the semantic paradoxes. Analysis of the paradoxes has led to various proposals for avoiding them. All of these proposals are restrictive in one way or another of the ‘‘naive’’ concepts that enter into the derivation of the paradoxes. Russell noted the self-reference present in all the paradoxes and suggested that every object Introduction xvii
- 23. must have a definite nonnegative integer as its ‘‘type.’’ Then an expression ‘‘x is a member of the set y’’ is to be considered meaningful if and only if the type of y is one greater than the type of x. This approach, known as the theory of types and systematized and devel- oped in Principia Mathematica by Whitehead and Russell (1910–1913), is suc- cessful in eliminating the known paradoxes,y but it is clumsy in practice and has certain other drawbacks as well. A different criticism of the logical para- doxes is aimed at their assumption that, for every property P(x), there exists a corresponding set of all objects x that satisfy P(x). If we reject this assumption, then the logical paradoxes are no longer derivable.z It is necessary, however, to provide new postulates that will enable us to prove the existence of those sets that are needed by the practicing mathematician. The first such axiomatic set theory was invented by Zermelo (1908). In Chapter 4 we shall present an axiomatic theory of sets that is a descendant of Zermelo’s system (with some new twists given to it by von Neumann, R. Robinson, Bernays, and Gödel). There are also various hybrid theories combining some aspects of type theory and axiomatic set theory—for example, Quine’s system NF. A more radical interpretation of the paradoxes has been advocated by Brouwer and his intuitionist school (see Heyting, 1956). They refuse to accept the universality of certain basic logical laws, such as the law of excluded middle: P or not-P. Such a law, they claim, is true for finite sets, but it is invalid to extend it on a wholesale basis to all sets. Likewise, they say it is invalid to conclude that ‘‘There exists an object x such that not-P(x)’’ follows from the negation of ‘‘For all x, P(x)’’; we are justified in asserting the existence of an object having a certain property only if we know an effective method for constructing (or finding) such an object. The paradoxes are not derivable (or even meaningful) if we obey the intuitionist strictures, but so are many important theorems of everyday mathematics, and, for this reason, intuitionism has found few converts among mathematicians. Exercises 0.1 Use the sentence (*) This entire sentence is false or 2 þ 2 ¼ 5 to prove that 2 þ 2 ¼ 5. Comment on the significance of this proof. 0.2 Show how the following has a paradoxical result. The smallest positive integer that is not denoted by a phrase in this book. y Russells’s paradox, for example, depends on the existence of the set A of all sets that are not members of themselves. Because, according to the theory of types, it is meaningless to say that a set belongs to itself, there is no such set A. z Russell’s paradox then proves that there is no set A of all sets that do not belong to themselves. The paradoxes of Cantor and Burali-Forti show that there is no universal set and no set that contains all ordinal numbers. The semantic paradoxes cannot even be formulated, since they involve notions not expressible within the system. xviii Introduction
- 24. Whatever approach one takes to the paradoxes, it is necessary first to examine the language of logic and mathematics to see what symbols may be used, to determine the ways in which these symbols are put together to form terms, formulas, sentences, and proofs, and to find out what can and cannot be proved if certain axioms and rules of inference are assumed. This is one of the tasks of mathematical logic, and, until it is done, there is no basis for comparing rival foundations of logic and mathematics. The deep and devastating results of Gödel, Tarski, Church, Rosser, Kleene, and many others have been ample reward for the labor invested and have earned for mathematical logic its status as an independent branch of mathematics. For the absolute novice a summary will be given here of some of the basic notations, ideas, and results used in the text. The reader is urged to skip these explanations now and, if necessary, to refer to them later on. A set is a collection of objects.* The objects in the collection are called elements or members of the set. We shall write ‘‘x 2 y’’ for the statement that x is a member of y. (Synonymous expressions are ‘‘x belongs to y’’ and ‘‘y contains x’’.) The negation of ‘‘x 2 y’’ will be written ‘‘x = 2 y.’’ By ‘‘x y’’ we mean that every member of x is also a member of y (synonymously, that x is a subset of y, or that x is included in y). We shall write ‘‘t ¼ s’’ to mean that t and s denote the same object. As usual, ‘‘t 6¼ s’’ is the negation of ‘‘t ¼ s.’’ For sets x and y, we assume that x ¼ y if and only if x y and y x—that is, if and only if x and y have the same members. A set x is called a proper subset of a set y, written ‘‘x y’’ if x y but x 6¼ y. (The notation x 6 y is often used instead of x y.) The union x [ y of sets x and y is defined to be the set of all objects that are members of x or y or both. Hence, x [ x ¼ x, x [ y ¼ y [ x, and (x [ y) [ z ¼ x [ (y [ z). The intersection x y is the set of objects that x and y have in common. Therefore, x x ¼ x, x y ¼ y x, and (x y) z ¼ x (y z). Moreover, x (y [ z) ¼ (x y) [ (x z) and x [ (y z) ¼ (x [ y) (x [ z). The rela- tive complement x y is the set of members of x that are not members of y. We also postulate the existence of the empty set (or null set) ;—that is, a set that has no members at all. Then x ; ¼ ;, x [; ¼ x, x ; ¼ x, ; x ¼ ;, and x x ¼ ;. Sets x and y are called disjoint if x y ¼ ;. Givenanyobjectsb1, . . . , bk,thesetthatcontainsb1, . . . , bk asitsonlymembers is denoted b1, . . . , bk f g. In particular, {x, y} is a set having x and y as its only members and, if x 6¼ y, is called the unordered pair of x and y. The set {x, x} is identical with {x} and is called the unit set of x. Notice that {x, y} ¼ {y, x}. By hb1, . . . , bki we mean the ordered k-tuple of b1, . . . , bk. The basic property of ordered k-tuples is that hb1, . . . , bki ¼ hc1, . . . , cki if and only if * Which collections of objects form sets will not be specified here. Care will be exercised to avoid using any ideas or procedures that may lead to the paradoxes; all the results can be formalized in the axiomatic set theory of Chapter 4. The term ‘‘class’’ is sometimes used as a synonym for ‘‘set,’’ but it will be avoided here because it has a different meaning in Chapter 4. If a property P(x) does determine a set, that set is often denoted {xjP(x)}. Introduction xix
- 25. b1 ¼ c1, b2 ¼ c2, . . . , bk ¼ ck. Thus, hb1, b2i ¼ hb2, b1i if and only if b1 ¼ b2. Ordered 2-tuples are called ordered pairs. The ordered 1-tuple hbi is taken to be b itself. If X is a set and k is a positive integer, we denote by Xk the set of all ordered k-tuples hb1, . . . , bki of elements b1, . . . , bk of X. In particular, X1 is X itself.IfYandZaresets,thenbyY Zwedenotethesetofallorderedpairshy, zi such that y 2 Y and z 2 Z. Y Z is called the Cartesian product of Y and Z. An n-place relation (or a relation with n arguments) on a set X is a subset of Xn —that is, a set of ordered n-tuples of elements of X. For example, the 3-place relation of betweenness for points on a line is the set of all 3-tuples hx, y, zi such that the point x lies between the points y and z. A 2-place relation is called a binary relation; for example, the binary relation of father- hood on the set of human beings is the set of all ordered pairs hx, yi such that x and y are human beings and x is the father of y. A 1-place relation on X is a subset of X and is called a property on X. Given a binary relation R on a set X, the domain of R is defined to be the set of all y such that hy, zi 2 R for some z; the range of R is the set of all z such that hy, zi 2 R for some y; and the field of R is the union of the domain and range of R. The inverse relation R1 of R is the set of all ordered pairs hy, zi such that hz, yi 2 R. For example, the domain of the relation on the set v of nonne- gative integers* is v, its range is v {0}, and the inverse of is . Notation: Very often xRy is written instead of hx, yi 2 R. Thus, in the example just given, we usually write x y instead of hx, yi 2 . A binary relation R is said to be reflexive if xRx for all x in the field of R; R is symmetric if xRy implies yRx; and R is transitive if xRy and yRz imply xRz. Examples: The relation on the set of integers is reflexive and transitive but not symmetric. The relation ‘‘having at least one parent in common’’ on the set of human beings is reflexive and symmetric, but not transitive. A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation. Examples of equivalence relations are: (1) the identity relation IX on a set X, consisting of all pairs hx, xi, where x 2 X; (2) given a fixed positive integer n, the relation x y (mod n), which holds when x and y are integers and x y is divisible by n; (3) the congruence relation on the set of triangles in a plane; (4) the similarity relation on the set of triangles in a plane. Given an equivalence relation R whose field is X, and given any y 2 X, define [y] as the set of all z in X such that yRz. Then [y] is called the R-equivalence class of y. Clearly, [u] ¼ [v] if and only if uRv. Moreover, if [u] 6¼ [v], then [u] [v] ¼ ;; that is, different R-equivalence classes have no elements in common. Hence, the set X is completely partitioned into the R-equivalence classes. In example (1) above, the equivalence classes are just the unit sets {x}, where x 2 X. In example (2), there are n equivalence classes, the kth equivalence class (k ¼ 0, 1, . . . , n 1) being the set of all integers that leave the remainder k upon division by n. * v will also be referred to as the set of natural numbers. xx Introduction
- 26. A function f is a binary relation such that hx, yi 2 f and hx, zi 2 f imply y ¼ z. Thus, for any element x of the domain of a function f, there is a unique y such that hx, yi 2 f; this unique y is denoted f(x). If x is in the domain of f, then f(x) is said to be defined. A function f with domain X and range Y is said to be a function from X onto Y. If f is a function from X onto a subset of Z, then f is said to be a function from X into Z. For example, if the domain of f is the set of integers and f(x) ¼ 2x for every integer x, then f is a function from the set of integers onto the set of even integers, and f is a function from the set of integers into the set of integers. A function whose domain consists of n-tuples is said to be a function of n arguments. A total function of n arguments on a set X is a function f whose domain is Xn . It is customary to write f x1, . . . , xn ð Þ instead of f hx1, . . . , xni ð Þ, and we refer to f x1, . . . , xn ð Þ as the value of f for the arguments x1, . . . , xn. A partial function of n arguments on a set X is a function whose domain is a subset of Xn . For example, ordinary division is a partial, but not total, function of two arguments on the set of integers, since division by 0 is not defined. If f is a function with domain X and range Y, then the restriction fz of f to a set Z is the function f (Z Y). Then fZ(u) ¼ v if and only if u 2 Z and f(u) ¼ v. The image of the set Z under the function f is the range of fz. The inverse image of a set W under the function f is the set of all u in the domain of f such that f(u) 2 W. We say that f maps X onto (into) Y if X is a subset of the domain of f and the image of X under f is (a subset of) Y. By an n-place operation (or operation with n arguments) on a set X we mean a function from Xn into X. For example, ordinary addition is a binary (i.e., 2-place) operation on the set of natural numbers {0, 1, 2, . . . }. But ordinary subtraction is not a binary operation on the set of natural numbers. The composition f g (sometimes denoted fg) of functions f and g is the function such that (f g)(x) ¼ f(g(x)); (f g)(x) is defined if and only if g(x) is defined and f(g(x)) is defined. For example, if g(x) ¼ x2 and f(x) ¼ x þ 1 for every integer x, then (f g)(x) ¼ x2 þ 1 and (g f)(x) ¼ (x þ 1)2 . Also, if h(x) ¼ x for every real number x and f (x) ¼ ffiffiffi x p for every nonnegative real number x, then (f h)(x) is defined only for x 4 0, and, for such x, (f h)(x) ¼ ffiffiffiffiffiffiffi x p . A function f such that f(x) ¼ f(y) implies x ¼ y is called a 1–1 (one–one) function. For example, the identity relation IX on a set X is a 1–1 function, since IX(y) ¼ y for every y 2 X; the function g with domain v, such that g(x) ¼ 2x for every x 2 v, is 1–1 (one–one); but the function h whose domain is the set of integers and such that h(x) ¼ x2 for every integer x is not 1–1, since h(1) ¼ h(1). Notice that a function f is 1–1 if and only if its inverse relation f 1 is a function. If the domain and range of a 1–1 function f are X and Y, then f is said to be a 1–1 correspondence between X and Y; then f1 is a 1–1 correspondence between Y and X, and f1 f ¼ IX and f f1 ¼ IY. If f is a 1–1 correspondence between X and Y and g is a 1–1 correspondence between Y and Z, then g f is a 1–1 correspondence between X and Z. Sets X and Y are said to be equinumerous (written X ffi Y) if and only if there is a 1–1 correspondence between X and Y. Clearly, X ffi X, X ffi Y implies Y ffi X, and X ffi Y and Y ffi Z implies X ffi Z. It is somewhat harder to Introduction xxi
- 27. show that, if X ffi Y1 Y and Y ffi X1 X, then X ffi Y (see Bernstein’s the- orem in Chapter 4). If X ffi Y, one says that X and Y have the same cardinal number, and if X is equinumerous with a subset of Y but Y is not equinumer- ous with a subset of X, one says that the cardinal number of X is smaller than the cardinal number of Y.* A set X is denumerable if it is equinumerous with the set of positive integers. A denumerable set is said to have cardinal number Q0, and any set equinu- merous with the set of all subsets of a denumerable set is said to have the cardinal number 2Q0 (or to have the power of the continuum). A set X is finite if it is empty or if it is equinumerous with the set {1, 2, . . . , n} of all positive integers that are less than or equal to some positive integer n. A set that is not finite is said to be infinite. A set is countable if it is either finite or denumerable. Clearly, any subset of a denumerable set is countable. A denumerable sequence is a function s whose domain is the set of positive integers; one usually writes sn instead of s(n). A finite sequence is a function whose domain is the empty set or {1, 2, . . . , n} for some positive integer n. Let P x, y1, . . . , yk ð Þ be some relation on the set of nonnegative integers. In particular, P may involve only the variable x and thus be a property. If P 0, y1, . . . , yk ð Þ holds, and, if, for every n, P n, y1, . . . , yk ð Þ implies P n þ 1, y1, . . . , yk ð Þ, then P x, y1, . . . , yk ð Þ is true for all nonnegative integers x (principle of mathematical induction). In applying this principle, one usually proves that, for every n, P n, y1, . . . , yk ð Þ implies P n þ 1, y1, . . . , yk ð Þ by assum- ing P n, y1, . . . , yk ð Þ and then deducing P n þ 1, y1, . . . , yk ð Þ; in the course of this deduction, P n, y1, . . . , yk ð Þ is called the inductive hypothesis. If the relation P actually involves variables y1, . . . , yk other than x, then the proof is said to proceed by induction on x. A similar induction principle holds for the set of integers greater than some fixed integer j. An example is: to prove by mathematical induction that the sum of the first n odd integers 1 þ 3 þ 5 þ þ (2n 1) is n2 , first show that 1 ¼ 12 (that is, P(1)), and then, that if 1 þ 3 þ 5 þ þ (2n 1) ¼ n2 , then 1 þ 3 þ 5 þ þ (2n 1) þ (2n þ 1) ¼ (n þ 1)2 (that is, if P(n) then P(n þ 1)). From the principle of math- ematical induction one can prove the principle of complete induction: If, for every nonnegative integer x the assumption that P u, y1, . . . , yk ð Þ is true for all u x implies that P x, y1, . . . , yk ð Þ holds, then, for all nonnegative integers x, P x, y1, . . . , yk ð Þ is true, (Exercise: Show by complete induction that every integer greater than 1 is divisible by a prime number.) A partial order is a binary relation R such that R is transitive and, for every x in the field of R, xRx is false. If R is a partial order, then the relation R0 that is the union of R and the set of all ordered pairs hx, xi, where x is in the field of R, we shall call a reflexive partial order; in the literature, ‘‘partial order’’ is used for either partial order or reflexive partial order. Notice that (xRy and yRx) is * One can attempt to define the cardinal number of a set X as the collection [X] of all sets equinumerous with X. However, in certain axiomatic set theories, [X] does not exist, whereas in others [X] exists but is not a set. xxii Introduction
