Logic 3rd Semester Notes.pdf. Bhu 3rd sem logic

Pranav__k91
Philosophy
Logic : BAO – 212
3rd
Semester
Unit - I
Introduction : Nature of Logic, Propositions, Arguments and their forms,
Deduction and Induction, Truth and Validity.
Categorical Propositions and Classes, Quality, quantity and distribution of
terms, Traditional Square of Opposition, Immediate inference.
Introduction
The word ‘logic’ comes from the Greek word logos, literally meaning,
word, thought, speech, reason, energy and fire. But in due course of
time these literal meanings were given up to make way for more
accurate meaning hinting at what we actually learn when we do logic.
This is how it came to be understood as a discipline dealing with
thought, reasoning and argument at different points of time.
➢ Logic is the study of method and principle used to distinguish
correct from incorrect reasoning.

Pranav__k91
Traditionally logic has been classified into two types 1) Formal and 2)
Material logic. Formal logic is otherwise known as deductive logic and
material logic as inductive logic. Formal logic is concerned with the form
or structure of argument whereas material logic is concerned with the
matter or content of argument. When matter is irrelevant, material truth
also is irrelevant. What matters in deductive logic is formal truth. By
formal truth we mean logical relation between the premises and the
conclusion. It is possible to know this kind of truth without knowing the
content of the argument. In this case, it is sufficient if the argument
follows the rules of the game. This whole explanation can be put in a
nut-shell in this manner. An argument consisting of only true
propositions can very well be invalid whereas an argument consisting of
only false propositions can very well be valid. It also means that in our
study of deductive logic it is possible to know whether an argument is
valid or not without knowing the contents of the argument (and many
times this is what precisely happens) provided we are in a position to
decide whether the argument has followed all the rules are not.
However, the case of material logic is different. In this case it is possible
to judge the truth or falsity of the conclusion only when we kow what
the argument is all about.
Nature of Logic :-
Science or Art
Questions have been raised on the issue whether logic is a science or an
art or both. Let us stay for a while on this problem. In ancient times
science just meant a systematic study of anything. But today the term
science has developed into a discipline distinct from several other
activities of mankind. Science has been defined as that branch of
knowledge which aims at explanation of phenomena. Used in this

Pranav__k91
technical sense, logic is no science at all. Does this mean that logic is an
art? Art is concerned with doing something. Logic, if defined as an art, is
so only in-derivative sense. In order to decide whether or not logic is an
art we have to consider the aim of logic. Is the aim of logic to give us
knowledge about valid argument forms or to make us better thinkers?
No one will deny that a study of logic results in improving our reasoning
ability. But there is a restriction. Just like a moralist who may not himself
be moral as a person, a logician may not be logical in his reasoning. We
can say that the effect of such a study is the acquisition of knowledge
regarding valid argument forms. It is not for logic to consider whether or
not this knowledge is put into practice. In view of this feature we can say
that logic is a science and not an art. It is a science not in the technical
sense, but in a general sense.
POSITIVE SCIENCE OR NORMATIVE SCIENCE?
Granted that logic is a science, what type of science is it? Science has
been classified into two types, viz., 1) positive Science and 2) normative
Science. Positive science describes what the case is. Normative science,
on the other hand, tells us what ought to be the case. Let us now
examine whether logic is a positive science or a normative science. Some
logicians consider logic to be a formal science and regard it as a
normative science. Just like object thought is made up of form and
matter. According to Latta & Macbeath ‘the form of thought is the way in
which we think of things, the matter of thought is the various particular
objects we think of. A form is something which may remain uniform and
unaltered, while the matter thrown into that form may change and vary.
A normative science attempts to find out the nature of forms (standards)
on which our judgments of value depend. Normative sciences have
before them a standard with reference to which everything within the

