Logic paper

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A STANDARDIZED LOGIC NOTATION FOR EVERYDAY CLASSROOM USE
Abstract: There are two main purposes of this paper. The first is to describe a simple and intuitive standardization of
logic notation, both for foundational courses which develop symbolic logic and proof styles, and for general everyday
use in undergraduate and even high school mathematics courses, particularly calculus though there is no reason
students cannot learn it and benefit from it in their algebra and geometry courses as well. This leads to the second
purpose of this paper, which is to argue that courses which do not usually include logic notation formally could
nonetheless benefit from widespread use of it, particularly if it is standard and thus truly portable. Several examples
are given. (The author assumes the reader has seen elementary treatments of logic symbols and truth tables, for which
there are many simple texts and online sources, including the author’s own homepage listed at the end of the paper.)
For students studying undergraduate mathematics, it is common and proper to encounter
logical symbols, such as connectives , Λ, V, →, and ↔, in courses specifically
containing logic and truth tables as topics, in other courses relying on the ability to
manipulate these symbols, or just incidentally as an instructor uses them to help
communicate mathematical ideas in lectures for any course. Some introduction to these
symbols, along with truth tables, historically occurred in public education in other
countries such as Taiwan as a standard topic by the end of secondary school. The
usefulness and beauty of symbolic logic often motivate U.S. instructors to introduce the
subject at some low level to students, even in the secondary schools, though it is rare for
the actual classroom texts to discuss abstract, symbolic logic except in courses involving
proofs, and then only well into the university level. In particular, algebra and calculus
texts almost completely avoid the issue, apparently appealing instead to the students’
“common sense.”
Done well, the use of symbolic logic can bring much needed clarity to a topic or course,
while done casually or incoherently it can deepen confusion. To further confound
matters, there is no universal standard for notation or its use, so what little a student
absorbs in one course might not be as portable as he or she would like. I address all of
these problems with my proposed standard, but I first address the problems themselves in
more detail.
Most authors and instructors who employ symbolic logic in their teaching do use the
connectives , Λ, V or minor variations similarly. However, some then use ⟹ and ⟺
where others use → and ↔. To indicate logical equivalence of compound statements,
some authors use the symbol ≡ (used elsewhere to indicate function identities or equality
by definition), while others use ↔, ⟺, or “iff” meaning “if and only if.” (Other
notations can also be found but are less common.) In lecture notes, it is not unheard of to
see ⟹ when ⟺ is also correct and therefore more precise (or even intended), and when
it would be very instructive to the students to see the differences between the two;

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similarly if ↔ and → are used instead. Indeed, often in some computational argument a
string of equivalence is broken, as in f (x) = x ∙ x ⟺ f (x) = x2
⟹ f ′(x) = 2x , where of
course we cannot replace ⟹ with ⟺. In a calculus context it seems inadvisable to use
the symbol → in-line because of its ubiquitous use in limit arguments. Even when we
restrict the discussion to the use of logic symbols and their meanings, we have contexts
where P → Q is supposed to be understood to be a tautology (always true), and others
where it is meant to have truth values that may be T or F, depending upon the context.
Similarly with P ↔ Q.
The simple notational proposal put forward here is this:
That we use → and ↔ as connectives, i.e., logical operations, which can
return either truth values T or F, and ⟹ and ⟺ only when the
corresponding connectives return tautologies.
This is not wholly original (see for instance Finite Mathematics and Its Applications,
Goldstein, Schneider, and Siegel, Prentice Hall), though I developed the convention
independently (being later pointed to the earlier development), arguably because it seems
like the next reasonable step in everyday logic notation that can be used throughout
mathematical presentation and writing within the classroom. However, while it has been
tested and used in my Calculus 1 course repeatedly, it has clearly not yet gained the
promulgation to or acceptance in the classroom advocated here. This seems to be because
of its present obscurity, perhaps due in part to the fact that most professional
mathematicians and mathematics educators who have had some exposure to logic and
truth tables understand an author’s intent in using logic symbols from the context, and so
one standard (or even no standard) might seem as legitimate as another. It is often only
after attempting to use them coherently in the classroom that one discovers we mean, for
instance, different things by “implies” in different contexts. It is my hope that some
demonstrations of this standard’s usefulness in calculus, and even algebra, as seen here
will motivate instructors to consider adopting it in the future.
The connectives will be referred to here as “single-line arrows” and the other symbols as
“double-line arrows.” Thus one may write
P ↔ Q ⟺ (P → Q) Λ (Q → P) (1)
P ↔ Q ⟹ P → Q. (2)
While the first of these (1) would normally be proved using truth tables by showing that
P ↔ Q and (P → Q) Λ (Q → P) have the same truth values for each of the 22
= 4 possible