- 28. impossible if R is a partial order, whereas (xRy and yRx) implies x ¼ y if R is a reflexive partial order. A (reflexive) total order is a (reflexive) partial order such that, for any x and y in the field of R, either x ¼ y or xRy or yRx. Examples: (1) the relation on the set of integers is a total order, whereas is a reflexive total order; (2) the relation on the set of all subsets of the set of positive integers is a partial order but not a total order, whereas the relation is a reflexive partial order but not a reflexive total order. If B is a subset of the field of a binary relation R, then an element y of B is called an R-least element of B if yRz for every element z of B different from y. A well-order (or a well-ordering relation) is a total order R such that every nonempty subset of the field of R has an R-least element. Examples: (1) the relation on the set of nonnegative integers is a well-order; (2) the relation on the set of nonnegative rational numbers is a total order but not a well-order; (3) the relation on the set of integers is a total order but not a well- order. Associated with every well-order R having field X there is a complete induction principle: if P is a property such that, for any u in X, whenever all z in X such that zRu have the property P, then u has the property P, then it follows that all members of X have the property P. If the set X is infinite, a proof using this principle is called a proof by transfinite induction. One says that a set X can be well-ordered if there exists a well-order whose field is X. An assumption that is useful in modern mathematics but about the validity of which there has been considerable controversy is the well-ordering principle: every set can be well- ordered. The well-ordering principle is equivalent (given the usual axioms of set theory) to the axiom of choice: for any set X of nonempty pairwise disjoint sets, there is a set Y (called a choice set) that contains exactly one element in common with each set in X. Let B be a nonempty set, f a function from B into B, and g a function from B2 into B. Write x0 for f(x) and x y for g(x, y). Then hB, f , gi is called a Boolean algebra if B contains at least two elements and the following conditions are satisfied: 1. x y ¼ y x for all x and y in B 2. (x y) z ¼ x (y z) for all x, y, z in B 3. x y0 ¼ z z0 if and only if x y ¼ x for all x, y, z in B. Let x [ y stand for x0 y0 ð Þ0 , and write x 4 y for x y ¼ x. It is easily proved that z z0 ¼ w w0 for any w and z in B; we denote the value of z z0 by 0. Let 1 stand for 00 . Then z [ z0 ¼ 1 for all z in B. Note also that is a reflexive partial order on B, and hB, f, [i is a Boolean algebra. (The symbols , [ , 0, 1 should not be confused with the corresponding symbols used in set theory and arithmetic.) An ideal J in hB, f , gi is a nonempty subset of B such that (1) if x 2 J and y 2 J, then x [ y 2 J, and (2) if x 2 J and y 2 B, then x y 2 J. Clearly, {0} and B are ideals. An ideal different from B is called a proper ideal. A maximal ideal is a proper ideal that is included in no other proper ideal. It can be shown Introduction xxiii
- 29. that a proper ideal J is maximal if and only if, for any u in B, u 2 J or u0 2 J. From the axiom of choice it can be proved that every Boolean algebra contains a maximal ideal, or, equivalently, that every proper ideal is included in some maximal ideal. For example, let B be the set of all subsets of a set X; for Y 2 B, let Y0 ¼ X Y, and for Y and Z in B, let Y Z be the ordinary set-theoretic intersection of Y and Z. Then hB,0 , i is a Boolean algebra. The 0 of B is the empty set ;, and 1 is X. For each element u in X, the set Ju of all subsets of X that do not contain u is a maximal ideal. For a detailed study of Boolean algebras, see Sikorski (1960), Halmos (1963), and Mendelson (1970). xxiv Introduction
- 30. 1 The Propositional Calculus 1.1 Propositional Connectives. Truth Tables Sentences may be combined in various ways to form more complicated sentences. We shall consider only truth-functional combinations, in which the truth or falsity of the new sentence is determined by the truth or falsity of its component sentences. Negation is one of the simplest operations on sentences. Although a sen- tence in a natural language may be negated in many ways, we shall adopt a uniform procedure: placing a sign for negation, the symbol :, in front of the entire sentence. Thus, if A is a sentence, then :A denotes the negation of A. The truth-functional character of negation is made apparent in the follow- ing truth table: A :A T F F T When A is true, :A is false; when A is false, :A is true. We use T and F to denote the truth values true and false. Another common truth-functional operation is the conjunction: ‘‘and.’’ The conjunction of sentences A and B will be designated by A ^ B and has the following truth table: A B A ^ B T T T F T F T F F F F F A ^ B is true when and only when both A and B are true. A and B are called the conjuncts of A ^ B. Note that there are four rows in the table, correspond- ing to the number of possible assignments of truth values to A and B. In natural languages, there are two distinct uses of ‘‘or’’: the inclusive and the exclusive. According to the inclusive usage, ‘‘A or B’’ means ‘‘A or B or both,’’ whereas according to the exclusive usage, the meaning is ‘‘A or B, but 1
- 31. not both,’’ We shall introduce a special sign, _, for the inclusive connective. Its truth table is as follows: A B A _ B T T T F T T T F T F F F Thus, A _ B is false when and only when both A and B are false. ‘‘A _ B’’ is called a disjunction, with the disjuncts A and B. Another important truth-functional operation is the conditional: ‘‘if A, then B.’’ Ordinary usage is unclear here. Surely, ‘‘if A, then B’’ is false when the antecedent A is true and the consequent B is false. However, in other cases, there is no well-defined truth value. For example, the following sentences would be considered neither true nor false: 1. If 1 þ 1 ¼ 2, then Paris is the capital of France. 2. If 1 þ 1 6¼ 2, then Paris is the capital of France. 3. If 1 þ 1 6¼ 2, then Rome is the capital of France. Their meaning is unclear, since we are accustomed to the assertion of some sort of relationship (usually causal) between the antecedent and the conse- quent. We shall make the convention that ‘‘if A, then B’’ is false when and only when A is true and B is false. Thus, sentences 1–3 are assumed to be true. Let us denote ‘‘if A, then B’’ by ‘‘A ) B.’’ An expression ‘‘A ) B’’ is called a conditional. Then ) has the following truth table: A B A ) B T T T F T T T F F F F T This sharpening of the meaning of ‘‘if A, then B’’ involves no conflict with ordinary usage, but rather only an extension of that usage.* * There is a common non-truth-functional interpretation of ‘‘if A, then B’’ connected with causal laws. The sentence ‘‘if this piece of iron is placed in water at time t, then the iron will dissolve’’ is regarded as false even in the case that the piece of iron is not placed in water at time t—that is, even when the antecedent is false. Another non-truth-functional usage occurs in so-called counterfactual conditionals, such as ‘‘if Sir Walter Scott had not written any novels, then there would have been no War Between the States.’’ (This was Mark Twain’s contention in Life on the Mississippi: ‘‘Sir Walter had so large a hand in making Southern character, as it existed before the war, that he is in great measure responsible for the war.’’) This sentence might be asserted to be false even though the antecedent is admittedly false. However, causal laws and counter- factual conditions seem not to be needed in mathematics and logic. For a clear treatment of conditionals and other connectives, see Quine (1951). (The quotation from Life on the Mississippi was brought to my attention by Professor J.C. Owings, Jr.) 2 Introduction to Mathematical Logic
- 32. A justification of the truth table for ) is the fact that we wish ‘‘if A and B, then B’’ to be true in all cases. Thus, the case in which A and B are true justifies the first line of our truth table for ), since (A and B) and B are both true. If A is false and B true, then (A and B) is false while B is true. This corresponds to the second line of the truth table. Finally, if A is false and B is false, (A and B) is false and B is false. This gives the fourth line of the table. Still more support for our definition comes from the meaning of statements such as ‘‘for every x, if x is an odd positive integer, then x2 is an odd positive integer.’’ This asserts that, for every x, the statement ‘‘if x is an odd positive integer, then x2 is an odd positive integer’’ is true. Now we certainly do not want to consider cases in which x is not an odd positive integer as counterexamples to our general assertion. This supports the second and fourth lines of our truth table. In addition, any case in which x is an odd positive integer and x2 is an odd positive integer confirms our general assertion. This corresponds to the first line of the table. Let us denote ‘‘A if and only if B’’ by ‘‘A , B.’’ Such an expression is called a biconditional. Clearly, A , B is true when and only when A and B have the same truth value. Its truth table, therefore is: A B A , B T T T F T F T F F F F T The symbols :, ^ , _, ), and , will be called propositional connectives.* Any sentence built up by application of these connectives has a truth value that depends on the truth values of the constituent sentences. In order to make this dependence apparent, let us apply the name statement form to an expres- sion built up from the statement letters A, B, C, and so on by appropriate applications of the propositional connectives. 1. All statement letters (capital italic letters) and such letters with numerical subscriptsy are statement forms. 2. If b and c are statement forms, then so are (:b), (b ^ c), (b _ c), (b ) c), and (b , c). 3. Only those expressions are statement forms that are determined to be so by means of conditions 1 and 2.z * We have been avoiding and shall in the future avoid the use of quotation marks to form names whenever this is not likely to cause confusion. The given sentence should have quotation marks around each of the connectives. See Quine (1951, pp. 23–27). y For example, A1, A2, A17, B31, C2, . . .. z This can be rephrased as follows: c is a statement form if and only if there is a finite sequence b1, . . . , bn (n 1) such that bn ¼ c and, if 1 i n, bi is either a statement letter or a negation, conjunction, disjunction, conditional, or biconditional constructed from previous expressions in the sequence. Notice that we use script letters a, b, c, . . . to stand for arbitrary expressions, whereas italic letters are used as statement letters. The Propositional Calculus 3
- 33. Some examples of statement forms are B, :C2 ð Þ, D3 ^ (:B) ð Þ, :B1 ð Þ _ B2 ð Þ ) ð A1 ^ C2 ð ÞÞ, and (((:A) , A) , (C ) (B _ C))). For every assignment of truth values T or F to the statement letters that occur in a statement form, there corresponds, by virtue of the truth tables for the propositional connectives, a truth value for the statement form. Thus, each statement form determines a truth function, which can be graphically represented by a truth table for the statement form. For example, the state- ment form (((:A) _ B) ) C) has the following truth table: A B C (:A) ((:A) _ B) (((:A) _ B) ) C) T T T F T T F T T T T T T F T F F T F F T T T T T T F F T F F T F T T F T F F F F T F F F T T F Each row represents an assignment of truth values to the statement letters A, B, and C and the corresponding truth values assumed by the statement forms that appear in the construction of (((:A) _ B) ) C). The truth table for ((A , B) ) ((:A) ^ B)) is as follows: A B (A , B) (:A) ((:A) ^ B) ((A , B) ) ((:A) ^ B)) T T T F F F F T F T T T T F F F F T F F T T F F If there are n distinct letters in a statement form, then there are 2n possible assignments of truth values to the statement letters and, hence, 2n rows in the truth table. A truth table can be abbreviated by writing only the full statement form, putting the truth values of the statement letters underneath all occurrences of these letters, and writing, step by step, the truth values of each component statement form under the principal connective of the form.* As an example, for ((A , B) ) ((:A) ^ B)), we obtain ((A , B) ) ((:A) ^ B)) T T T F FT F T F F T T TF T T T F F T FT F F F T F F TF F F * The principal connective of a statement form is the one that is applied last in constructing the form. 4 Introduction to Mathematical Logic
- 34. Exercises 1.1 Let designate the exclusive use of ‘‘or.’’ Thus, A B stands for ‘‘A or B but not both.’’ Write the truth table for . 1.2 Construct truth tables for the statement forms ((A ) B) _ (:A)) and ((A ) (B ) C)) ) ((A ) B) ) (A ) C))). 1.3 Write abbreviated truth tables for ((A ) B) ^ A) and ((A _ (:C)) , B). 1.4 Write the following sentences as statement forms, using statement letters to stand for the atomic sentences—that is, those sentences that are not built up out of other sentences. (a) If Mr Jones is happy, Mrs Jones is not happy, and if Mr Jones is not happy, Mrs Jones is not happy. (b) Either Sam will come to the party and Max will not, or Sam will not come to the party and Max will enjoy himself. (c) A sufficient condition for x to be odd is that x is prime. (d) A necessary condition for a sequence s to converge is that s be bounded. (e) A necessary and sufficient condition for the sheikh to be happy is that he has wine, women, and song. (f) Fiorello goes to the movies only if a comedy is playing. (g) The bribe will be paid if and only if the goods are delivered. (h) If x is positive, x2 is positive. (i) Karpov will win the chess tournament unless Kasparov wins today. 1.2 Tautologies A truth function of n arguments is defined to be a function of n arguments, the arguments and values of which are the truth values T or F. As we have seen, any statement form containing n distinct statement letters determines a corresponding truth function of n arguments.* * To be precise, enumerate all statement letters as follows: A, B, . . . , Z; A1, B1, . . . , Z1; A2, . . . ,. If a statement form contains the i1th , . . . , inth statement letters in this enumeration, where i1 in, then the corresponding truth function is to have xi1 , . . . , xin , in that order, as its arguments, where xij corresponds to the ijth statement letter. For example, (A ) B) generates the truth function: x1 x2 f x1, x2 ð Þ T T T F T T T F F F F T whereas (B ) A) generates the truth function: x1 x2 g x1, x2 ð Þ T T T F T F T F T F F T The Propositional Calculus 5
- 35. A statement form that is always true, no matter what the truth values of its statement letters may be, is called a tautology. A statement form is a tautology if and only if its corresponding truth function takes only the value T, or equivalently, if, in its truth table, the column under the statement form contains only Ts. An example of a tautology is (A _ (:A)), the so-called law of the excluded middle. Other simple examples are (:(A ^ (:A))), (A , (:(:A))), ((A ^ B) ) A), and (A ) (A _ B)). b is said to logically imply c (or, synonymously, c is a logical consequence of b) if and only if every truth assignment to the statement letters of b and c that makes b true also makes c true. For example, (A ^ B) logically implies A, A logically implies (A _ B), and (A ^ (A ) B)) logically implies B. b and c are said to be logically equivalent if and only if b and c receive the same truth value under every assignment of truth values to the statement letters of b and c. For example, A and (:(:A)) are logically equivalent, as are (A ^ B) and (B ^ A). PROPOSITION 1.1 (a) b logically implies c if and only if (b ) c) is a tautology. (b) b and c are logically equivalent if and only if (b , c) is a tautology. Proof (a) (i) Assume b logically implies c. Hence, every truth assignment that makes b true also makes c true. Thus, no truth assignment makes b true and c false. Therefore, no truth assignment makes (b ) c) false, that is, every truth assignment makes (b ) c) true. In other words, (b ) c) is a tautology. (ii) Assume (b ) c) is a tautology. Then, for every truth assignment, (b ) c) is true, and, therefore, it is not the case that b is true and c false. Hence, every truth assignment that makes b true makes c true, that is, b logically implies c. (b) (b , c) is a tautology if and only if every truth assignment makes (b , c) true, which is equivalent to saying that every truth assign- ment gives b and c the same truth value, that is, b and c are logically equivalent. By means of a truth table, we have an effective procedure for determining whether a statement form is a tautology. Hence, by Proposition 1.1, we have effective procedures for determining whether a given statement form logic- ally implies another given statement form and whether two given statement forms are logically equivalent. To see whether a statement form is a tautology, there is another method that is often shorter than the construction of a truth table. 6 Introduction to Mathematical Logic