Pranav__k91
scope of science is to be judged. A normative science gives us judgments
of value, i.e., it tells us what ought to be the case. Logic has an important
normative aspect; but a norm or ideal in logic has a special meaning. The
main business of logic is to discover the general conditions on which the
validity of inference depends. In our discussion of logic we try to force
these conditions on the way of arguing. We do so because there are
certain objective relations between statements. This means that
statements must possess a certain structure and there must be certain
objective relations between them if our inferences are to be valid. These
structures of statements and their mutual relations are pure forms,
which serve as norms in logic. Traditional logicians while considering
logic to be a normative science meant that it is a science concerned with
those principles which ought to be followed in order to attain the ideal
of truth. Some other logicians consider logic to be a descriptive science
or a positive science and not a normative science since it does not lay
down any norm for thinking. Its nature is description as it aims at
describing and classifying various types of arguments.
In fact the classification of sciences into positive and
normative cannot be applied to logic.
Logic cannot be characterized either as positive or as normative science.
If logic were a positive
Sentence and Proposition
Sentence:- It is meaningful arrangement of words and it has a subject,
Predicate and a copula
Ex:- Shops are closed on Sunday.
⬇️ ⬇️ ⬇️

Pranav__k91
Subject Copula Predicate
Proposition :- A proposition is the basic building block of logic. It is
defined as a declarative sentence that is either True or False, but not
both.
Categories Of Proposition :-
Tautology : A proposition which is always true
Ex- Cats are mammals.
Contradictory: A proposition which is always false
Ex – All men are immortal
Contingent : A proposition which is true in some cases and false in some
cases Ex- It is raining.

Pranav__k91
Argument
With propositions as building blocks, we construct arguments. In any
argument we affirm one proposition on the basis of some other
propositions. In doing this, an inference is drawn.
Inference: A process by which one proposition is arrived at and affirmed
on the basis of some other proposition or propositions.

Pranav__k91
➢ In logic, argument refers strictly to any group of propositions of
which one is claimed to follow from the others, which are
regarded as providing support for the truth of that one. For every
possible inference there is a corresponding argument.
Premises : In an argument, the propositions upon which inference is
based; the propositions that are claimed to provide grounds or reasons
for the conclusion.
Conclusion : In any argument, the proposition to which the other
propositions in the argument are claimed to give support, or for which
they are given as reasons.
EXAMPLE : A well-regulated militia being necessary to the security of a
free state, the right of the people to keep and bear arms shall not be
infringed.
SOLUTION Premise: A well-regulated militia is necessary for the security
of a free state.
Conclusion: The right of the people to keep and bear arms shall not be
infringed.
Deductive and Inductive Argument
Every argument makes the claim that its premises provide grounds for
the truth of its conclusion; that claim is the mark of an argument.
However, there are two very different ways in which a conclusion may be
supported by its premises, and thus there are two great classes of
arguments: the deductive and the inductive.
Deductive :- An argument said to be deductive if its conclusion is
claimed to be necessarily follow from its premises.

Pranav__k91
➢ Conclusion is already in the premises
➢ Universal to particular
Ex :- Premises: All plants with rainbow berries are poisonous. This plant
has rainbow berries.
Conclusion: This plant is poisonous.
Inductive :- In Inductive argument , conclusion probably follow from its
premises
➢ Particular to universal
Ex :- Premise: All known fish species in this genus have yellow fins.
Conclusion: Any newly discovered species in the genus is likely to have
yellow fins.