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combinations of truth values of the P and Q, it should be noted that it can also be proved
by showing that [P ↔ Q] ↔ [(P → Q) Λ (Q → P)] is a tautology:
P Q P ↔ Q P → Q Q → P
(P → Q)
Λ (Q → P)
[P ↔ Q]
↔ [(P → Q)
Λ (Q → P)]
T
T
F
F
T
F
T
F
T
F
F
T
T
F
T
T
T
T
F
T
T
F
F
T
T
T
T
T
Of course it is easier, and time saving, to note simply that the third and sixth columns of
the truth table have the same value, and to spare the effort of producing the final column.
However it is worth noting that showing two compound statements have the same truth
values in all cases is equivalent to showing that the connective ↔ operating on the
compound statements will always return T.
To rephrase, the spirit of the longer method of showing equivalence is the notion that ⟺
means that if its place is taken by ↔ we get a tautology, after adding necessary grouping
symbols. This can be extended to implications, in which ⟹ is taken to mean that, if
replaced by →, we get a tautology. So to prove (2) we construct a truth table as follows
(note how the second and third cases have ( )→( ) returning T “vacuously”):
P Q P ↔ Q P → Q (P ↔ Q) → (P → Q)
T
T
F
F
T
F
T
F
T
F
F
T
T
F
T
T
T
T
T
T
When verbalizing → versus ⟹ in the classroom, I usually simply use “implies” if there
is no ambiguity, and “single-line implies” versus “double-line implies” if both are
present. Perhaps better diction can be developed eventually, such as “implies” versus
“strongly implies.” For ↔ versus ⟺, one could use “bi-implies” versus “if and only if,”
while I am careful to reserve “is equivalent to” for ⟺. Another acceptable terminology
could use “logically implies” and “is logically equivalent to” for ⟹ and ⟺, respectively,
though no doubt there are other good terminologies possible. The symbols , Λ, and V
of course are verbalized as “not,” “and” and “or,” respectively.

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Now consider compound statements P = P (P1, P2..., Pn) and Q = Q (P1, P2..., Pn), where
the P1, P2..., Pn are assumed to be independent component statements, with 2n
possible
truth value combinations of T and F. When it is appropriate to write P ⟺ Q, or P ⟹ Q
for compound statements P , Q, I call these, respectively, a valid equivalence and a valid
implication. Otherwise they would be fallacies, and perhaps, unlike statements using the
connectives ↔ and →, we would usually avoid writing the fallacies (just as we avoid
writing 2 = 3) except for demonstration purposes. We can use this notation to display
valid argument styles in-line. For instance consider modus tollens, i.e.,
which becomes (P → Q) Λ ( Q) ⟹ P. A check with a truth table construction shows
[(P → Q) Λ ( Q)] → ( P) is indeed a tautology. (I have found that using extra
“grouping” characters ( ) and [ ] expedites the learning of logic by my calculus students,
and thus defers the whole discussion of an “order of operations” until they are more
comfortable with the notation. Note that in written English the standard grouping
symbols are single and double quotation marks, which limits the practical level of nested
grouping available. Of course spoken English is often quite ambiguous.)
The symbols ⟺ and ⟹ are robust enough to use in many contexts, including algebraic
contexts, in which they can very much enhance student understanding of the logical
nature of simple mathematics. For example,
Students witness where we “broke” our equivalence, allowing us (at this point) only to
conclude that , and thus we have a need to
“check” our candidate answers, discovering that will work in the original while
will not, so we finally conclude . Of course this kind
of thing happens all the time, and in fact we can prove (as above) that P ⟹ P V Q, as for
example .
We can also point to cases in algebra and elsewhere for which there is no need to check
solutions (except for actual errors), as for instance when we have a polynomial equation
that we solve by factoring, or a linear equation we solve by steps which only yield