- 36. Examples 1. Determine whether ((A , ((:B) _ C)) ) ((:A) ) B)) is a tautology. Assume that the statement form sometimes is F (line 1). Then (A , ((:B) _ C)) is T and ((:A) ) B) is F (line 2). Since ((:A) ) B) is F, (:A) is T and B is F (line 3). Since (:A) is T, A is F (line 4). Since A is F and (A , ((:B) _ C)) is T, ((:B) _ C) is F (line 5). Since ((:B) _ C) is F, (:B) and C are F (line 6). Since (:B) is F, B is T (line 7). But B is both T and F (lines 7 and 3). Hence, it is impossible for the form to be false. ((A , ((:B) _ C)) ) ((:A) ) B)) F 1 T F 2 T F 3 F F 4 F 5 F F 6 T 7 2. Determine whether ((A ) (B _ C)) _ (A ) B)) is a tautology. Assume that the form is F (line 1). Then (A ) (B _ C)) and (A ) B) are F (line 2). Since (A ) B) is F, A is T and B is F (line 3). Since (A ) (B _ C)) is F, A is T and (B _ C) is F (line 4). Since (B _ C) is F, B and C are F (line 5). Thus, when A is T, B is F, and C is F, the form is F. Therefore, it is not a tautology. ((A ) (B _ C)) _ (A ) B)) F 1 F F 2 T F 3 T F 4 F F 5 Exercises 1.5 Determine whether the following are tautologies. (a) (((A ) B) ) B) ) B) (b) (((A ) B) ) B) ) A) (c) (((A ) B) ) A) ) A) (d) (((B ) C) ) (A ) B)) ) (A ) B)) (e) ((A _ (:(B ^ C))) ) ((A , C) _ B)) (f) (A ) (B ) (B ) A))) (g) ((A ^ B) ) (A _ C)) (h) ((A , B) , (A , (B , A))) (i) ((A ) B) _ (B ) A)) (j) ((:(A ) B)) ) A) The Propositional Calculus 7
- 37. 1.6 Determine whether the following pairs are logically equivalent. (a) ((A ) B) ) A) and A (b) (A , B) and ((A ) B) ^ (B ) A)) (c) ((:A) _ B) and ((:B) _ A) (d) (:(A , B)) and (A , (:B)) (e) (A _ (B , C)) and ((A _ B) , (A _ C)) (f) (A ) (B , C)) and ((A ) B) , (A ) C)) (g) (A ^ (B , C)) and ((A ^ B) , (A ^ C)) 1.7 Prove: (a) (A ) B) is logically equivalent to ((:A) _ B). (b) (A ) B) is logically equivalent to (:(A ^ (:B))). 1.8 Prove that b is logically equivalent to c if and only if b logically implies c and c logically implies b. 1.9 Show that b and c are logically equivalent if and only if, in their truth tables, the columns under b and c are the same. 1.10 Prove that b and c are logically equivalent if and only if (:b) and (:c) are logically equivalent. 1.11 Which of the following statement forms are logically implied by (A ^ B)? (a) A (b) B (c) (A _ B) (d) ((:A) _ B) (e) ((:B) ) A) (f) (A , B) (g) (A ) B) (h) ((:B) ) (:A)) (i) (A ^ (:B)) 1.12 Repeat Exercise 1.11 with (A ^ B) replaced by (A ) B) and by (:(A ) B)), respectively. 1.13 Repeat Exercise 1.11 with (A ^ B) replaced by (A _ B). 1.14 Repeat Exercise 1.11 with (A ^ B) replaced by (A , B) and by (:(A , B)), respectively. A statement form that is false for all possible truth values of its statement letters is said to be contradictory. Its truth table has only Fs in the column under the statement form. One example is (A , (:A)): A (:A) (A , (:A)) T F F F T F Another is (A ^ (:A)). Notice that a statement form b is a tautology if and only if (:b) is contradictory, and vice versa. 8 Introduction to Mathematical Logic
- 38. A sentence (in some natural language like English or in a formal theory)* that arises from a tautology by the substitution of sentences for all the statement letters, with occurrences of the same statement letter being replaced by the same sentence, is said to be logically true (according to the propositional calculus). Such a sentence may be said to be true by virtue of its truth-functional structure alone. An example is the English sentence, ‘‘If it is raining or it is snowing, and it is not snowing, then it is raining,’’ which arises by substitution from the tautology (((A _ B) ^ (:B)) ) A). A sentence that comes from a contradictory statement form by means of substitution is said to be logically false (according to the propositional calculus). Now let us prove a few general facts about tautologies. PROPOSITION 1.2 If b and (b ) c) are tautologies, then so is c. Proof Assume that b and (b ) c) are tautologies. If c took the value F for some assignment of truth values to the statement letters of b and c, then, since b is a tautology, b would take the value T and, therefore, (b ) c) would have the value F for that assignment. This contradicts the assumption that (b ) c) is a tautology. Hence, c never takes the value F. PROPOSITION 1.3 If t is a tautology containing as statement letters A1, A2, . . . , An, and b arises from t by substituting statement forms s1, s2, . . . , sn for A1, A2, . . . , An, respectively, then b is a tautology; that is, substitution in a tautology yields a tautology. Example Let t be ((A1 ^ A2) ) A1), let s1 be (B _ C) and let s2 be (C ^ D). Then b is (((B _ C) ^ (C ^ D)) ) (B _ C)). Proof Assume that t is a tautology. For any assignment of truth values to the statement letters in b, the forms s1, . . . , sn have truth values x1, . . . , xn (where each xn is T or F). If we assign the values x1, . . . , xn to A1, . . . , An, * By a formal theory we mean an artificial language in which the notions of meaningful expres- sions, axioms, and rules of inference are precisely described (see page 25). The Propositional Calculus 9
- 39. respectively, then the resulting truth value of t is the truth value of b for the given assignment of truth values. Since t is a tautology, this truth value must be T. Thus, b always takes the value T. PROPOSITION 1.4 If c1 arises from b1 by substitution of c for one or more occurrences of b, then ((b , c) ) (b1 , c1)) is a tautology. Hence, if b and c are logically equivalent, then so are b1 and c1. Example Let b1 be (c _ D), let b be c, and let c be (:(:c)). Then c1 is ((:(:c)) _ D). Since c and (:(:c)) are logically equivalent, (c _ D) and ((:(:c)) _ D) are also logically equivalent. Proof Consider any assignment of truth values to the statement letters. If b and c have opposite truth values under this assignment, then (b , c) takes the value F, and, hence, ((b , c) ) (b1 , c1)) is T. If b and c take the same truth values, then so do b1 and c1, since c1 differs from b1 only in contain- ing c in some places where b1 contains b. Therefore, in this case, (b , c) is T, (b1 , c1) is T, and, thus, ((b , c) ) (b1 , c1)) is T. Parentheses It is profitable at this point to agree on some conventions to avoid the use of so many parentheses in writing formulas. This will make the reading of complicated expressions easier. First, we may omit the outer pair of parentheses of a statement form. (In the case of statement letters, there is no outer pair of parentheses.) Second, we arbitrarily establish the following decreasing order of strength of the connectives: :, ^, _, ), ,. Now we shall explain a step-by-step process for restoring parentheses to an expression obtained by eliminating some or all parentheses from a statement form. (The basic idea is that, where possible, we first apply parentheses to negations, then to conjunctions, then to disjunctions, then to conditionals, and finally to biconditionals.) Find the leftmost occurrence of the strongest connective that has not yet been processed. (i) If the connective is : and it precedes a statement form b, restore left and right parentheses to obtain (:b). (ii) If the connective is a binary connective C and it is preceded by a statement form b and followed by a statement form d, restore left and right parentheses to obtain (b C d). 10 Introduction to Mathematical Logic
- 40. (iii) If neither (i) nor (ii) holds, ignore the connective temporarily and find the leftmost occurrence of the strongest of the remaining unpro- cessed connectives and repeat (i)–(iii) for that connective. Examples Parentheses are restored to the expression in the first line of each of the following in the steps shown: 1. A , (:B) _ C ) A A , ((:B) _ C) ) A A , (((:B) _ C) ) A) (A , (((:B) _ C) ) A)) 2. A ) :B ) C A ) (:B) ) C (A ) (:B)) ) C ((A ) (:B)) ) C) 3. B ) ::A B ) :(:A) B ) (:(:A)) (B ) (:(:A))) 4. A _ :(B ) A _ B) A _ :(B ) (A _ B)) A _ (:(B ) (A _ B))) (A _ (:(B ) (A _ B)))) Not every form can be represented without the use of parentheses. For example, parentheses cannot be further eliminated from A ) (B ) C), since A ) B ) C stands for ((A ) B) ) C). Likewise, the remaining parentheses cannot be removed from :(A _ B) or from A ^ (B ) C). Exercises 1.15 Eliminate as many parentheses as possible from the following forms. (a) ((B ) (:A)) ^ C) (b) (A _ (B _ C)) (c) (((A ^ (:B)) ^ C) _ D) (d) ((B _ (:C)) _ (A ^ B)) (e) ((A , B) , (:(C _ D))) (f) ((:(:(:(B _ C)))) , (B , C)) (g) (:((:(:(B _ C))) , (B , C))) (h) ((((A ) B) ) (C ) D)) ^ (:A)) _ C) 1.16 Restore parentheses to the following forms. (a) C _ :A ^ B (b) B ) :::A ^ C The Propositional Calculus 11
- 41. (c) C ) :(A ^ B ) C) ^ A , B (d) C ) A ) A , :A _ B 1.17 Determine whether the following expressions are abbreviations of state- ment forms and, if so, restore all parentheses. (a) ::A , A , B _ C (b) :(:A , A) , B _ C (c) :(A ) B) _ C _ D ) B (d) A , (:A _ B) ) (A ^ (B _ C))) (e) :A _ B _ C ^ D , A ^ :A (f) ((A ) B ^ (C _ D) ^ (A _ D)) 1.18 If we write :b instead of (:b), )bc instead of (b ) c), ^bc instead of (b ^ c), _bc instead of (b _ c), and ,bc instead of (b , c), then there is no need for parentheses. For example, ((:A) ^ (B ) (:D))), which is ordinarily abbreviated as :A ^ (B ) :D), becomes ^:A ) B:D. This way of writing forms is called Polish notation. (a) Write ((C ) (:A)) _ B) and (C _ ((B ^ (:D)) ) C)) in this notation. (b) If we count ), ^, _, and , each as þ1, each statement letter as 1 and : as 0, prove that an expression b in this parenthesis-free notation is a statement form if and only if (i) the sum of the symbols of b is 1 and (ii) the sum of the symbols in any proper initial segment of b is nonnegative. (If an expression b can be written in the form cd, where c 6¼ b, then c is called a proper initial segment of b.) (c) Write the statement forms of Exercise 1.15 in Polish notation. (d) Determine whether the following expressions are statement forms in Polish notation. If so, write the statement forms in the standard way. (i) : ) ABC _ AB:C (ii) )) AB )) BC ) :AC (iii) _ ^ _:A:BC ^ _AC _ :C:A (iv) _ ^ B ^ BBB 1.19 Determine whether each of the following is a tautology, is contra- dictory, or neither. (a) B , (B _ B) (b) ((A ) B) ^ B) ) A (c) (:A) ) (A ^ B) (d) (A ) B) ) ((B ) C) ) (A ) C)) (e) (A , :B) ) A _ B (f) A ^ (:(A _ B)) (g) (A ) B) , ((:A) _ B) (h) (A ) B) , :(A ^ (:B)) (i) (B , (B , A)) ) A (j) A ^ :A ) B 1.20 If A and B are true and C is false, what are the truth values of the following statement forms? 12 Introduction to Mathematical Logic
- 42. (a) A _ C (b) A ^ C (c) :A ^ :C (d) A , :B _ C (e) B _ :C ) A (f) (B _ A) ) (B ) :C) (g) (B ) :A) , (A , C) (h) (B ) A) ) ((A ) :C) ) (:C ) B)) 1.21 If A ) B is T, what can be deduced about the truth values of the following? (a) A _ C ) B _ C (b) A ^ C ) B ^ C (c) :A ^ B , A _ B 1.22 What further truth values can be deduced from those shown? (a) :A _ (A ) B) F (b) :(A ^ B) , :A ) :B T (c) (:A _ B) ) (A ) :C) F (d) (A , B) , (C ) :A) F T 1.23 If A , B is F, what can be deduced about the truth values of the following? (a) A ^ B (b) A _ B (c) A ) B (d) A ^ C , B ^ C 1.24 Repeat Exercise 1.23, but assume that A , B is T. 1.25 What further truth values can be deduced from those given? (a) (A ^ B) , (A _ B) F F (b) (A ) :B) ) (C ) B) F 1.26 (a) Apply Proposition 1.3 when t is A1 ) A1 _ A2, s1 is B ^ D, and s2 is :B. (b) Apply Proposition 1.4 when b1 is (B ) C) ^ D, b is B ) C, and c is :B _ C. 1.27 Show that each statement form in column I is logically equivalent to the form next to it in column II. I II (a) A ) (B ) C) (A ^ B) ) C (b) A ^ (B _ C) (A ^ B) _ (A ^ C) (Distributive law) (c) A _ (B ^ C) (A _ B) ^ (A _ C) (Distributive law) The Propositional Calculus 13
- 43. (d) (A ^ B) _ :B A _ :B (e) (A _ B) ^ :B A ^ :B (f) A ) B :B ) :A (Law of the contrapositive) (g) A , B B , A (Biconditional commutativity) (h) (A , B) , C A , (B , C) (Biconditional associativity) (i) A , B (A ^ B) _ (:A ^ :B) (j) :(A , B) A , :B (k) :(A _ B) (:A) ^ (:B) (De Morgan’s law) (l) :(A ^ B) (:A) _ (:B) (De Morgan’s law) (m) A _ (A ^ B) A (n) A ^ (A _ B) A (o) A ^ B B ^ A (Commutativity of conjunction) (p) A _ B B _ A (Commutativity of disjunction) (q) (A ^ B) ^ C A ^ (B ^ C) (Associativity of conjunction) (r) (A _ B) _ C A _ (B _ C) (Associativity of disjunction) (s) A B B A (Commutativity of exclusive ‘‘or’’) (t) (A B) C A (B C) (Associativity of exclusive ‘‘or’’) (u) A ^ (B C) (A ^ B) (A ^ C) (Distributive law) 1.28 Show the logical equivalence of the following pairs. (a) t ^ b and b, where t is a tautology. (b) t _ b and t, where t is a tautology. (c) f ^ b and f, where f is contradictory. (d) f _ b and b, where f is contradictory. 1.29 (a) Show the logical equivalence of :(A ) B) and A ^ :B. (b) Show the logical equivalence of :(A , B) and (A ^ :B) _ (:A ^ B). (c) For each of the following statement forms, find a statement form that is logically equivalent to its negation and in which negation signs apply only to statement letters. (i) A ) (B , :C) (ii) :A _ (B ) C) (iii) A ^ (B _ :C) 1.30 (Duality) (a) If b is a statement form involving only :, ^, and _, and b0 results from b by replacing each ^ by _ and each _ by ^, show that b is a tautology if and only if :b0 is a tautology. Then prove that, if b ) c is a tautology, then so is c0 ) b0 , and if b , c is a tautology, then so is b0 , c0 . (Here c is also assumed to involve only :, ^, and _.) (b) Among the logical equivalences in Exercise 1.27, derive (c) from (b), (e) from (d), (l) from (k), (p) from (o), and (r) from (q). (c) If b is a statement form involving only :, ^, and _, and b* results from b by interchanging ^ and _ and replacing every statement letter by its negation, show that b* is logically equivalent to :b. 14 Introduction to Mathematical Logic
- 44. Find a statement form that is logically equivalent to the negation of (A _ B _ C) ^ (:A _ :B _ D), in which : applies only to statement letters. 1.31 (a) Prove that a statement form that contains , as its only connective is a tautology if and only if each statement letter occurs an even number of times. (b) Prove that a statement form that contains : and , as its only connectives is a tautology if and only if : and each statement letter occur an even number of times. 1.32 (Shannon, 1938) An electric circuit containing only on–off switches (when a switch is on, it passes current; otherwise it does not) can be represented by a diagram in which, next to each switch, we put a letter representing a necessary and sufficient condition for the switch to be on (see Figure 1.1). The condition that a current flows through this network can be given by the statement form (A ^ B) _ (C ^ :A). A statement form representing the circuit shown in Figure 1.2 is (A ^ B) _ ((C _ A) ^ :B), which is logically equivalent to each of the following forms by virtue of the indicated logical equivalence of Exercise 1.27. ((A ^ B) _ (C _ A)) ^ ((A ^ B) _ :B) (c) ((A ^ B) _ (C _ A)) ^ (A _ :B) (d) ((A ^ B) _ (A _ C)) ^ (A _ :B) (p) (((A ^ B) _ A) _ C) ^ (A _ :B) (r) (A _ C) ^ (A _ :B) (p), (m) A _ (C ^ :B) (c) Hence, the given circuit is equivalent to the simpler circuit shown in Figure 1.3. (Two circuits are said to be equivalent if current flows through A B C A FIGURE 1.1 A A B C B FIGURE 1.2 The Propositional Calculus 15
- 45. one if and only if it flows through the other, and one circuit is simpler if it contains fewer switches.) (a) Find simpler equivalent circuits for those shown in Figures 1.4 through 1.6. (b) Assume that each of the three members of a committee votes yes on a proposal by pressing a button. Devise as simple a circuit as you can that will allow current to pass when and only when at least two of the members vote in the affirmative. (c) We wish a light to be controlled by two different wall switches in a room in such a way that flicking either one of these switches will turn the light on if it is off and turn it off if it is on. Construct a simple circuit to do the required job. 1.33 Determine whether the following arguments are logically correct by representing each sentence as a statement form and checking whether FIGURE 1.3 A B C FIGURE 1.4 C A A C B C B C FIGURE 1.5 A C C C B B B A FIGURE 1.6 A C C D D B A A B 16 Introduction to Mathematical Logic