Pranav__k91
Truth and Validity
An argument is valid if the conclusion follows from the premises. In logic,
truth is a property of statements, i.e. premises and conclusions, whereas
validity is a property of the argument itself.
Premise + Conclusion = Argument
True premises and a valid argument guarantee a true conclusion. An
argument which is valid and has true premises is said to be sound
(adjective) or have the property of soundness (noun).
Terms and Distribution of Terms
Term:- A term is a word or group of words which is either subject or a
predicate of a categorical proposition.
❖ There are two terms subject and predicate in each categorical
proposition A, E , I and O.
Distribution of Terms:-
□ A term is said to be distributed if it refers to all the members of a class.
For example, in this proposition “All dogs are animals”. The term ‘dog’ is
distributed for a it refers to all the members of the class dog.
□ If a term refers to few members of a class then it is called
Undistributed term. For example, in the proposition “Some students are

Pranav__k91
intelligent.” The term ‘students' are undistributed for it refers to only
some members of the class student.
Quality :- Quality of a categorical proposition describes whether the
proposition affirms or denies the inclusion of a subject within the class of
the predicate.
➡️ Affirmative
➡️ Negative
Quantity :- Quantity of a categorical proposition refers to the number
of members of the subject class that are used in the proposition.
➡️ Universal - All
➡️ Particular - Some

Pranav__k91
Square Of Opposition
❖ The doctrine of square of opposition originated with Aristotle in the
4th
century BC.
❖It is a diagram in the form of a square in which the four types of
categorical ( A,E ,I and O ) are situated at the corners , exhibiting the
logical relations (called opposition) among these Proposition.
➢ There are four types of relationship described by this square of
opposition
• Contradictory relationship

Pranav__k91
• Contrary relationship
• Subcontrary relationship
• Subalterns relationship
. ‘Every S is P’ and ‘Some S is not P’ are contradictories.
. ‘No S is P’ and ‘Some S is P’ are contradictories.
. ‘Every S is P’ and ‘No S is P’ are contraries.
. ‘Some S is P’ and ‘Some S is not P’ are subcontraries.
. ‘Some S is P’ is a subaltern of ‘Every S is P’.
. ‘Some S is not P’ is a subaltern of ‘No S is P’.
➢ Two propositions are contradictory iff they cannot both be true
and they cannot both be false.
➢ Two propositions are contraries iff they cannot both be true but
can both be false.
➢ Two propositions are subcontraries iff they cannot both be false
but can both be true.
➢ A proposition is a subaltern of another iff it must be true if its
superaltern is true, and the superaltern must be false if the
subaltern is false.
Given the truth, or the falsehood, of any one of the four standard-
form categorical propositions, the truth or falsehood of some or all
of the others can be inferred immediately.
• A is given as true: E is false; I is true; O is false.
• E is given as true: A is false; I is false; O is true.
• I is given as true: E is false; A and O are undetermined
• O is given as true: A is false ; E and I are undetermined.

Pranav__k91
• A is given as false: O is true; E and I are undetermined.
• E is given as false: I is true ; A and O are undetermined.
• I is given as false: A is false ; E is true; O is true.
• O is given as false: A is true; E is false; I is true.
Immediate Inference
➢ Inference is a process of drawing conclusion from certain
premise or premises.
Immediate inference :-An inference that is drawn directly from
one Premise without the Mediation of any other premise.
Mediate inference :- Any inference drawn from more than one
Premise.
There are three other important kinds of immediate inference:
Conversion, Obversion, and Contraposition. These are not associated
directly with the square of opposition. Each is explained below:
Conversion
Conversion is an inference that proceeds by interchanging the
subject and predicate terms of a proposition. ”No men are angels”
converts to “No angels are men,” and these propositions may be
validly inferred from one another. Similarly, ”Some women are
writers” and “Some writers are women” are logically equivalent, and
by conversion either can be validly inferred from the other.
Conversion is perfectly valid for all E propositions and for all I
propositions. One standard-form categorical proposition is said to be