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equivalent statements. Knowing when we do not need to check the solutions except for
errors can be a useful skill in its own right.
At this point in the discussion it is interesting to show students how, if we have P ⟹ Q
and Q ⟹ P, then we can write P ⟺ Q. (It is reasonable on its face, and can also be
shown with creative use of truth tables.) This fact is also often useful, such as when we
show how the three “row operations” in Gauss-Jordan elimination produce systems with
the same solutions, because the operations are reversible. More specifically it is easy to
show that a solution of the original system is contained in the solution of the transformed
system, and that “going backwards” we see the inverse operation of the transformation
gives that the solution of the transformed system is contained in the original system’s
solution. However, differentiation–much like squaring both sides of an equation–is not
quite reversible.
Similarly, . This simple example reminds
students of two important, often forgotten ideas: that squaring both sides of the equation
can lose important information about the signs of the original expressions (the
technicality which can be buried in the process, as in our original algebraic example (3)),
and that . Students are also interested to note that we do not have
the same difficulty when cubing both sides: , assuming
.
More simple examples where we lose equivalence can be found in solving logarithmic
equations, such as the following:
A quick check shows is not a solution while is a solution. We lost
equivalence in combining the logarithms, since when we have
while . In fact, the first equivalence is only valid because (for the
direction ).
When dealing with quantifiers, it also seems appropriate to keep the single-line-arrow
connectives “inside” the quantified statements. For instance, since ,
we can write

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.
Perhaps less clear at first is the one-way nature of the implication
.
Once students learn about negating quantified statements and implications P → Q the
following is relatively easy, assuming f is defined:
In fact, to be precise we should note that is actually a statement about
f being defined at x and a, as well as the inequality holding true. Thus we can replace
in the continuity with the equivalent statement
or simply understand that it can be so replaced, and then the negation will include
common cases where is undefined, is undefined or . This
uses , and the obvious extension to three statements:
, which are unambiguous because of the
associative natures of and .
Of course, quantified statements do not conform to the usual truth table based analysis.
Indeed, a quantified statement is either true or false, i.e., a tautology or a contradiction. In
this standard the symbol is appropriate between any two “true” quantified statements,
or between any two “false” quantified statements.
It can also be useful to define a symbol to represent a generic tautology, and another for a
generic contradiction. For these purposes I define “script-T” and “script-F,” or and ,
respectively. For instance, ,
and so on. These allow for some interesting computations. For instance, if we wish to
reason one level above truth tables, we can use well known equivalences to prove say,
by showing :
Of course there are some results which surprise students at first, such as
, and . However some are quite reasonable, such as