- 46. the conclusion is logically implied by the conjunction of the assump- tions. (To do this, assign T to each assumption and F to the conclusion, and determine whether a contradiction results.) (a) If Jones is a communist, Jones is an atheist. Jones is an atheist. Therefore, Jones is a communist. (b) If the temperature and air pressure remained constant, there was no rain. The temperature did remain constant. Therefore, if there was rain, then the air pressure did not remain constant. (c) If Gorton wins the election, then taxes will increase if the deficit will remain high. If Gorton wins the election, the deficit will remain high. Therefore, if Gorton wins the election, taxes will increase. (d) If the number x ends in 0, it is divisible by 5. x does not end in 0. Hence, x is not divisible by 5. (e) If the number x ends in 0, it is divisible by 5. x is not divisible by 5. Hence, x does not end in 0. (f) If a ¼ 0 or b ¼ 0, then ab ¼ 0. But ab 6¼ 0. Hence, a 6¼ 0 and b 6¼ 0. (g) A sufficient condition for f to be integrable is that g be bounded. A necessary condition for h to be continuous is that f is integrable. Hence, if g is bounded or h is continuous, then f is integrable. (h) Smith cannot both be a running star and smoke cigarettes. Smith is not a running star. Therefore, Smith smokes cigarettes. (i) If Jones drove the car, Smith is innocent. If Brown fired the gun, then Smith is not innocent. Hence, if Brown fired the gun, then Jones did not drive the car. 1.34 Which of the following sets of statement forms are satisfiable, in the sense that there is an assignment of truth values to the statement letters that makes all the forms in the set true? (a) A ) B B ) C C _ D , :B (b) :(:B _ A) A _ :C B ) :C (c) D ) B A _ :B :(D ^ A) D 1.35 Check each of the following sets of statements for consistency by repre- senting the sentences as statement forms and then testing their conjunc- tion to see whether it is contradictory. (a) Either the witness was intimidated or, if Doherty committed suicide, a note was found. If the witness was intimidated, then Doherty did not commit suicide. If a note was found, then Doherty committed suicide. The Propositional Calculus 17
- 47. (b) The contract is satisfied if and only if the building is completed by 30 November. The building is completed by 30 November if and only if the electrical subcontractor completes his work by 10 November. The bank loses money if and only if the contract is not satisfied. Yet the electrical subcontractor completes his work by 10 November if and only if the bank loses money. 1.3 Adequate Sets of Connectives Every statement form containing n statement letters generates a correspond- ing truth function of n arguments. The arguments and values of the function are T or F. Logically equivalent forms generate the same truth function. A natural question is whether all truth functions are so generated. PROPOSITION 1.5 Every truth function is generated by a statement form involving the connect- ives :, ^, and _. Proof (Refer to Examples 1 and 2 below for clarification.) Let f(x1, . . . , xn) be a truth function. Clearly f can be represented by a truth table of 2n rows, where each row represents some assignment of truth values to the variables x1, . . . , xn, followed by the corresponding value of f(x1, . . . , xn). If 1 i 2n , let Ci be the conjunction Ui 1 ^ Ui 2 ^ . . . ^ Ui n, where Ui j is Aj if, in the ith row of the truth table, xj takes the value T, and Ui j is :Aj if xj takes the value F in that row. Let D be the disjunction of all those Cis such that f has the value T for the ith row of the truth table. (If there are no such rows, then f always takes the value F, and we let D be A1 ^ :A1, which satisfies the theorem.) Notice that D involves only :, ^, and _. To see that D has f as its corresponding truth function, let there be given an assignment of truth values to the statement letters A1, . . . , An, and assume that the correspond- ing assignment to the variables x1, . . . , xn is row k of the truth table for f. Then Ck has the value T for this assignment, whereas every other Ci has the value F. If f has the value T for row k, then Ck is a disjunct of D. Hence, D would also have the value T for this assignment. If f has the value F for row k, then Ck is not a disjunct of D and all the disjuncts take the value F for this assignment. Therefore, D would also have the value F. Thus, D generates the truth function f. 18 Introduction to Mathematical Logic
- 48. Examples 1. x1 x2 f (x1, x2) T T F F T T T F T F F T D is (:A1 ^ A2) _ (A1 ^ :A2) _ (:A1 ^ :A2). 2. x1 x2 x3 g(x1, x2, x3) T T T T F T T F T F T T F F T T T T F F F T F F T F F F F F F T D is (A1 ^ A2 ^ A3) _ (A1 ^ :A2 ^ A3) _ (:A1 ^ :A2 ^ A3) _ (:A1 ^ :A2 ^ :A3): Exercise 1.36 Find statement forms in the connectives :, ^, and _ that have the following truth functions. x1 x2 x3 f(x1, x2, x3) g(x1, x2, x3) h(x1, x2, x3) T T T T T F F T T T T T T F T T T F F F T F F F T T F F T T F T F F F T T F F F T F F F F T F T COROLLARY 1.6 Every truth function can be generated by a statement form containing as connectives only ^ and :, or only _ and :, or only ) and :. The Propositional Calculus 19
- 49. Proof Notice that b _ c is logically equivalent to :(:b ^ :c). Hence, by the second part of Proposition 1.4, any statement form in ^, _, and : is logically equivalent to a statement form in only ^ and : [obtained by replacing all expressions b _ c by :(:b ^ :c)]. The other parts of the corollary are similar consequences of the following tautologies: b ^ c , :(:b _ :c) b _ c , (:b ) c) b ^ c , :(b ) :c) We have just seen that there are certain pairs of connectives — for example, ^ and : — in terms of which all truth functions are definable. It turns out that there is a single connective, # (joint denial), that will do the same job. Its truth table is A B A # B T T F F T F T F F F F T A # B is true when and only when neither A nor B is true. Clearly, :A , (A # A) and (A ^ B) , ((A # A) # (B # B)) are tautologies. Hence, the adequacy of # for the construction of all truth functions follows from Corollary 1.6. Another connective, j (alternative denial), is also adequate for this purpose. Its truth table is A B AjB T T F F T T T F T F F T AjB is true when and only when not both A and B are true. The adequacy of j follows from the tautologies :A , (AjA) and (A _ B) , ((AjA)j(BjB)). PROPOSITION 1.7 The only binary connectives that alone are adequate for the construction of all truth functions are # and j. 20 Introduction to Mathematical Logic
- 50. Proof Assume that h(A, B) is an adequate connective. Now, if h(T, T) were T, then any statement form built up using h alone would take the value T when all its statement letters take the value T. Hence, :A would not be definable in terms of h. So, h(T, T) ¼ F. Likewise, h(F, F) ¼ T. Thus, we have the partial truth table: A B h(A, B) T T F F T T F F F T If the second and third entries in the last column are F, F or T, T, then h is # or j. If they are F, T, then h(A, B) , :B is a tautology; and if they are T, F, then h(A, B) , :A is a tautology. In both cases, h would be definable in terms of :. But : is not adequate by itself because the only truth functions of one variable definable from it are the identity function and negation itself, whereas the truth function that is always T would not be definable. Exercises 1.37 Prove that each of the pairs ), _ and :, , is not alone adequate to express all truth functions. 1.38 (a) Prove that A _ B can be expressed in terms of ) alone. (b) Prove that A ^ B cannot be expressed in terms of ) alone. (c) Prove that A , B cannot be expressed in terms of ) alone. 1.39 Show that any two of the connectives {^, ), ,} serve to define the remaining one. 1.40 With one variable A, there are four truth functions: A :A A _ :A A ^ :A T F T F F T T F (a) With two variable A and B, how many truth functions are there? (b) How many truth functions of n variables are there? 1.41 Show that the truth function h determined by (A _ B) ) :C generates all truth functions. 1.42 By a literal we mean a statement letter or a negation of a statement letter. A statement form is said to be in disjunctive normal form (dnf) if it is a disjunction consisting of one or more disjuncts, each of which is a con- junction of one or more literals—for example, (A ^ B) _ (:A ^ C), (A ^ B ^ :A) _ (C ^ :B) _ (A ^ :C), A, A ^ B, and A _ (B _ C). A form is in conjunctive normal form (cnf) if it is a conjunction of one or more con- juncts, each of which is a disjunction of one or more literals—for example, The Propositional Calculus 21
- 51. (B _ C) ^ (A _ B), (B _ :C) ^ (A _ D), A ^ (B _ A) ^ (:B _ A), A _ :B, A ^ B, A. Note that our terminology considers a literal to be a (degenerate) conjunction and a (degenerate) disjunction. (a) The proof of Proposition 1.5 shows that every statement form b is logically equivalent to one in disjunctive normal form. By applying this result to :b, prove that b is also logically equivalent to a form in conjunctive normal form. (b) Find logically equivalent dnfs and cnfs for :(A ) B) _ (:A ^ C) and A , ((B ^ :A) _ C). [Hint: Instead of relying on Proposition 1.5, it is usually easier to use Exercise 1.27(b) and (c).] (c) A dnf (cnf) is called full if no disjunct (conjunct) contains two occurrences of literals with the same letter and if a letter that occurs in one disjunct (conjunct) also occurs in all the others. For example, (A ^ :A ^ B) _ (A ^ B), (B ^ B ^ C) _ (B ^ C) and (B ^ C) _ B are not full, whereas (A ^ B ^ :C) _ (A ^ B ^ C) _ (A ^ :B ^ :C) and (A ^ :B) _ (B ^ A) are full dnfs. (i) Find full dnfs and cnfs logically equivalent to (A ^ B) _ :A and :(A ) B) _ (:A ^ C). (ii) Prove that every noncontradictory (nontautologous) statement form b is logically equivalent to a full dnf (cnf) c, and, if c contains exactly n letters, then b is a tautology (is contradictory) if and only if c has 2n disjuncts (conjuncts). (d) For each of the following, find a logically equivalent dnf (cnf), and then find a logically equivalent full dnf (cnf): (i) (A _ B) ^ (:B _ C) (ii) :A _ (B ) :C) (iii) (A ^ :B) _ (A ^ C) (iv) (A _ B) , :C (e) Construct statement forms in : and ^ (respectively, in : and _ or in : and )) logically equivalent to the statement forms in (d). 1.43 A statement form is said to be satisfiable if it is true for some assignment of truth values to its statement letters. The problem of determining the satisfiability of an arbitrary cnf plays an important role in the theory of computational complexity; it is an example of a so-called np-complete problem (see Garey and Johnson, 1978). (a) Show that b is satisfiable if and only if :b is not a tautology. (b) Determine whether the following are satisfiable: (i) (A _ B) ^ (:A _ B _ C) ^ (:A _ :B _ :C) (ii) ((A ) B) _ C) , (:B ^ (A _ C)) (c) Given a disjunction d of four or more literals: L1 _ L2 _ . . . _ Ln, let C1, . . . , Cn2 be statement letters that do not occur in d, and con- struct the cnf e: (L1 _ L2 _ C1) ^ (:C1 _ L3 _ C2) ^ (:C2 _ L4 _ C3) ^ . . . ^ (:Cn3 _ Ln1 _ Cn2) ^ (:Cn2 _ Ln _ :C1) 22 Introduction to Mathematical Logic
- 52. Show that any truth assignment satisfying d can be extended to a truth assignment satisfying e and, conversely, any truth assignment satisfying e is an extension of a truth assignment satisfying d. (This permits the reduction of the problem of satisfying cnfs to the corre- sponding problem for cnfs with each conjunct containing at most three literals.) (d) For a disjunction d of three literals L1 _ L2 _ L3, show that a form that has the properties of e in (c) cannot be constructed, with e a cnf in which each conjunct contains at most two literals (R. Cowen). 1.44 (Resolution) Let b be a cnf and let C be a statement letter. If C is a disjunct of a disjunction d1 in b and :C is a disjunct of another disjunction d2 in b, then a nonempty disjunction obtained by elimin- ating C from d1 and :C from d2 and forming the disjunction of the remaining literals (dropping repetitions) is said to be obtained from b by resolution on C. For example, if b is (A _ :C _ :B) ^ (:A _ D _ :B) ^ (C _ D _ A) the first and third conjuncts yield A _ :B _ D by resolution on C. In addition, the first and second conjuncts yield :C _ :B _ D by resolution on A, and the second and third conjuncts yield D _ :B _ C by resolution on A. If we conjoin to b any new disjunctions obtained by resolution on all variables, and if we apply the same procedure to the new cnf and keep on iterating this operation, the process must eventually stop, and the final result is denoted res(b). In the example, res(b) is (A _ :C _ :B) ^ (:A _ D _ :B) ^ (C _ D _ A) ^ (:C _ :B _ D) ^ (D _ :B _ C) ^ (A _ :B _ D) ^ (D _ :B) (Notice that we have not been careful about specifying the order in which conjuncts or disjuncts are written, since any two arrangements will be logically equivalent.) (a) Find res(b) when b is each of the following: (i) (A _ :B) ^ B (ii) (A _ B _ C) ^ (A _ :B _ C) (iii) (A _ C) ^ (:A _ B) ^ (A _ :C) ^ (:A _ :B) (b) Show that b logically implies res(b). (c) If b is a cnf, let bC be the cnf obtained from b by deleting those conjuncts that contain C or :C. Let rC(b) be the cnf that is the conjunction of bC and all those disjunctions obtained from b by resolution on C. For example, if b is the cnf in the example above, then rC(b) is (:A _ D _ :B) ^ (A _ :B _ D). Prove that, if rC(b) is satisfiable, then so is b (R. Cowen). (d) A cnf b is said to be a blatant contradiction if it contains some letter C and its negation :C as conjuncts. An example of a blatant The Propositional Calculus 23
- 53. contradiction is (A _ B) ^ B ^ (C _ D) ^ :B. Prove that if b is unsatis- fiable, then res(b) is a blatant contradiction. [Hint: Use induction on the number n of letters that occur in b. In the induction step, use (c).] (e) Prove that b is unsatisfiable if and only if res(b) is a blatant contra- diction. 1.45 Let b and d be statement forms such that b ) d is a tautology. (a) If b and d have no statement letters in common, show that either b is contradictory or d is a tautology. (b) (Craig’s interpolation theorem) If b and d have the statement letters B1, . . . , Bn in common, prove that there is a statement form c having B1, . . . , Bn as its only statement letters such that b ) c and c ) d are tautologies. (c) Solve the special case of (b) in which b is (B1 ) A) ^ (A ) B2) and d is (B1 ^ C) ) (B2 ^ C). 1.46 (a) A certain country is inhabited only by truth-tellers (people who always tell the truth) and liars (people who always lie). Moreover, the inhabitants will respond only to yes or no questions. A tourist comes to a fork in a road where one branch leads to the capital and the other does not. There is no sign indicating which branch to take, but there is a native standing at the fork. What yes or no question should the tourist ask in order to determine which branch to take? [Hint: Let A stand for ‘‘You are a truth-teller’’ and let B stand for ‘‘The left-hand branch leads to the capital.’’ Construct, by means of a suitable truth table, a statement form involving A and B such that the native’s answer to the question as to whether this statement form is true will be yes when and only when B is true.] (b) In a certain country, there are three kinds of people: workers (who always tell the truth), businessmen (who always lie), and students (who sometimes tell the truth and sometimes lie). At a fork in the road, one branch leads to the capital. A worker, a businessman and a student are standing at the side of the road but are not identifiable in any obvious way. By asking two yes or no questions, find out which fork leads to the capital (Each question may be addressed to any of the three.) More puzzles of this kind may be found in Smullyan (1978, Chapter 3; 1985, Chapters 2, 4–8). 1.4 An Axiom System for the Propositional Calculus Truth tables enable us to answer many of the significant questions concern- ing the truth-functional connectives, such as whether a given statement form is a tautology, is contradictory, or neither, and whether it logically implies or 24 Introduction to Mathematical Logic
- 54. is logically equivalent to some other given statement form. The more com- plex parts of logic we shall treat later cannot be handled by truth tables or by any other similar effective procedure. Consequently, another approach, by means of formal axiomatic theories, will have to be tried. Although, as we have seen, the propositional calculus surrenders completely to the truth table method, it will be instructive to illustrate the axiomatic method in this simple branch of logic. A formal theory s is defined when the following conditions are satisfied: 1. A countable set of symbols is given as the symbols of s.* A finite sequence of symbols of s is called an expression of s. 2. There is a subset of the set of expressions of s called the set of well- formed formulas (wfs) of s. There is usually an effective procedure to determine whether a given expression is a wf. 3. There is a set of wfs called the set of axioms of s. Most often, one can effectively decide whether a given wf is an axiom; in such a case, s is called an axiomatic theory. 4. There is a finite set R1, . . . , Rn of relations among wfs, called rules of inference. For each Ri, there is a unique positive integer j such that, for every set of j wfs and each wf b, one can effectively decide whether the given j wfs are in the relation Ri to b, and, if so, b is said to follow from or to be a direct consequence of the given wfs by virtue of Ri.y A proof in s is a sequence b1, . . . , bk of wfs such that, for each i, either bi is an axiom of s or bi is a direct consequence of some of the preceding wfs in the sequence by virtue of one of the rules of inference of s. A theorem of s is a wf b of s such that b is the last wf of some proof in s. Such a proof is called a proof of b in s. Even if s is axiomatic—that is, if there is an effective procedure for checking any given wf to see whether it is an axiom—the notion of ‘‘the- orem’’ is not necessarily effective since, in general, there is no effective procedure for determining, given any wf b, whether there is a proof of b. A theory for which there is such an effective procedure is said to be decidable; otherwise, the theory is said to be undecidable. From an intuitive standpoint, a decidable theory is one for which a machine can be devised to test wfs for theoremhood, whereas, for an undecidable theory, ingenuity is required to determine whether wfs are theorems. * These ‘‘symbols’’ may be thought of as arbitrary objects rather than just linguistic objects. This will become absolutely necessary when we deal with theories with uncountably many symbols in Section 2.12. y An example of a rule of inference will be the rule modus ponens (MP): c follows from b and b ) c. According to our precise definition, this rule is the relation consisting of all ordered triples hb, b ) c, ci, where b and c are arbitrary wfs of the formal system. The Propositional Calculus 25