Pranav__k91
the converse of another when we derive it by simply interchanging
the subject and predicate terms of that other proposition. The
proposition from which it is derived is called the convertend. Thus,
”No idealists are politicians” is the converse of ”No politicians are
idealists,” which is its convertend.
The conversion of an O proposition is not valid. The O
proposition, “Some animals are not dogs,” is plainly true; its converse
is the proposition, “Some dogs are not animals,” which is plainly
false. An O proposition and its converse are not logically equivalent.
The A proposition presents a special problem here. Of course, the
converse of an A proposition does not follow from its convertend.
From “All dogs are animals” we certainly may not infer that “All
animals are dogs.” Traditional logic recognized this, of course, but
asserted, nevertheless, that something like conversion was valid for
A propositions. On the traditional square of opposition, one could
validly infer from the A proposition, “All dogs are animals,” its
subaltern I proposition, “Some dogs are animals.” The A proposition
says something about all members of the subject class (dogs); the I
proposition makes a more limited claim, about only some of the
members of that class. It was held that one could infer “Some S is P”
from “All S is P.” And, as we saw earlier, an I proposition may be
converted validly; if some dogs are animals, then some animals are
dogs.
So, if we are given the A proposition, “All dogs are animals,” we
first infer that “Some dogs are animals” by subalternation, and from
that subaltern we can by conversion validly infer that “Some animals
are dogs.” Hence, by a combination of subalternation and
conversion, we advance validly from “All S is P” to”Some P is S.” This
pattern of inference, called conversion by limitation (or conversion
per accidens), proceeds by interchanging subject and predicate

Pranav__k91
terms and changing the quantity of the proposition from universal to
particular.
In all conversions, the converse of a given proposition contains
exactly the same subject and predicate terms as the convertend,
their order being reversed, and always has the same quality (of
affirmation or denial). A complete picture of this immediate
inference as traditionally understood is given by the following table:
Obversion
Obversion is an immediate inference that is easy to explain once
the concept of a term complement is understood. To obvert a
proposition, we change its quality(affirmative to negative or negative
to affirmative) and replace the predicate term with its complement.
However, the subject term remains unchanged, and so does the
quantity of the proposition being obverted. For example, the A
proposition, “All residents are voters,” has as its obverse the E

Pranav__k91
proposition, ”No residents are non voters.” These two are logically
equivalent propositions, and either may be validly inferred from the
other.
Obversion is a valid immediate inference when applied to any
standard-form categorical proposition:
• The E proposition, “No umpires are partisans,” has as its
obverse the logically equivalent A proposition, “All umpires are
non partisans.”
• The I proposition, “Some metals are conductors,” has as its
obverse the O proposition, ”Some metals are non conductors.”
• The O proposition, “Some nations were not belligerents,” has
as its obverse The I proposition, ”Some nations were non
belligerents.”
The proposition serving as premise for the obversion is called the
obvertend ; the conclusion of the inference is called the obverse.
Every standard-form categorical proposition is logically equivalent to
its obverse, so obversion is a valid form of immediate inference for
all standard-form categorical propositions. To obtain the obverse of
any proposition, we leave the quantity (universal or particular)and
the subject term unchanged; we change the quality of the

Pranav__k91
proposition and replace the predicate term with its complement.
The following table gives a complete picture of all valid obversions:
Contraposition
Another type of immediate inference, contraposition, can be reduced to
the first two, conversion and obversion. To form the contrapositive of a
given proposition, we replace its subject term with the complement of
its predicate term, and we replace its predicate term with the
complement of its subject term. Neither the quality nor the quantity of
the original Proposition is changed , so the contrapositive of the of an A
proposition is an A proposition, the contrapositive of an O proposition is
an O proposition, and so forth.
For example, the contrapositive of the A proposition, “All members
are voters,” is the A proposition, “All nonvoters are nonmembers.” These
are logically equivalent propositions, as will be evident on reflection.
Contraposition is plainly a valid form of immediate inference when
applied to A propositions. It really introduces nothing new, because we
can get from any A proposition to its contrapositive by first obverting it,
next applying conversion, and then applying obversion again. Beginning
with “All S is P,” we obvert it to obtain “No S is non-P,” which converts
validly to “No non-P is S,” whose obverse is “All non-P is non-S.” The
contrapositive of any A proposition is the obverse of the converse of the
obverse of that proposition.
Contraposition is a valid form of immediate inference when applied to O
propositions also, although its conclusion may be awkward to express.
The contrapositive of the O proposition, “Some students are not
idealists,” is the somewhat cumbersome O proposition, “Some
nonidealists are not nonstudents,” which is logically equivalent to its