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, which is perhaps best shown with a truth table, which will have
possible truth value combinations to check, since can have two truth values while
has only one:
T
F
F
F
F
T
F
T
T
T
In fact, a quick glance at the truth table reveals that , another interesting
computational result for student meditation.
I have already begun to make my case for the second proposition offered here, which is
this:
That secondary or early undergraduate mathematics education include a
standard and coherent introduction to symbolic logic, to improve the
clarity of the material and general discussion within textbooks, lecture
discussions and homework.
I have already pointed out how some confusing topics in algebra and calculus can be
clarified by the arrows showing the flow of implication, whether it be two way or
one way . In my own calculus courses over the last four years, I have seen enough
benefits to justify spending four complete lectures on an outline of symbolic logic
notation and its use. The outline I have used for those four lectures is as follows:
1. the operators and truth tables with examples from everyday life;
2. logical equivalence with several examples which are mostly intuitive upon
reflection;
3. valid implications and arguments along with symbols for generic tautologies
and contradictions ;
4. and finally a short introduction to quantified statements and their negations, some
from mathematics and some from everyday life, such as how to negate “for every
man there is a woman who loves him,” using quantifiers. (Students seem to
benefit from such contextual examples in all four lectures.)
With these and a few other innovations which work well with the notation, much clarity
can be gained. For instance, the definitions of continuity and a finite limit at a point are
less mysterious after some brief experience with quantifiers:1
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One could also argue that we should make some distinction between equivalence by definition and
regular logical equivalence. This suggests a possible third standard, that be only used for definitions,
which is similar in spirit to its use when we say, for instance, in other contexts:

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f continuous at
.
From here students can see how the limit definition is silent at the point (since
means , and thus is vacuously
true), while continuity includes the case . They can also see how the part of in the
limit definition is played by in the continuity definition. Both definitions cut
through the seemingly ad hoc nature of the explanations one often needs to demonstrate
the idea of limits in all the various cases without using - definitions. Many questions
can be answered by referring back to the definitions, thus achieving coherence across
examples, at least for the cases of a finite limit at a point.
In fact I have used symbolic logic in - proofs for continuity at points with some
additional success (compared to not using it) in Calculus 1. For instance, suppose I wish
my students to prove that the function is continuous at the point .
The exercise is broken into two parts: the “scratch work” and the “proof.”
Scratch-work: We want to follow from our choice of . We work
backwards from that statement, with .
Now many texts explain that because we have equivalences, we found our . However
this is not a good strategy if we look ahead to nonlinear functions, so instead it is useful
to write a stand-alone proof.
.
Of course, where we have “ ” as a logic symbol, we also have “ .”

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Proof: For any choose . Then exists and satisfies
Within the proof, students can trace the definition of continuity piece-by-piece, though
some “pieces” are longer than others.
Suppose instead we wish to prove that is continuous at . We can attempt
the same strategy but it quickly falls short, until we remember which direction of “ ”
we actually need.
Scratch-work: Here and . We therefore want to choose such that
Working backwards as before we get
Of course is not constant, so we are stuck until we realize the direction we really
need is in the above i.e., We can also have
statements before and after our antecedent if the arrows continue to flow the
correct direction. Of course we accomplish our implication by employing some a priori
assumption, for instance that . Then
If this last quantity is less than when , we will have our proof, and that can
happen if , and this last part is true if . We note that we keep
thinking, “that can happen if,” and so we are in fact working backwards again from
, except that (1) we do not have equivalence, but our “arrows” are
pointing towards , and (2) we in fact require two things to be present
to accomplish . Indeed, we need to accomplish both
, and we do this by taking . From there we have our proof.