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. were placed, side by side, that fifty could go up abreast. The blue Voltigeur flag, now full of holes, was planted on the parapet. A tide of brave Americans overflowed the fort. Resistance was vain. A little before half-past nine Bravo gave up his diamond-hilted sword, and the tricolor, that had been waving placidly amidst the uproar, came down with a jerk.[17] Fire was opened then upon the Mexicans at the gateway below, and fearless Captain Roberts of Casey’s storming party, at the head of all the troops on the causeway and supported by General Smith’s brigade, carried the gateway batteries. Many from Quitman’s and Smith’s commands rushed to the summit, dealing with flying enemies as they went. Scott himself came up—the hero of Chippewa and Lundy’s Lane. The men pressed round him. He told them how glad he was, and how proud of them; and how proud their country, their wives, their sisters and their sweethearts would be; and it seemed as if such cheering had never been heard, anywhere in the world, before.[17] Exultant but weary, the soldiers now looked about them as they took breath. From this eyrie the whole wonderful Valley of Mexico could be surveyed. All round the west the great wall of rugged mountains closed it in, and two vast, snowy peaks guarded its portal on the east. As if reluctantly the mountains gradually subsided into verdant hills and a wide plain, enamelled in a thousand soft hues. The broad, smooth lakes gleamed like molten silver. The gold of ripening grain, penciled lines of pale-green maguey, cottages radiant in the sun like the sails of distant ships, country-houses and villas half hiding in foliage, and many straight, converging avenues, lined with trees, delighted the eye. In the midst, clear-cut as a medallion, lay the city of Mexico, the capital, its roofs and towers black with people; and there, just yonder, stood the Halls of the Montezumas, the Jerusalem of these ardent young crusaders. Unfortunately breastworks, redoubts, cannon and a Mexican army were still to be reckoned with. Santa Anna had probably lost not more than 1800 killed, wounded and missing this day, and apparently Scott’s loss had been about one fourth as great.[18]
- 57. QUITMAN’S OPERATIONS The Citadel in 1840. But the Americans quickly prepared to advance—first of all, Quitman. Naturally a certain discretion had been given to the commanding generals, and he intended to make the most of it. Looking from the hill along the Belén causeway, he saw a wide avenue divided through the middle by a stone aqueduct some eight feet wide and fifteen feet high, resting on heavy arches and pillars of masonry. Owing to fine weather the road was unusually firm. A small number of troops, fleeing in the utmost confusion, could be seen upon it, but at only one point fortifications. Borrowing all of Pillow’s troops except the Fifteenth Infantry, which remained to hold Chapultepec and guard the prisoners, he quietly gave orders that his men should assemble near the main gateway. At once the inspiring words began to circulate, “Quitman’s division to the city!” and as soon as possible the Rifles, in their crimson sashes, were leading the march forward. About a mile on, a two-gun battery, with a field redan at its right on the marsh, blocked the way. For an hour or so Drum used a small gun upon it. Then the Rifles, after creeping along the aqueduct from arch to arch, took it by assault, and the march continued toward the fortifications at the garita.[19] As at the other garitas, no gates existed here, but a ditch and a parapet blocked one half of the causeway and a zigzag redoubt the other. Just at the north was the stone house intended for guards and customs officials, beyond which lay the wide Paseo (Promenade). South, on the Piedad road, were artillery and infantry that could fire through the arches. Inside the garita, buildings extending toward the east offered shelter, and in open ground a little more toward the north and about 300 yards distant, the extensive edifice called the citadel, protected with a wall and a wet ditch, constituted a serious obstacle. [19]
- 58. Santa Anna, after acting like a madman when Chapultepec fell, came to this garita. General Terrés, a brave old Spaniard, commanded here with about 180 infantry and some artillerymen. Santa Anna gave him three guns of medium power, and stationed General Ramírez in the Paseo, Brevet General Argüelles on the opposite side, and General Perdigón Garay and Colonel Barrios in the rear with substantial reserves.[19] On approaching this formidable position, Quitman encountered a withering storm of bullets, grape and solid shot from both sides and the front, and suffered rather severely. But Drum and Benjamin, iron men, bringing up as soon as possible a long 18-pounder and a 24- pound howitzer on the opposite sides of the aqueduct, dampened the ardor of the Mexicans not a little, and splinters from the masonry did havoc among the sheltered artillerymen at the garita. Some troops already beaten at Chapultepec and at the intermediate battery soon became demoralized. At about one o’clock rumors crept in that Americans from the southern front were turning the position. Ramírez, Garay, Argüelles and Barrios retired without the formality of saying good-by; and Terrés, whose cannon ammunition had failed, withdrew prudently to the citadel with two of the guns and about seventy panicky men, the remnant of his garrison. The Rifles now dashed over the parapet; and at exactly twenty minutes past one a tall, slender man with short, bristling, grayish hair stood on it, smoking a cigar and waving a red handkerchief tied to a rifle. It was Quitman, self-possessed but exultant; and in a few moments the Palmetto colors and the green banner of the Rifles, with its blazing gold eagle, were flying at the portal of the city.[19] The advance then continued for some little distance, and, as the ammunition of our two heavy guns had been exhausted, the captured Mexican 8-pounder was made to do good service. But Santa Anna, who had thought the position safe and gone on to San Cosme, soon arrived with ordnance and troops. The citadel was reinforced, and infantry and cannon were placed at other points. Quitman’s last artillery cartridges were used, and under the enemy’s fire no more could be brought up. Solid shot cut down both Drum
- 59. THE SAN COSME APPROACH and Benjamin. Our infantry had to retire to the vicinity of the garita. Attempts were then made to strengthen the position; but they did not accomplish very much. Ammunition gave out entirely, and firing ceased. The enemy grew bolder. Again and again they charged, and though repulsed they did not appear to be discouraged. By this time every member of Quitman’s staff, Beauregard, his engineer officer, and all his artillery officers had been killed or wounded, and he longed anxiously for night.[19] Meanwhile, events had occurred on Scott’s other wing. Trousdale’s command, supplemented with Jackson’s guns, pushed along the road and aqueduct on the north side of the rectangle, and the latter distinguished himself by fearlessly attacking a one-gun redoubt, which, supported by infantry and by fire from the summit of the hill, barred the way. To check Mexican reinforcements and threaten the enemy—particularly the troops in Quitman’s front—Scott now had Worth, Garland’s brigade, C. F. Smith’s battalion, Duncan’s battery, the rest of Magruder’s battery and Sumner’s dragoons pursue the same route. The one-gun redoubt was flanked and occupied; and Worth’s forces arrived at the northeast corner of the rectangle in time to annoy the retreat of Rangel and other departing Mexicans. [20] Here began the broad, straight Verónica causeway—closely similar to that of Belén— which extended almost north for nearly two miles (3530 yards) to the English cemetery, and there joined the San Cosme highway at approximately a right angle. Understanding the difficulties of the Belén approach, Scott intended to make only a feint in that quarter, and let his left wing break into the city. He therefore sent the brigades of Clarke and Cadwalader and also Huger with siege guns to Worth. To organize the attacking column, replenish the ammunition, make other needed preparations, and sweep away the resistance encountered at several minor fortifications, especially near the cemetery, required time; but at about four o’clock Worth found himself on the straight highway about half a mile from the San Cosme garita.[20]
- 60. This entrance to the city had been included in the general scheme of defence, but on account of its remoteness from pressing danger few workmen had been employed here; and when Chapultepec fell, it lay entirely open except for a small parapet without a ditch extending partly across the highway some 250 yards to the west. General Peña, however, coming this way from Chapultepec, stopped at the parapet, and Rangel placed at the garita such troops as he could assemble. Santa Anna, who displayed on this occasion reckless valor and an almost fiendish activity, sent three available cannon and brought additional troops. The roofs of buildings in the vicinity were occupied. A redoubt with embrasures was hastily erected at the garita, the near arches of the aqueduct were stopped up with sand- bags, and some guns in the Paseo were prepared to coöperate.[20] On attempting to advance, therefore, Worth found the highway swept with bullets, canister, grape and shells. Garland, however, was ordered to creep forward under the protection of the arches, and endeavor to reach the south flank of the garita, and Clarke to burrow through the continuous line of buildings on the other side, and strike the northern flank. Lieutenant U. S. Grant, who was reported as acquitting himself at this time “most nobly,” waded some ditches with a party of men and a mountain howitzer, and planted the gun on the roof of a church at the right; and Lieutenant Raphael Semmes of the navy performed a similar exploit on the left. Artillery fire compelled Peña, who—reinforced by Santa Anna with two companies of the Eleventh Infantry—was fighting gallantly, to leave the parapet; and Hunt, of Duncan’s battery, though he lost more than half his men in dashing 150 yards at full speed, landed a gun at that point, where he could load in safety and then fire from the one embrasure.[20] By five o’clock these preparations were complete. On the other hand Rangel had been severely wounded, and his principal gun, a 24- pound howitzer, had become unserviceable. Suddenly, to his utter astonishment, Americans appeared on the top of a three-story house that commanded the interior of his redoubt, and with a single volley disposed of almost every gunner and artillery mule. Then some of
- 61. them hurried down to the front door of the house, burst it open, and rushed into the redoubt, where they met Americans just arrived by a flanking movement from the other side of the highway. In a panic the Mexicans fled, literally sweeping away Santa Anna and a body of troops, who had come at all speed from Belén to support the position. Many of them scattered, but with no little difficulty others were conducted to the citadel. By six o’clock Worth entered Mexico. Near the garita his forces were safely housed, and by way of “good- night” and good advice, Huger dropped a few shells in the vicinity of the palace.[20] The end, however, was not yet in view. Santa Anna had some 5000 infantry and fifteen cannon at the citadel, with probably about 7000 more troops not far away, and the Americans, besides having lost many in the day’s fighting, were now fearfully divided. Not only Worth but Quitman, who planted three heavy guns in battery during the night, intended to advance in the morning, and apparently a day of carnage was to ensue.[20] But Santa Anna probably began to feel the reaction that always followed his great efforts. Funds and provisions were scanty. The army was demoralized, and the mass of the people felt disheartened. Within the town there were no fortifications, and it looked as if another battle under these conditions might scatter the troops, and involve the loss of nearly all the war material. Besides, leading persons in the city had always been strenuously anxious to prevent bombardment and assault; and the President was urged now, as four months previously, to spare it. Early in the evening, therefore, he briefly discussed the situation with Olaguíbel, the minister of war and three generals. The Governor was for acting deliberately; but Santa Anna, declaring that honor had been satisfied and the city could not be defended successfully, ordered immediate evacuation; and by one o’clock the troops retired in a somewhat orderly fashion to Guadalupe Hidalgo. About three hours later a commission of the city council (ayuntamiento) offered terms of capitulation at the American headquarters in Tacubaya. These were
- 62. THE AMERICANS CAPTURE MEXICO of course rejected, for the town lay at our mercy; but Scott gave informally the usual assurances of good treatment.[21] So when the first thin streak of dawn glimmered forth behind the gray volcanoes, and our cannon at Belén garita were on the point of opening fire, a white flag and an invitation to enter the capital reached Quitman. First making sure there was no deception, he advanced; and after stopping about half an hour at the citadel he moved forward under a splendid sun to the grand plaza, which fronted the palace and the cathedral, with Smith’s brigade, the Marines, the New York volunteers and Steptoe’s battery. As a triumphal procession the command looked rather strange. Quitman and Smith marched at its head on foot—the former with only one shoe; and behind them came troops decorated with mud, the red stains of battle and rough bandages, carrying arms at quite haphazard angles. Not less astonishing looked the city, for sidewalks, windows, balconies and housetops were crowded with people. Except for the silence, the countless white handkerchiefs and the foreign flags, it might have been thought a holiday. Before the palace, which filled the east side of the plaza, the troops formed in line of battle. Officers took their places at the front, and when Captain Roberts hoisted a battle-scarred American flag on the staff of the palace at seven o’clock, arms were presented and the officers saluted.[22] Soon loud cheering was heard. A few squares away the commander- in-chief, escorted by cavalry with drawn swords, had reached Worth’s command, which had stopped at six o’clock by orders opposite the high ash trees of the Alameda. A clatter of galloping hoofs followed; and in another moment, amidst the involuntary applause of the Mexicans, General Scott, dressed in full uniform and mounted on a tall, heavy bay charger, dashed with his staff and Harney’s dragoons into the grand plaza—his noble figure, gold epaulets and snowy plumes, resplendent under the brilliant sun, fitly typifying the invisible glory of his unkempt and limping army. Uncovering, he rode slowly along the line of battle to the music of
- 63. our national airs; the troops, presenting arms again, cheered and hurrahed till it seemed as if the earthquake-proof cathedral must be shaking, and the cavalry escort waved high their flashing blades.[22] In stentorian tones the commander-in-chief appointed Quitman governor of the city; and then, dismounting in the courtyard, he clanked up the broad stairway of the palace, to indite congratulations on the “many glorious victories” of his army. Presently cross-belted American Marines were calmly patrolling the Halls of the Montezumas as if they owned them, while the rest of the troops gazed with profound exultation at the long pinkish façade and the endless balconies of the upper story, and the people gazed silently at the troops. “They are all and each of them heroes,” commented a foreigner present, and others in the world thought the same.