Pranav__k91
premise. This also can be shown to be the outcome of first obverting,
then converting, then obverting again. “Some S is not P” obverts to
“Some S is non-P,” which converts to “Some non-P is S,” which obverts to
“Some non-P is not non-S.”
For I propositions, however, contraposition is not a valid form of
inference The true I proposition,”Some citizens are nonlegislators,” has
as its contrapositive the false proposition,”Some legislators are
noncitizens.” The reason for this invalidity becomes evident when we try
to derive the contrapositive of the I proposition by successively
obverting, converting, and obverting. The obverse of the original I
proposition, “Some S is P,” is the O proposition, “Some S is not non-P,”
but (as we saw earlier) the converse of an O proposition does not follow
validly from it.
In the case of E propositions, the contrapositive does not follow validly
from the original, as can be seen when, if we begin with the true
proposition,”No wrestlers are weaklings,” we get, as its contrapositive,
the obviously false propo sition, “No nonweaklings are nonwrestlers.”
The reason for this invalidity we will see, again, if we attempt to derive it
by successive obversion, conversion,and obversion. If we begin with the
E proposition,”No S is P,” and obvert it, we obtain the A proposition, “All S
is non-P”-which in general cannot be validly converted except by
limitation. If we do then convert it by limitation to obtain”Some non-P is
S,” we can obvert this to obtain “Some non-P is not non-S.” This outcome
we may call the contrapositive by limitation-and this too we will consider
further in the next section.
Contraposition by limitation, in which we infer an O proposition from an
E proposition (for example, we infer”Some non-P is not non-S” from “No
S is P”),has the same peculiarity as conversion by limitation, on which it
depends. Because a particular proposition is inferred from a universal

Pranav__k91
proposition, the resulting contrapositive cannot have the same meaning
and cannot be logically equivalent to the proposition that was the
original premise. On the other hand, the contrapositive of an A
proposition is an A proposition, and the contrapositive of an O
proposition is an O proposition, and in each of these cases the
contrapositive and the premise from which it is derived are equivalent.
Contraposition is thus seen to be valid only when applied to A and
O propositions. It is not valid at all for I propositions, and it is valid for E
propositions only by limitation. The complete picture is exhibited in the
following table:

Pranav__k91
UNIT-II
Categorical Syllogism: Standard form of Categorical Syllogism, The
Formal Nature of Syllogistic Argument,
Venn-Diagram Technique for Testing Syllogism, Rules and Fallacies.
Categorical Syllogism
Syllogism : Any deductive argument in which a conclusion is inferred
from two premises.
Categorical syllogism : A deductive argument consisting of three
Categorical propositions that contain exactly three terms, each of which
occurs in exactly two of the propositions.
Standard form The form in which a Syllogism is said to be when its
premises and conclusion are all Standard-form Categorical propositions
(A, E, I, or O) and are arranged in standard Order (major premise, then
minor premise, then conclusion).
Terms of the Syllogism: Major, Minor, and Middle

Pranav__k91
In a categorical syllogism there are three and only three propositions,
accordingly there are only three terms – Middle term, Major term, and
Minor term.
Major term (P) : The term that occurs as the predicate term of the
conclusion in a standard-form Categorical syllogism.
Minor term (S) : The term that occurs as the subject term of the
conclusion in a standard-form categorical syllogism.
Middle term (M) : In a standard-form categorical syllogism (which must
contain exactly three terms), the term that appears in both premises but
does not appear in the Conclusion.
Example: No politicians are Professors.
Some Doctors are Professors.
Therefore, Doctors are politicians.
Here, ‘Professors’ is Middle term. ‘Politicians’ is Major term and ‘Doctors’
is Minor term.
Premises in Categorical Syllogism
Major premise: In a standard-form categorical syllogism, the premise
that contains the major term is called major premise.