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Proof: For , choose . Then exists and satisfies
The level of sophistication required of calculus students to understand such a proof, and
moreover reproduce such a proof, is less remarkable when the mathematics is put
precisely and concisely with the aid of such a system of symbolic logic. Unfortunately
topics such as - proofs are more and more commonly excluded in high school calculus
since the Advanced Placement calculus exams no longer cover these. In both high school
and college, this loss dulls the aura of sophistication of the calculus. I am left to conclude
that the successful re-introduction of these topics, particularly with the aid of this system
of symbolic logic, has the potential to improve the mathematical sophistication of our
students and to prepare them better for more challenging mathematics in future studies.
However, there are pitfalls to avoid when introducing symbolic logic to students for the
first time in, say, a first-semester engineering calculus or a senior high school calculus
course. First, it should be done with some completeness; a piecemeal introduction,
particularly “on the fly,” will likely cause more confusion with most students than it is
worth. Second, while most students will enthusiastically embrace symbolic logic, they
will also need some correction, as the students often write “ ” where what is meant is
“=,” for instance, so some patience is necessary, but in that sense symbolic logic is no
different from other topics. Third, some theorems and arguments instructors take for
granted can have surprisingly sophisticated logical structure when forced into symbolic
form, and the instructors will need to work these things out carefully before presenting
them. While students can learn to produce an - proof of continuity for a linear case
reasonably quickly (though not immediately), it becomes much more interesting for the
nonlinear cases, where must be restricted a priori. Part of the solution is to keep the
“scratch-work” seeking and the actual - proof of continuity separate, which
is less necessary but still a good practice for the linear case. (This practice of separating
scratch-work from the final proof is hardly unique.) Fourth, some rewriting of lecture
notes and some reshuffling of order and priorities may be required to take full advantage
of the notation. I have had reasonable success by introducing the more intuitive (in this
context) topic of continuity before the less intuitive limits, requiring some - proofs of
continuity at points (the hardest part of the entire semester for some), then defining
continuity on intervals and giving the usual theorems there (including the Intermediate
Value Theorem and using it to solve inequalities), and then “breaking” the continuity and
introducing limits to describe the behavior of discontinuous functions, as well as the
continuous ones, but mostly excluding - proofs for limits in the exercises. Finally,
some time is lost initially by including a primer on logic, so some consolidation may be
required. For instance, I often introduce some easily grasped calculus facts in earlier
contexts than usual--such as the fact that on an interval implies is (strictly)

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increasing on that interval while implies decreasing--and use these for some
simple graphing problems long before the Mean Value Theorem is introduced to prove
them. In other words, introducing logic does not have to turn calculus into rigorous
analysis, though it naturally lends itself to increasing the rigor somewhat. An instructor
can judge the level of rigor appropriate for individual topics and classes.
These pitfalls are also arguments for introducing symbolic logic earlier, perhaps in
geometry but one could argue it is not inappropriate even within Algebra 1. Indeed, it is
a topic which stands alone, as many who teach mathematics for liberal arts students
discover when teaching college mathematics. Students who are afraid of algebra are not
necessarily afraid of symbolic logic. While exposure to symbolic logic might not be a
sufficient cure for this fear of algebra, it can help students to think mathematically, which
bodes well for their future success.
In the end, an introduction to logic symbols and their manipulations can help students to
clarify their own thinking, help illuminate the flow of a mathematical argument, and
better illustrate the idea that mathematics is a language of sorts. This is assuming the
logic is introduced in a somewhat complete and coherent fashion. It would also help if it
were “portable,” i.e., standardized, as well. Simply using as connectives, and
when the respective connectives would return tautologies as well as between
quantified statements where appropriate, can do much towards standardizing the
classroom-level use of symbolic logic, with the potential to make symbolic logic a
powerful tool for mathematics instruction.
One criticism of this approach is that we cannot expect students to perform the
calculations better it they are constantly worrying whether they should connect them with
. This criticism has some merit. A possible relaxation of the approach would be
to use throughout the process, until the problem is more or less complete, and then to
step back and notice where we in fact have . Even if is never used in a particular
classroom, using can give direction on the road map of a given argument (though then
the opportunity to explain symbolically why we sometimes must “check” the apparent
solutions, and other times we need not except for errors, is lost). When useful, the
standard presented here gives the instructor some structure to fall back upon, even if the
symbols are only resorted to when a worked solution can use some clarification.

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I would like to thank the reviewers for their comments and suggestions, one of which was
inserted almost verbatim into the text. This feedback certainly improved the content of
the manuscript. I would also like to thank Jacob Pool for typesetting my LaTeX
manuscript into Microsoft Word format for this publication. Readers wishing to see this
standard written more for student consumption are referred to my calculus textbook
project, Chapter 1, which can be found from my homepage
http://faculty.swosu.edu/michael.dougherty.

Logic paper

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