- 64. “Light up your homes, O fathers, For those young hero bands, Whose march is still through vanquished towns And over conquered lands, Whose valor, wild, impetuous In all its fiery glow, Pours onward like a lava-tide, And sweeps away the foe!”[22]
- 65. XXIX FINAL MILITARY OPERATIONS January, 1847–April, 1848 At the north, after the Buena Vista campaign and the embarrassments growing out of it came to an end, Taylor probably wished, in what an officer calledto advance as far as San Luis Potosí, and retained troops urgently needed by Scott; but by the middle of June, 1847, he doubtless realized that effective operations on so long a line, especially through hostile and much of the way through barren territory, were impracticable, and advised that Scott’s column alone should act on the offensive. A month later orders of a corresponding tenor were issued at Washington, and then some 3000 surplus troops of the northern army proceeded toward the capital, though too late, of course, to assist in the decisive struggle.[1] Valencia, during his brief stay at San Luis Potosí in the early summer of 1847, not only requested permission to move against Saltillo, but planned that General Filisola, aided by a brigade under Avalos, then lying at Matehuala, by Reyes, the comandante general of Zacatecas, and by Urrea—who still commanded theand could easily pass across the Sierra Madre from Tula—should threaten, if not attack, Saltillo and Monterey, and at least keep the Americans on the defensive. Some disquieting movements of these troops resulted; but Valencia was soon called to Mexico, and various difficulties, chiefly a lack of means resulting from the American occupation, proved fatal to this ambitious enterprise, besides hindering the Mexican preparations to receive Taylor at San Luis Potosí.[1] During the winter of 1846–47 and to some extent later, the garrison of Tampico was menaced by plans for an uprising, to be assisted by outside forces, and sometimes it was feared that a move to capture the city would be launched from Tula in the hope of embarrassing Scott’s communications; but the Americans, though not strong in numbers there, were vigilant and well protected by fortifications. Besides, the authorities of Tamaulipas, now living on fairly good terms with the invaders, had little wish to take part in active hostilities. They quarrelled bitterly with Urrea, who naturally attempted to draw supplies and money from the region, and in November, 1847, with a view to bringing about harmony, that officer was removed. Scott’s victories and especially the fall of Mexico had no little effect in this quarter; the prospect of serious operations entirely disappeared; and early in November, 1847, General Taylor, who had reached the conclusion some time before that his country wanted him for President, and had laid aside his old brown coat in favor of checked shirt sleeves, set out for home on a leave of absence, which actually
- 66. FIGHTING IN THE CAPITAL continued until the close of the war. Wool took his place; but nothing occurred in this region except guerilla affairs, of which a due account will be given presently.[1] In the northwest, meantime, Price, who commanded in New Mexico and was disturbed by rumors of danger from the south, decided on his own responsibility, ignoring instructions to do otherwise, that he must assume the aggressive. Early in March, 1848, the city of Chihuahua was therefore reoccupied; and on the sixteenth of that month, after a little brisk fighting, the town of Rosales, about sixty miles to the southeast, which Angel Trias held with some 800 men, chiefly National Guards, was captured by assault with a trifling loss. But this campaign had no general effect on the war—indeed, the treaty of peace had already been signed—and Price was ordered by Marcy to retire.[1] In Scott’s department the final military operations began very promptly. Immediately after the Americans took possession of the grand plaza at Mexico on the morning of September 14, a multitude of blanketed léperos crowded closely upon them. Already these miscreants had tasted the disorder they loved, for the palace had been left unguarded, and they had sacked it; and now they showed signs of turbulence. The plaza was cleared, however, and no further trouble seemed likely. But when our troops began to march away to their quarters, a shot was heard. A bullet probably intended for Worth struck Garland, and almost instantly firing from street corners, windows and the tops of houses became general, though not systematic. Thousands of convicts from the jail supported the populace, and in one way or another not a few of the better class coöperated. By Tornel’s order paving stones had been taken to many of the azoteas with a view to resisting the invader step by step, and these, like every other sort of weapon, were now used.[2] Though surprised, the Americans promptly accepted the challenge. Skirmishers drove back the mobs. Grape and canister swept the streets. As a rule, every house from which a shot flew became a target for our heavy cannon, which seemed to shake the very foundations of the city, and when breached was immediately sacked; and sharpshooters worked effectively on towers and roofs. Scott threatened even sterner measures; and the city authorities not only put up notices, embodying his threats and imploring the people to desist from a vain and imprudent contest, but interceded personally with them in the streets. By about noon the Americans held all the points of vantage, and as evening approached, the firing died away. A fearful night ensued. It was dark and cold. No lights relieved the gloom. Wild mobs ran shouting through the streets, and the hoof-beats of American patrols resounded from square to square.[2] Santa Anna, finding it impossible on the morning of the fourteenth to subsist his army at Guadalupe, had ordered the infantry and heavy guns to Querétaro under General Herrera, and proceeded with four small pieces and the cavalry to San Cristóbal, a point about fourteen miles northeast of the capital. After seeing the
- 67. people of Mexico view with indifference his efforts of the previous day, he expected nothing of them; but on learning of the outbreak he marched back to Guadalupe, and at a late hour sent into Mexico a small force of cavalry and infantry to investigate and assist. This met Duncan’s battery and retreated; but Santa Anna, assured that on the next day there would be a rising en masse, erected a breastwork at the Peralvillo garita on the north side of the town, and waited.[2] As soon as day broke, gloomy and wet, the shooting was in fact resumed, at least in the northern quarters. But he soon perceived that no general movement was taking place, and again marched away. This disheartened the people still more; the efforts of the authorities influenced them greatly; and by the end of the afternoon, realizing that much was to be suffered and nothing gained, they generally abandoned hope. During the next day or two scattering shots could be heard, but real fighting was over. Extravagant hopes of destroying the small American army were still entertained by lightheaded men.wrote one of these, and attempts were made by military officers to organize a real conspiracy; but lack of courage, means, confidence and mutual trust—as well as the watchfulness of the Americans—made success impossible. Scott repeatedly warned his troops to be vigilant and orderly, to keep together, and to refrain from drinking. As the danger grew less menacing, however, they became less careful, and for probably a month assassinations were frequent. From first to last several hundred Americans perished in the hostilities, and no doubt far more of the enemy. But by the middle of October the city was tranquil.[2] The concluding field operations in Scott’s department resembled for the most part the fighting just described, for they had to do chiefly with guerillas. That style of warfare suited the national character. It had figured prominently in the Spanish struggle against Napoleon and in the Mexican war of independence; and when the public began to see clearly that battles could not stop the Americans, it was invoked—even though by universal military practice in Europe those who robbed and fought at will, while pretending to be inoffensive, were considered brigands and assassins—as the one hope.[3] Thoughtful persons like J. F. Ramírez and General Mora pointed out serious dangers: the impossibility of discipline, the relaxation of morale, the destruction of all standards, and the certainty that a spirit of violence and rapine would grow by what it fed upon; and they recognized the improbability that such methods could prevail against the strength, equipment, compactness and skill of the Americans. But the obvious advantages of the guerilla system, which it required far less intelligence to perceive and appreciate, counted powerfully on the other side. How much the Spanish themselves had suffered from their irregulars during the hostilities against Napoleon was not understood, and patriotic pride in the war of independence had tended to draw a veil over its horrors.[3]
- 68. THE GUERILLAS OF THE NORTHEAST The dagger, said the official newspaper, was the favorite weapon of the people. Unarmed men could burn wagons and intercept communications, it was pointed out. Even women and children could help. A thorough knowledge of the country, its mountains and its by-paths, would evidently constitute an enormous advantage. Light corps of the abstemious rancheros, embarrassed with no baggage, could travel quickly day and night, concentrate in large numbers against an American detachment, strike, vanish, and then, when least expected, reappear, making the most of all neglects, all mistakes, nullifying superior strength by avoiding it, and nullifying discipline by fighting in a style that had no need of discipline. Situated even more favorably than Spain for such warfare, the Mexicans were to outdo her example.[3] This is what will save us, proclaimed in effect the legislature of México state.exclaimed the congress of Vera Cruz. Santa Anna endorsed the plan. Salas organized thewhich were to make” “in every manner imaginable”; and in April, 1847, the government, pinning its faith to the system, set it on foot in earnest. Scott, thewas to be routed after all.[3] In the north February, 1847, was the golden month of the irregulars, for the approach of the Mexican army under Santa Anna encouraged the rancheros to lay aside the habits of peace. Canales boasted of 161 Americans killed that month, and Urrea with his combined force of regulars and guerillas, besides engaging in other operations, captured a train of wagons at Agua Negra, and horribly slaughtered a large number of guards and teamsters. To avenge this butchery a party of Rangers, teamsters and other civilians murdered twenty-four men in a village not far distant. Upon this Canales declared what he called martial law, announcing that every American, armed and unarmed, and every Mexican living peaceably would be shot; and many were led by fear or a lust for plunder to take up arms.[4] The American trains in particular seemed likely to be easy prey. As they commonly stretched out for some two miles and were guarded only—for so the character of the road usually dictated—at the ends, the Mexicans, trained to charge at full speed through an ordinary thicket, could readily attack them from ambush at about the middle point, create a stampede, and do a great deal of mischief. Infantry could not pursue the guerillas with success, and the number of our mounted men was always comparatively small, for every Mexican ranchero had at least one smart pony. In September, 1847, a band even attacked Mier. Governor Aguirre of Coahuila exerted himself particularly to organize forces of this character, and not only alcaldes but priests aided the cause.[4] The American leaders, however, pursuing a course that was now conciliatory, now severe, and in many instances technically unjust, succeeded in coping with a system that was itself unjust. Taylor levied on the people of Nuevo Léon a tax of
- 69. GUERILLAS IN VERA CRUZ STATE $96,000, the estimated value of the goods destroyed at Agua Negra, but suspended it indefinitely, when representative authorities proved the substantial innocence of the population and begged for mercy. Cavalry patrols and detachments pursuing culprits fairly wore out their horses. Villages, if even suspected of harboring thewere burned. Contributions were imposed wherever connivance appeared probable. By April, 1847, Canales was in despair.[5] Then Wool determined to stamp out the evil, and announced in July that any guerillas caught by him would be executed. In December, 1847, he issued his famous Order 11, which not only made the Mexican authorities and their towns responsible for all damages done, but required them to hunt down theAguirre attempted to retaliate, but in vain. The Americans had power enough to carry out threats, whereas he had not; and he admitted his failure. Besides, the mass of the population were indolent in mind as well as body, and looked upon submission as preferable to danger. In February, 1848, finding the guilty rancheros were anxious to give up the business, Wool enabled them to resume peaceful occupations by declaring an amnesty, and in the following May he stated that the country had never before been so free from highway robbery.[5] In the south, Vera Cruz, a state of mountains, gorges, thickets and forests threaded with blind paths, was the chief home of the guerilla, and it looked as if Scott’s line of communication might be virtually destroyed. Not only many hardy, hot-blooded and unscrupulous natives, but a great many desperadoes hailing from Cuba were ready to enlist. After the fall of Vera Cruz, and still more after the battle of Cerro Gordo, a large number of regular officers, to say nothing of privates, could scarcely find bread, and some men, like the ex-divinity student, ex-Carlist, Jarauta—whose small, close beard, fierce black eyes, braided jacket, graceful cloak and gold-laced sombrero gave him a romantic air—had acquired in Spain a taste for this adventurous, reckless life; but a vastly greater number were prosaic felons, liberated from prison under a pledge to rob and murder. Nominally J. C. Rebolledo, a fine looking man of rather humane instincts, was the chief in this district, but the 800 or so persons belonging to many small bands, while occasionally acting more or less in concert, were mainly independent. The decree under which all goods coming from points occupied by the Americans were lawful booty opened possibilities of large gains, and Rebolledo’s capture of ten loaded wagons in April, 1847, set the people aflame with cupidity.[6] Brevet Colonel McIntosh and his inexperienced officers, who left Vera Cruz for the interior—it will be recalled—about the first of June, 1847, with a well-advertised convoy including a large amount of specie, dependent on wild mustangs under raw, half-mutinous drivers largely ignorant of English, received the full benefit of this ambitious feeling. Near Tolomé and at Paso de Ovejas he lost men, wagons and pack-mules; and at the national bridge there was a genuine skirmish, in which a number of Americans were killed or wounded. Out of about 130 wagons twenty-
- 70. four had to be abandoned in the low country; and a little way above Jalapa, though strongly reinforced, the troops had to fight again. General Pierce, who left Vera Cruz about six weeks later than McIntosh, had similar experiences. Early in August Major Lally set out from the coast with a few more than 1000 soldiers, two 6- pounders and sixty-four wagons. He lost no merchandise, but his four fights cost him nearly 100 men killed, wounded and missing; and Captain Wells, who followed Lally with some 200 recruits and additional ammunition, lost forty and had to retreat. These and other affairs proved that irregulars, favored by the geography of the region, were capable of doing substantial harm.[7] But in Vera Cruz, as in every other quarter where they operated, though perhaps nowhere else in so marked a degree, the lack of morale, which enabled the guerilla system to exist, proved the cause of its failure. Poor arms, poor ammunition, poor marksmanship, and the want of artillery might have been remedied, or at least might have been offset by the counter-balancing advantages; but this defect was fatal. The Mexican guerillas were very different from what the guerillas of Spain had been. They fought like savages without the excuse of savages, for they knew better. Infuriated by their treacheries and cruelties, the Americans were persistent and unsparing in severity. Patrols who seemed never to sleep hunted out their nests in the mountains. On the march, flanking parties would force their way through the woods five miles or more from the road to catch them between two fires. The torch was applied with much liberality on suspicion, and sometimes on general principles, to huts and villages; and in the end a black swath of devastation, leagues in width, marked the route.[8] Scott ordered that in every case of outrage the nearest alcalde, if he failed to deliver up the guilty, should be fined at least $300 for a murder or the value of the stolen property for a robbery, and that any robber or murderer and any person belonging to a known party of such miscreants might, when caught, be summarily tried by three officers, and either flogged or executed. This plan, however, did not quite satisfy those on the ground—especially the Texas troops. Captain Walker, on his cream-colored horse, and Colonel Hays, in his blue roundabout, black trousers and black leather cap, impressed themselves on the Mexican imagination as the agents of diabolical wrath; and in general it was a tale of merciless atrocities followed by merciless reprisals.[8] At the same time this lack of morale deprived the guerillas of Mexican support. By taking bribes for letting merchandise pass up to the interior and sometimes even guarding it, they violated the laws on which their existence rested. Mostly they were brave only where they felt safe. When laden with booty they would scatter to their homes, no matter how important the business in hand. Rivalries and even hostilities between parties operating in the same district arose. Cooperation could seldom be reckoned upon, and hardly any would face the climate far above Jalapa. Soon learning that it was more wholesome to waylay Mexicans than Americans, they plundered their fellow-countrymen without ceremony; and they would rob
- 71. GUERILLAS OF THE CENTRE even old women or young children of their needful clothing. Sheafs of complaints against them piled up in the state and national archives. People organized to fight them, and sometimes appealed to the Americans against the very men who were to have been their champions.said an American officer.[9] In the states of Puebla, México and Oaxaca also guerillas were organized, and in Puebla all these parties could find an opportunity. General Rea, a pupil of Morelos and the Mexican revolution, had the discredit of the chief command, though Bravo, who stood at the summit of the social scale, was mainly responsible for their iniquities, since during his brief term as comandante general of Puebla he issued a great number of patents to unfit leaders. What Rea did particularly in this regard was to combine individuals and small groups, and place them under some kind of supervision. He loved to answer critics by saying that his guerillas were in the field because honorable men were not; and that, had not the government condoned their crimes, they would have served the Americans as counter-guerillas. After a time his officers adopted a set of rules which aimed to regulate operations, but even this measure seems to have accomplished little. The guerillas robbed the people, seized funds belonging to the state, and pillaged even churches. Some gangs were large enough to attack haciendas. One party called themselves the “Lancers of the Poisoned Spear.”[10] Soon after Scott left Puebla for Mexico early in August, 1847, these banditti and every individual ruffian of that vicinity hurried to the city. Mexicans and even foreign residents were robbed and outraged, and about the first of September, in the hope of more booty, the Americans also were attacked. Two thousand soldiers were needed for a garrison, and Colonel Childs, the civil and military governor, actually had 2193; but 1800 of these were in hospitals. His effectives consisted of about fifty cavalry, 100 artillery, 250 of the First Pennsylvania volunteers, and a small spy company of Mexicans.[11] Headquarters, Lieutenant Colonel Black of Pennsylvania and the main body of troops occupied thea large rectangular building on the eastern side of the town, which had a plaza of its own opening toward a public promenade called the Tivoli. To this position five howitzers were allotted, and within a hundred and fifty yards of it all the sick were placed. Half a mile or so from the town on a hill stood Loreto fort, a stone affair equipped with two 12-pound field guns and a 10-inch mortar, where Major Gwynn of the Sixth Infantry commanded; and not far distant, on a higher point of the same hill, was Guadalupe church, now protected with mountain howitzers, a ditch and an earthen wall, under Captain Morehead of the Pennsylvania regiment. But the chief element of the defence was the large, robust, finely-featured Childs, a skilful and veteran officer, cold in manner, clear in judgment, and inflexible in courage. September 13 thebegan in earnest, and from that day on there was a continual small-arm attack, particularly at night, upon San José, which replied with a musket and howitzer fire that kept the assailants at a
- 72. THEOF PUEBLA respectful distance. What was more serious than guerilla shooting, all supplies were now cut off. Such was the state of things at the second city of Mexico when Santa Anna retired from the first.[11] Santa Anna’s real intention was probably to seek an asylum in Guatemala. But many of his friends urged that he could make himself dictator as the sole hope of the country, and it was clear that, if he should recover Puebla and cut off Scott, he would still be able to boast of a triumph. His cavalry, though greatly reduced by desertion, included some 2000 men backed with four light guns. Alvarez, who was ordered to Puebla, still had about 600 foot and horse. Rea, Santa Anna understood, commanded 600 irregulars; 2500 National Guards lay near him with two field pieces, it was reported; and the Pueblans were described as eager to fight. Six thousand men and six guns appeared quite enough to dispose ofas Mexicans described the garrison; and he therefore presented himself at Puebla on September 21. Two days later Alvarez arrived there. But between these two events Childs appeared at a second-story balcony, as a soldier expressed it, and announced that Scott had taken the capital. Evidently, therefore, the Mexican President was not greatly to be feared.[11] After looking about, Santa Anna concluded that it would not be easy to capture the American positions by assault, and appealed to the minister of war—wherever that official might be—for 1000 infantry, a 16-pounder, a 12-pounder, ammunition and supplies. He now had ten cannon, but all of them were light; owing to desertion his force included only some 4000 men; and the citizens had no arms, he reported. Probably, too, the annoyances and outrages inflicted upon them by him and his troops, and his appointing the guerilla chief military commandant of the city dampened whatever ardor they had possessed.[12] On September 23 and 24 unsuccessful attempts were made at Guadalupe, and the next day Santa Anna summoned Childs, describing his army as 8000 strong, and graciously announcing thatthe Americans might retirewith the honors of war. Childs replied as was proper, and then, riding to the posts, gave notice amid cheers that no surrender need be expected. To add the touch of humor that soldiers love, an American flag was manufactured out of an old Mexican uniform, and raised aloft; and the garrison settled down to severe duty, stern discipline, short rations and incessant watchfulness at all hours. The Mexicans tried to approach San José by throwing up successive breastworks at night in the streets leading that way, but shot, shell and rockets from Loreto kept them back. September 30 Santa Anna learned that no ammunition could be provided for the heavy cannon demanded of the minister, and resolved apparently to make a bold effort. With two 6-pounders he fired all day on the weak, plaza face of San José. But Childs, anticipating such a manoeuvre, had brought a 12-pounder from Loreto the night before; and this, protected with bags of tobacco, made an assault impracticable.[12]
- 73. THE FIGHT AT HUAMANTLA A new factor now entered the military situation. About the middle of September GeneralLane, one of Taylor’s chief officers at Buena Vista, arrived at Vera Cruz from the Rio Grande, and on the nineteenth his brigade set out for the interior. Aware of the situation at Puebla but not aware what was to be encountered on the route, the General had not made adequate preparations, and on meeting guerillas at the national bridge he was obliged to send back for ammunition and supplies. By October 1, however, he managed to leave Jalapa.[13] It was a hard march that ensued. Torrents of rain deluged the troops. Sometimes the road lay deep under water. For dinner they had a thin slice of beef, a couple ofand some coffee; for supper, after darkness fell upon them with tropical abruptness, the same without the beef; and perhaps mud for a couch. But Lane, a hearty westerner with a stout frame and unbounded vigor, led on unshrinkingly in his black hat and old blue overcoat, and the rest followed him eagerly. October 5, after incorporating additional troops at Perote, he left that place with a force of about 3300 and seven guns, and marched on across hot plains, where water sold for five dollars a drink, and men died of sheer fatigue.[13] Santa Anna, informed by spies that 1000 Americans were approaching, and anxious, not only to prevent them from joining Childs, but still more to win the glory of routing them, had set out from Puebla four days earlier with perhaps 3500 men, leaving Rea to continue the fighting. Desertion played havoc with his command, especially when the strength of Lane’s force was ascertained; but, after sending back a large part of the faithful in order to keep control of them, he took possession of El Pinal, where the national highway passed between a precipitous mountain and a ravine, with about 1000 cavalry and six guns, and made preparations to ambush Lane’s rear. This done, he moved to Huamantla, a sizable town eight miles distant, and waited.[13] Early on October 9 the drums and bugles awoke Lane’s troops at the hacienda of San Antonio Tamaris, approximately ten miles from Huamantla and twelve from El Pinal; and the men, leaping from the damp grass and buckling their muddy belts, found the white walls of the hacienda, the church towers of neighboring villages, the dark woods on the hillsides, and the distant, snowy peaks all aglow under a splendid sun. Never, perhaps, did soldiers feel more like having an adventure. Santa Anna had just marched from Huamantla to conceal his force at El Pinal, leaving behind him with no scouts or outposts his six guns, a very small guard for them and a party of irregulars; but a spy reported to the Americans that he was at Huamantla, and Lane moved off to attack him. First rode four mounted companies, and at their head a rather short, slender, spare, slouchy man, with reddish hair, a small reddish beard, mild blue eyes and a quiet, kindly manner, whom nobody would have picked out as a fearless, indomitable fighter, the scourge of the guerillas, but in fact he was Captain Walker; and then marched Lane with five guns and some 1800 men.[14]
- 74. When about three miles from their destination, Walker and his 200 cavalry, seeing a party of Mexican horse approach the town, dashed ahead. Entering Huamantla they formed in fours, and then with a yell, a flash of sabres and a thunder of hoofs they swept through to the plaza. The Mexicans had time to get four of the guns away, but the others were captured, and most of the American troopers, concluding their work had been finished, scattered to drink, loot or hunt for cannon and ammunition. But now Santa Anna, who had observed Lane’s movement from a church tower near El Pinal, appeared with his full command. They were a beautiful sight—galloping horses, red and green uniforms, brilliant pennons and a billowy sea of flashing lance points; but they were enemies, and the Americans accepted their challenge.[14] “Take it cool, my boys, but run like the devil!” cried Lane. Every nerve was taxed. Blood gushed from nostrils. The Mexicans, lashing their steeds into foam, reached the goal first, however, and the American troopers found themselves attacked on all sides. Walker was shot from a house, and soon expired; but he lived long enough to give a final order: “Don't surrender boys; the infantry will soon be here.” And so they were—“with a shout and a bound,” said one of them. The tide was quickly turned, and giving up the town, as the soldiers loved to call Santa Anna, passed the night some distance away. So ended the Tale of Huamantla or The Biter Bitten, which received no little applause at the time.[14] While these events were taking place, the garrison of Puebla continued to be hemmed in, starved and harassed. Their casualties numbered in all only fifty-two, but they felt severely the effect of so long a strain. Though a number of sorties were made, and their persecutors had to retire from several annoying positions, the Americans were not strong enough to do more. Their day of deliverance was approaching, however. October 10 Lane moved forward, dogged and somewhat annoyed by Santa Anna. Two days later his men saw the spires of Puebla, dominated by the sombre towers of the cathedral, and set off by white volcanoes veiled with clouds. And now and again the numberless bells of the city, great and small, pealed forth harmonious tones of many colors, that seemed to blend and interweave in rich and varied tapestries of sound, hung out in the mediaeval style to honor their triumphal approach.[15] At about one o’clock, announced by the bells of Guadalupe, they entered the suburbs—not a few of them at a run. One column then advanced by the main street, while another flanked the town by the left. For two hours there was considerable firing from houses, though Rea’s guerillas had begun to leave their posts the night before; but at length Lane extended histo Childs, with a sunny smile on his rather hard features, and the garrison joyously welcomed their deliverers. In the main plaza a bugler playedand all sang the chorus: “The star-spangled banner, Oh, long may it wave O’er the land of the free and the home of the brave!”[15]
- 75. POLITICAL CHAOS IN MEXICO Now followed the punishment of Rea. Some twenty-five miles from Puebla toward the southwest lay beautiful Atlixco, a defensible point that not only reconciled the climates of the temperate and the cold zones of Mexico, and controlled a region fertile in grains, flocks and herds, but, while fairly remote from the Americans, gave convenient access to important roads. Here, in the midst of flowers, fruits and snowy mountains, the government of Puebla had found a refuge, when the Americans under Worth approached the state capital; and to this asylum Rea now withdrew. De facto, at least, the guerilla chief was the most important person on the ground. The authorities did not relish his prominence; they detested his men and his methods; and on October 18, tired of spending money fruitlessly on the National Guards for Rea to command, they dissolved the corps. But many of the irregulars proposed to make the best of what appeared to be a good situation, in which they could live on plunder, if not paid.[16] October 18 Lane, who apparently never slept nor expected his followers to sleep, ordered them to be ready in the morning for an expedition. Many of the soldiers were barefoot, but they borrowed shoes; and at about nine o’clock, cheered by the fife and the drum, some 1500 men set out round the base of Popocatepetl under a hot sun. At about four in the afternoon, after making twenty miles or so, they came in sight of the enemy, and a running fight began. Blistered feet and parching tongues were now forgotten. The Mexicans, holding some good position and protected by chaparral, could make a stand against cavalry, but when the infantry came up they always fled. Shortly after sunset Lane reached Atlixco, which stood on the slope of a lofty hill. As it was unsafe to risk a street fight in an unknown town at night, he ordered the artillery to open. The moon was full. Marks were easily selected. By their burning fuses the shells could be traced until they fell amidst the shadows; and then a burst of red fire, the crash of roofs and walls, and the cries of the people told the rest.[17] After about an hour of cannonading, the troops advanced into the town—which surrendered at once—and there slept as best they could. Rea, with two guns and the disordered remnants of his force, retreated to Izucar de Matamoros, about thirty miles farther down the valley; but from that point he was routed a month later. These and other exploits of Lane’s discouraged as well as dispersed the chief guerilla forces of the plateau, and in February, 1848, Rea asked permission of the Mexican authorities to leave the country.[17] Neither in these affairs nor in any other military operations did Santa Anna figure at this time, and there was a good reason for his inactivity. Officially he no longer existed. As General Scott had feared, our entering the capital had resulted in the destruction of the Mexican government. September 16 Santa Anna resigned, explaining that it was advisable to preserve the chief magistracy from the hazards of war, and fix it near the centre of wealth and population, whereas he proposed to continue the hostilities wherever that should
- 76. SANTA ANNA ELIMINATED be possible. The same proclamation or decree assigned the executive power to a triumvirate: the president of the supreme court, General Herrera and General Alcorta, and Santa Anna then ceased actually to exercise any civil authority.[18] But as Congress was not in session to accept his resignation, some doubted whether it became effective; the presidency of the supreme court was vacant on account of the incumbent’s death; the appointment of Herrera and Alcorta needed to be made, or at least confirmed, by the council of government, a body no longer acting; and it was denied broadly that Santa Anna had the power to issue such a decree. Peña y Peña, to be sure, was regarded as a member of the court, and, if he was, he could claim by right of seniority to act as the chief justice; but the legality of his membership was questioned, and the presidency of that body was really an elective office. Peña was old, feeble and even timid; his ill-success as Herrera’s minister of relations doubtless weighed heavily upon him; and he was now living, almost as a recluse, in the country. Indeed there was really no organic law even, for the amended constitution of 1824, though formally adopted, had not come into effect. In short, chaos reigned, and the states were officially “resuming” their individual sovereignty.[18] But a number of good and able men, particularly Cuevas and Couto, determined to ward off ruin, and awakened others. Peña, drawn from his retirement, consented for patriotic reasons to override all the technical difficulties; and on September 22 he announced formally that, in order to give the nation a head, he would act as the Executive until an interim President could somehow be chosen. At the small city of Toluca, capital of the state of México, just outside the Valley, this fiction of a government pitched its tent; and perhaps it gained some feeling of security from the vast bastioned, battlemented ridge between it and the Americans, from snowy Mt. Miguel towering above the city, and from the peacefully shining lagoons of the intervening meadows. What was more important, Herrera, Olaguíbel and many others of the best men rallied to the support of Peña, the representatives of neutral governments recognized him, and the states began to concur. Early in October, however, he removed to Querétaro, a safer yet central place, and with Luis de la Rosa as sole minister addressed himself to his task.[19] The programme that he announced was honorable and straightforward. My tenure of office will be extremely brief, he said in effect, for Congress will be assembled as soon as possible; I will usurp no powers, but will not be turned from the path of duty by insurrections; the closest economy will be practised, the necessary taxes laid fairly, and all interests respected; union and harmony will be the watchwords, and the national rights will be maintained. His most urgent problem, of course, was to deal with Santa Anna, who not only held the chief military command, but insisted that he could resume the Presidential authority by simply withdrawing his resignation; and in this matter the government showed a decision that earned it no little prestige. All Santa Anna’s protests against political
- 77. effacement were disregarded, and on October 7 he was instructed both to give up his troops and to submit, as did other unsuccessful commanders, to a military trial. [20] At about the time this order overtook him, the Huamantla affair occurred. From a military point of view he was now prostrate. He saw it himself, and knew that the country would see it. Evidently his countless political enemies would make the most of his complete failure, and he was doubtless aware that his military reports had offended many officers. His chief executive merits—decision and activity—had led only to a useless expenditure of life and money, it was now pointed out, and his ostensible patriotism was attributed to passion and obstinacy. Even his confidence in himself broke down. Unable to understand why failure had attended all his efforts, he fell into a sombre depression, and without a struggle he placed his troops at the orders of General Reyes, who joined him on October 11 with about 1000 men. His part in the war was over; and in the following January, realizing that nothing could be gained through intrigue or conspiracy and fearing the Americans would make him a prisoner, he asked for permission to leave the country. Both his own government and our authorities consented. And after giving a dinner at El Encero to the American officers of that vicinity, who had treated him with distinguished consideration, he sailed once more, about the first of April, from what he regarded as an ungrateful country.[21] Santa Anna being now eliminated, the government had to face its military difficulties without his assistance. In general the problem was to make bricks with neither straw nor clay. Almost every good cannon had been taken by the Americans, and the muskets had nearly all been captured, thrown away or sold. Ammunition was almost wholly wanting. The engineering material had been lost or destroyed. Vast sums of money were needed to provide fortifications as well as replace all this equipment, and the government could hardly obtain enough, day by day, to cover its minimum expenses. Even officers had to sell their shoes for bread. [22] As for an army, Santa Anna and Alvarez together had some 2000 troops the first week of October, Reyes had about 1000, about 3000 from Mexico City concentrated at Querétaro under Herrera, about 1000 from Jalisco were on their way to the same point, and small detachments existed at various other places. But nearly all of these men were utterly demoralized. “Almost useless,” they were officially termed; and the army as a whole felt the crushing weight of general contempt. Herrera, the commander-in-chief, became so disgusted over the uncontrolled excesses of the troops that he resigned. Rincón declined on the ground of ill-health to serve. Arista, when summoned to Querétaro, declared he would not command a soldier until exonerated for his conduct on the Rio Grande. No officers of high distinction, indeed, were available except the aged, torpid and infirm Bustamante and “the old woman,” Filisola, as Bancroft described him.[22]
- 78. THE AMERICAN POLICY Attempts were made to lay plans of campaign, but an expert summed up one of them by saying it appeared excellent—only it was based upon things as they should have been, not as they were; and all the others had the same defect. Schemes were devised to reform, reorganize and build up the forces, and quotas amounting to 16,000 were assigned to the states; but México, which had been expected to furnish nearly a quarter of these men, promptly answered that she could not, and other states did not even reply. In fact, the regular forces decreased instead of multiplying, for sometimes a general could not feed his troops, and frequently, when soldiers were let out of the barracks on service, they vanished; and the people, instead of helping to support the Mexican troops, even dreaded to see them approach, for their coming was liable to draw an American attack, and more than liable to mean extortion, outrage and robbery. Nowhere on the military horizon could a glimmer of light be seen.[22] Over against this pitiful government stood the United States—wealth against poverty, strength against weakness; and the antithesis was complete, for while the Mexicans could only plan, that was the hardest thing for us to do. The idea of retiring to a defensive line still persisted. Taylor himself adhered to it. But in addition to the other overwhelming objections to this project, it seemed improbable that a majority in Congress could agree where to draw the line. Even Calhoun, though qualified to make a better argument for an untenable proposition than any other man in the country, was unable to present this policy in such a manner as to satisfy either the friends or the opponents of the war. Some advised holding, in addition to the territory thus to be cut off, the chief ports of Mexico; and some advocated retaining the capital also, and the line to Vera Cruz. Others favored the occupation of still more cities; and many were for subjugating and holding the entire country.[23] To this last plan, however, even had it been practicable to levy all the costs upon Mexico, there were tremendous objections. It would have involved keeping under arms 80,000 or possibly 100,000 young men, seriously needed at home for the most part, in order to be sure of having effectives enough at the front. The troops in Mexico would have become corrupted both physically and morally; and the commanders would have acquired the ideas and vices of proconsuls. It seemed to be almost an insoluble problem. No final decision was made. But the government determined to occupy the capital, hold the line to Vera Cruz, retain the chief ports, and extend our holdings according to circumstances.[23] To Scott, however, the lack of a definitive plan signified little. Not one reinforcement entered the capital until after the first of November, and even at the end of that month he was barely able to garrison Mexico and Chapultepec. December 4 his army included only about 8000 privates, of whom a quarter were sick. During the next three weeks Generals Patterson, Butler and Cushing, Colonel Hays, Lieutenant
- 79. Colonel Johnston and Major Lally, each with troops, arrived; and the forces then numbered about 11,000 effectives and 3000 sick.[24] Scott therefore announced, with no doubt a strategic purpose as well as a rhetorical flourish, that our army was “about to spread itself over and to occupy the Republic of Mexico.” What he really intended was to take possession successively of the principal mining regions—those of Zacatecas and San Luis Potosí—and the capitals of such important states as lay within easy reach. Even for the former purpose, however, two columns of some 5000 effectives each were needed, and the men as well as clothing for them could not be provided. The only immediate operations, therefore, aside from the establishment of new posts on the road to Vera Cruz, were the peaceful occupation of Pachuca, a mining town about fifty miles northeast of Mexico, Toluca, about thirty-eight miles distant in the opposite direction, and Cuernavaca, the key to the Acapulco region, a little farther away toward the southwest.[24] February 6, 1848, an expedition of more consequence marched, by Scott’s orders, from Vera Cruz. Most of the guerillas who infested the road to the interior lived and found a market at or near Córdoba, a city about sixty-five miles to the southwest, and Orizaba, sixteen or eighteen miles beyond it in the same direction; and Bankhead was instructed to occupy those towns. Very different from Lane’s rough trips on the plateau was this march. Near Córdoba flourished such genuine tropical wonders as the bread tree, the butter tree, the milk tree, and a kind of palm called “the traveller’s friend,” which covered the wayfarer’s head with a tent, and quenched his thirst with abundant sap. Going on, one found enormous masses of vegetation—thick, matted, boundlessly prolific—moulded into astonishing yet harmonious forms by the bays and promontories of the rapidly mounting foothills; terraces of luxuriant foliage piled on sheer cliffs, castles on the terraces, and cathedrals on the castles; verdure, verdure everywhere, dripping, flowing, spurting, tumbling in every hue and shade of green, with a dark, velvety mist in the gorges that became clear sapphire when the sun touched it, and here and there a cascade letting fall its crystal thread from a mossy crag.[25] Then came the rich Orizaba valley, hemmed in with jungles, and winding off between sombre, precipitous mountains until lost in the dreamy distance; and above it the sparkling snows of the vast peak sent down a torrent of gray glacier water, that leaped into mid-air, and then, gathering itself below, wound on through splendid, odorous trees full of parrots, canaries and mocking-birds, hurried past fragrant orange groves and still more fragrant blossoms, poured through the arches of a noble old bridge, and buried itself in the woods. But the Americans did not forget their orders. Both cities were occupied without resistance, and both were garrisoned; and the guerillas now found their proceedings considerably hampered. [25]
- 80. THE REMOVAL OF SCOTT The final military operations of Scott seemed thus rather tame, as was natural; but Polk executed one that could be termed startling, if not exactly brilliant. His principal assistants were Pillow, Worth and Duncan; and in different ways each had excellent qualifications for the work. Pillow was not “The Lie Incarnate,” as Trist believed, nor even “a perfect ass,” as many thought; but vanity, ambition, lack of probity, and a gift for dark and cunning methods characterized him. His instincts and talents, indeed, were those of the criminal lawyer who minds nothing about his case except the verdict. When the President’s brother shot a man down in the street at Nashville, Pillow got him off. With reference to his work in helping bring about Polk’s nomination at Baltimore he wrote, “The fatal blow was given, but it was not seen nor known what produced such a result—nor where the blow came from.” “I feel as boyant as the air,” he said in December, 1846, when great dissatisfaction with Taylor prevailed at Washington, because I know “that I have done the work.... I have paid him in full” for his treatment of me. And one could seldom get a finger on Pillow’s back, when he was not wriggling actively toward some object of selfish desire.[26] Without a particle of real military ability or success to his credit, he now stood second in our army, and hence logically enough saw no reason why he might not, by some devious path, arrive at the first position and even at the Presidency. “Modesty,” said Burke, “does not long survive innocence.” To plant such a person, with urgent recommendations, at open, big-hearted Scott’s right hand, to win his confidence, to spy upon, criticise and undermine him, and inevitably to scheme for his place, was indecent; but Polk did it.[26] Very unlike Pillow was the courtly and fascinating Worth; but his mind was intense, narrow and self-centred. After the battle of Monterey he exclaimed, “I am satisfied with myself. The most vindictive foes crouch at my feet, and my friends choke with joy and delight.” And there is one sin of which even angels are capable, we have been told. All his military recognition he owed to Scott, but probably the debt weighed heavily on his proud and restive nature; and, while apparently reciprocating the genuine affection of his chief, he had inwardly rejected Scott’s principles and methods nearly thirty years before the Mexican war. Regarding his friend, fellow New Yorker and brother Democrat, Marcy, an adroit politician, he felt very differently. “I would not give an ounce” of his wisdom, he wrote in June, 1846, for all Scott’s glory; Scott “is determined to sink and draw his friends down with him.”[27] At Vera Cruz the commander-in-chief, relying on their long intimacy, told Worth frankly that he believed the administration intended to ruin him, and the subordinate officer evidently determined not to be drawn down. At the same place a brother officer suggested to Worth a higher position than was even the highest in the army. The suggestion appears to have struck root. All military men believed the next President would be one of them, and what commander had acquired a more
- 81. THE REMOVAL OF SCOTT brilliant reputation? The New York Sun recommended him for the place; and the idea of his candidacy was favorably received by many. This prospect naturally turned him still more against his old friend, for either Scott or Taylor seemed almost certain to be the Whig nominee. Through a series of clashes, for which little—if any —justification can be seen, and in spite of Scott’s efforts to conciliate him, Worth proceeded then to gain emancipation from his burden of gratitude, and place himself in open antagonism to his former patron.[28] Duncan’s motives were different again. He was intimate with Worth; and Pillow, who offered to marry the Colonel to a rich and handsome widow, doubtless promised him the post of inspector general. At any rate he urged Polk to make the appointment, hinting at other reasons than mere qualifications, and it was made; and we know that Duncan gave himself much trouble to assist Pillow as a partisan supporter. The power of such a combination, headed by the President himself, to gather adherents from the many ambitious officers hardly needs to be pointed out; and finally there were, of course, jealous and envious men. “Since we cannot attain to greatness, let us revenge ourselves by railing at it,” said Montaigne for the benefit of such persons; and many of the officers knew that greatness was beyond their powers. None of them could monologue as Scott did; none could look in a cocked hat as he looked; none had won the Mexican war; and, moreover, he was the sole general-in-chief.[29] The result was a powerful movement against the prestige and authority properly belonging to Scott. Pillow’s reports on the battles of Contreras and Chapultepec tended to represent the General as a nonentity; and Worth not only did somewhat the same, but referred to the Commander in terms of ridicule and contempt. A letter, doubtless written directly or indirectly by Pillow over the signature of “Leonidas,” extolling Pillow shamelessly and belittling Scott, was trickily worked into the New Orleans Delta of September 10, 1847; and another letter, containing a passage intended to show that Worth and Duncan had saved Scott from choosing the wrong approach to the capital, appeared in the United States, then in a Tampico newspaper and finally at Mexico City. Both letters were grossly improper, especially since the army lay in the enemy’s country; and Scott found it necessary to act. As he well said, “The general-in-chief who once submits to an outrage from a junior, must lay his account to suffer the like from all the vicious under him,” and “even the great mass of the spirited, intelligent, and well affected, among his brothers in arms, would soon reduce such commander to utter imbecility, by holding him in just scorn and contempt” for his recreancy to himself and the country.[30] On November 12, therefore, he issued his General Orders 349, which aimed to stigmatize these offences in such a way as to prevent a recurrence of them. Duncan then assumed in a plainly defiant manner the paternity of the Tampico letter, although in fact the offensive passage had not been written by him. His
- 82. primary object in doing this was evidently to give Worth a handle, and the handle was promptly seized. One thing led to another; and in the end formal charges were brought by the commander-in-chief against Pillow, Worth and Duncan, and by the two generals against him; appeals—insulting to Scott—were made by Pillow and Worth to the government; and the technical “arrest” of the three officers followed. [31] The government then stepped in. Scott had no doubt given it offence during the campaign, for his letters had plainly enough revealed a conviction that Polk had broken faith with him, and purposely thrown difficulties and annoyances across his path; but the circumstances had appeared to warrant his complaints, and Marcy had at least “got even” by administering liberal censures in reply. The balance in fact—aside, perhaps, from a mere acerbity of language—was against the administration. Besides, having served the country well and saved the government from disaster, Scott was entitled to some indulgence for irritation caused by the peculiarly trying circumstances that surrounded him. He was a large man, had done a large work and merited large treatment. But there was nothing large about the administration. The confines of mediocrity hemmed it in. Pillow and Duncan were therefore by its orders relieved of arrest; Worth was not only released, but assigned to duty according to his highest brevet rank; and “in view of the present state of things in the army,” chiefly or entirely caused by Polk’s agent and Marcy’s friend, Scott was deposed. He had performed his task, said Robert E. Lee, and now was “turned out as an old horse to die.” April 22, 1848, amidst the lamentations, cheers and blessings of the army as a whole—trembling himself with emotion—he took his leave, and Major General Butler, who was a Democrat and looked well on a horse, bore sway at headquarters.[32]
- 83. XXX THE NAVAL OPERATIONS OF THE WAR 1845–1848 In January, 1846, the United States had available for naval hostilities one ship-of- the-line, seven frigates and razees, fifteen sloops-of-war, six brigs, one schooner and three steamers—that is to say, thirty-three war craft. As ships-of-the-line carried more than seventy guns, frigates about forty-four to fifty, sloops twenty, brigs ten and other vessels in proportion, this fleet had 1155 cannon. Two of the vessels, under Commodore James Biddle, were on the coast of Asia; several occupied the Brazil station; and five cruised in African waters to check the trade in slaves. The Pacific squadron, commanded by Commodore John D. Sloat, comprised on July 1, 1846, the frigate Savannah, the sloops Portsmouth, Levant, Warren and Cyane, the schooner Shark and the storeship Erie, to which the frigate Congress, the razee Independence and the sloops Dale, Saratoga and Preble were added later in the year, while the Levant went home; and substantially all the rest of the fleet, known as the Home Squadron, attended to the West Indies and Gulf service, under Commodore David Conner.[1] The appropriation for the year ending with June, 1846, was a little less than ten millions, but only about six and a half millions were expended. The war bill of May 13 permitted the completion of all vessels then building and the purchase of others; and by November, 1847, after suffering a number of losses, the navy had in commission five ships-of-the-line, one razee, four frigates, thirteen sloops, six brigs, eleven schooners, four bomb-vessels, twelve steamers and six storeships.[1] The peace establishment created by Congress in 1844 provided for 7500 petty officers, seamen, landsmen and boys, and in August, 1846, this number was raised to 10,000 for the period of the war; but owing to the remarkable activity of the merchant marine and the consequently high wages, men could not easily be obtained. During the most important year—November, 1846, to November, 1847— not over 8000 were in the fleet at any one time. The whole number of seamen employed in the course of the war did not exceed 7000; and hence plans to strengthen our forces in the Gulf and the Pacific had to be curtailed. The service, too, did not enjoy unqualified popularity. In the sailor’s decalogue appeared this commandment:
- 84. PRIVATEERING “Six days shalt thou work And do all thou art able, On the seventh thou shalt holystone The deck and scrape the cable”; and the cannon had to be rubbed with fragrant “sea pitch” from the bottom of the ocean until they shone like Japanese lacquer. Discipline, therefore, not reinforced by the enthusiasm and the necessities of war, fell considerably below its reputation, and the crews were eager to be free when their time expired. The officers, even, had become lax after thirty years of peace, and in too many instances their standards of conduct had given way.[2] In the administration of the navy, also, the effects of a long peace could be seen. The control of matters had fallen, though not by accident, into the hands of shrewd officers deeply interested in themselves and their friends. Supernumeraries abounded. Those who drew the most pay often rendered the least service. The pet ambition was for a safe, quiet and easy position. Shore billets were too numerous. No field officer of the Marines had cruised since his promotion, and one of them had been in the service more than a generation without going to sea. Secretary Bancroft, eager for distinction, undertook to eliminate the abuses, but only succeeded in eliminating himself. He had taught Greek, and was ridiculed by the naval men as undertaking to play the pedagogue over them. Having no dominating force of character nor even a commanding presence, he could not stand against the governing clique. The requirements of the war, which might have assisted an abler administrator to win the day, only increased his difficulties. The Senate refused to confirm some of his appointees; and early in September, 1846, he became our minister to England.[3] J. Y. Mason, who succeeded him, was a fat, easy, agreeable man, quite innocent of the desire to achieve reforms. Nobody disliked him, but nobody felt obliged to obey him; and as late as the twentieth of February, 1847, suddenly discovering that Scott had mentioned certain designs of the army against Vera Cruz, he awoke to the fact that his department had failed to give the anticipated assistance. Just what could be expected of the navy under all these prejudicial conditions was, therefore, in some minds, a little uncertain.[3] One of the most serious duties imposed upon it was to guard against privateering, for not only our commerce but the supplies required by our troops depended upon free lanes. About the middle of 1845 the government issued orders that any activity of such a kind on the part of Mexico should be considered the signal for war; and as a deterrent it was announced by the newspapers, though incorrectly, that privateersmen were to be regarded as pirates. Crews not predominantly composed of Mexicans, it was often asserted, could legally be “strung up to the
- 85. yard-arm,” since we were understood to have treaties that sanctioned this principle with most countries.[4] After the war actually began, a great deal of danger was apprehended. Desperate characters were believed to be waiting at New Orleans, and “piratical gangs” in the ports of Cuba, where Almonte seemed to be at work. News arrived early in August, 1846, that privateering regulations had been issued by Mexico, and suspicious craft soon appeared off Key West. In December the Mexican minister of war openly avowed that great hopes of injuring the United States in this manner were entertained. Blank certificates and commissions reached Washington; information regarding efforts to set vessels at work in various quarters arrived there; and finally the Carmelita of Bangor, Maine, was captured near Gibraltar by a felucca named El Unico, fitted out at Oran, Algeria, and run by Spanish desperadoes.[4] Mexico had not in reality, after studying the subject with deep interest, much expectation of accomplishing any large results by issuing letters of marque, and the regulations of July, 1846, were intended principally or wholly to annoy this country; but in September and October she took the matter up rather seriously. A new law provided that any foreigner entering her naval service might become a Mexican at once, and blank naturalization papers as well as thousands of privateering commissions, duly signed but not filled out, were carried by agents to the West Indies, Great Britain, France and Spain. Almonte did his best at Havana. J. N. Pareda, appointed Mexican chargé d’affaires at Madrid, appears to have circulated the documents actively in the Peninsular ports; and another privateer, a Spanish steamer named La Rosita, put out from Oran.[5] On the other hand, the representatives of the United States insisted upon our treaties and the obligations of neutrality. Polk’s annual Message of December, 1846, denounced the Mexican plan as inviting “all the freebooters upon earth,” who felt like paying for the privilege, to cruise against American commerce, announced that our own courts would say whether such papers could protect them from the pirate’s doom, recommended that Congress provide at once for the trial of Spanish subjects caught in such business, and suggested American privateers—intended mainly to recapture vessels taken under Mexican letters. An American force hastened to the Mediterranean, and our squadrons were expected to seize all the rovers putting out, as well as intercept all prizes on their way to the enemy’s ports. These precautions looked rather discouraging to enterprising desperadoes.[6] In England there was a feeling, as will appear later, that Mexico should be allowed the utmost license against us, and the Mexican minister at London received many applications for letters; but Great Britain did not really wish her supplies of cotton to be endangered, and all the seas to be filled with corsairs preying upon the trade of the world; and in October, 1845, her minister to Mexico was instructed to prevent that country, if he could, from issuing letters of marque indiscriminately. Bankhead protested also, as did the Spanish minister, against important features of
- 86. THE BLOCKADE the regulations. Palmerston himself, though he acted in a languid fashion, and gave notice at Washington that British subjects, found on Mexican privateers, could not be treated as pirates, announced that his government would faithfully do its duty.[7] France was prompt and active in responding to our demands. Spain, placed under stringent obligations by the treaty of October, 1795, promised full compliance with its requirements, captured El Unico, punished its crew, pursued La Rosita, and ordered O’Donnell, the captain general of Cuba, to act as her obligations required; but she accepted Pareda, the colporteur of what was piracy under her agreement, as consul of Mexico; and the captain general, while he convinced the American representative of his good-will and in fact would not permit an open violation of the treaty, suggested to the Mexicans ways—fortunately impracticable—of evading his own rules. But the risks of privateering under so many embarrassments and the virtual impossibility of converting a prize into cash, prevented all attempts except the feeble ones already mentioned. In this field, consequently, our navy, though incessantly watchful, could acquire no laurels.[7] Another aspect of the situation concerned it more seriously. On the day Congress passed the war bill (May 13, 1846) orders were issued to blockade the ports of Mexico. Several definite aims prompted this action. Primarily, of course, it was desired to prevent supplies of all kinds from reaching the enemy, and to deprive them of the almost indispensable revenues obtained in peace by taxing imports; but there were also hopes that loss of business would induce Great Britain and France, which had a profitable trade in that quarter, to urge upon Mexico the acceptance of our terms. The blockade was therefore to be enforced vigorously. At the same time neutrals were to be treated with all reasonable indulgence. Theoretically only their war vessels had the right of entering closed ports, but practically the intention was to broaden that narrow door considerably. Toward itself, however, the United States determined to be strictly faithful in observing its declared principles. Merely those ports where the order could become effective were in view. The announcement of blockade was to be made as public as it could be; and in particular the government required that a full warning should be given to neutral ships.[8] For the work thus imposed upon him Conner had ample time to prepare. As early as August, 1845, he was directed to blockade the Gulf ports in case of war; early in 1846 he knew of Mexico’s attitude regarding Slidell; before the end of March his vessels occupied convenient positions; and promptly on the outbreak of hostilities a blockade was announced at the chief harbors. By July, with some assistance from the revenue service, it extended from the Rio Grande to the Goatzacoalcos.[9] But the difficulty of making it continuously effective proved to be extraordinary. There were not vessels enough of the proper kind; occasionally a more or less complete concentration became necessary; and accidents of many sorts occurred.
- 87. Uncharted shoals and rocks, currents of unknown direction and force, the frequent haze, and the darkening of the lighthouses made extreme caution imperative. The suddenness and violence of the storms almost surpassed belief. At Vera Cruz the Somers was blown over and sunk before Semmes, her able commander, could take steps to avert the disaster. Even at the anchorages one would suddenly hear on a calm afternoon the clarion orders of the speaking trumpet; the ship would quiver and reverberate as the cable of the heaviest anchor ran swiftly out; in a moment the storm would burst; and for days it might be a question almost hourly of going ashore. At such times all sailing vessels on patrol duty had to make instantly for the open sea, and before they could return to their stations a lurking blockade runner could perhaps enter the port. Owing to such difficulties Alvarado and Frontera, for instance, could not be watched continuously.[9] Embarrassments also of a minor yet serious character had to be encountered. Our vessels, unlike those of England, were designed exclusively for war, and long confinement impaired the efficiency of the men. The government supplies of eatables needed to be eked out from New Orleans huckster boats and European merchant ships. Water could not be obtained readily from a hostile shore. At the Antigua River, in July, 1846, the boats going up with casks were fired upon, and such affairs had to be expected. Vessels were despatched long distances occasionally to obtain fresh provisions, but even then scurvy of a most serious nature broke out in the summer of 1846, disabling some of the largest and most efficient ships for several months. The Raritan had more than 200 cases. Nearly all on the Potomac suffered. The Falmouth had to go as far north as Boston to throw it off. Swampy shores and kelp rotting under the torrid sun produced myriads of poisonous as well as otherwise annoying insects. During a brief stay in the river off Tampico nearly all the officers and men contracted ague, and the yellow fever scourged a number of the vessels. More than two thirds of those on the Saratoga had the latter disease. In August, 1847, the Mississippi left her station with some 200 men suffering from it.[10] Being strangers and enemies, the Americans labored under peculiar disadvantages. The people gladly assisted blockade runners in every possible way. Spanish captains in particular, having friends on shore and pilots thoroughly familiar with the coast, could not be prevented from reaching harbor at night or in thick weather by way of the shoals. Sometimes it looked, for one or another of these many reasons, as if our officers were careless or incompetent. Army observers, not well informed regarding the conditions, felt disposed now and then to pronounce the blockade a humbug, and naturally some foreigners did so. This opinion had neither truth nor probability in its favor. But naturally, in view of all the circumstances, it proved more satisfactory to occupy the ports, and open them to commerce on the basis of a reasonable contributory tariff.[11] Besides cruising to watch for privateers and hovering off the chief harbors to maintain a blockade, our fleet was expected to share in the general offensive. For
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