Pranav__k91
Minor premise : In a standard-form Categorical syllogism, the premise
that contains the minor term is called Minor Premise.
Example: Major premise – AII M is P.
Minor premise- All S is M.
Conclusion – Therefore, All S is P.
Mood of the Syllogism
➢ Mood of syllogism means arrangement of premises and
conclusion.
➢ The mood of a categorical syllogism is a series of three letters
corresponding to the type of proposition the major premise,
the minor premise, and the conclusion.
➢ The mood of a syllogism is determined by the types of
categorical propositions contained in the argument, and the
order in which they occur. To determine the mood, put the
argument into standard form, and then simply list the types of
categorical (A, E, I, O) featured in the order they occur.
Example:
All animals are mammals. ➡️ A
All Dogs are animals. ➡️ A
Therefore, all dogs are mammals. ➡️ A
From top to bottom, we have an A, an A, and an I. So the mood of our
argument is AAA .It turns out that there are 64 possible moods—64 ways

Pranav__k91
of combining A, E, I, and O into unique three-letter combinations, from
AAA to OOO and everything in between.
Figure Of Categorical Syllogism
• The figure of a categorical syllogism is a number which corresponds
to the placement of the two middle terms.
• Position of the middle term in the premises decides the figure of a
Syllogism.
➢ In 1st
figure, Middle Term is subject in the major premises and
Predicate in the minor premises.
➢ In 2nd
figure, Middle term is predicate in both major and minor
Premises.
➢ In 3rd
figure, Middle Term is subject in both major and minor
premises.
➢ In the 4th
figure Middle term is predicate in the major premises and
subject in the minor premises.
EXAMPLE.

Pranav__k91
No nuclear-powered submarines are commercial vessels, so
no warships are commercial vessels, because all nuclear-powered
submarines are warships.
SOLUTION
Step 1. The conclusion is “No warships are commercial vessels.”
Step 2. “Commercial vessels” is the predicate term of this conclusion and
is therefore the major term of the syllogism.
Step 3. The major premise, the premise that contains this term, is “No
nuclear-powered submarines are commercial vessels.”
Step 4. The remaining premise, “All nuclear-powered submarines are
warships,” is indeed the minor premise, because it does contain the
subject term of the conclusion, “warships.”
Step 5. In standard form this syllogism is written thus:
No nuclear-powered submarines are commercial vessels.
All nuclear-powered submarines are warships.
Therefore no warships are commercial vessels.
Step 6. The three propositions in this syllogism are, in order, E, A, and E.
The middle term, “nuclear-powered submarines,” is the subject term of
both premises, so the syllogism is in the third figure. The
Mood and figure of the syllogism therefore are EAE–3.

Pranav__k91
Syllogistic Rules and Syllogistic Fallacies
▪ There is a list of six rules, each of which states a necessary
condition
for the validity of any categorical syllogism.
• Violating any of these rules involves committing one of the formal
Fallacies.
• There are two objectives of syllogistic rules & fallacies:
➢ To know the necessary condition for the validity of any categorical
Syllogism.
➢ Explain how violating any of these rules involves committing
fallacies.
Rule 1. Avoid four terms.
• A syllogistic reasoning must have three and-only three
unambiguous categorical terms. Middle term, Major term and
Minor Term.
• Each term must occur twice.
• Each term should be used in the same sense in both of its
occurrence.
The Fallacy of Four Terms:
If more than three terms are involved in the syllogism then syllogism
would be invalid and the fallacy committed is called ‘The fallacy of four
term’.

Pranav__k91
Rule 2. Distribute the middle term in at least one premise.
➢ In a valid categorical syllogism the middle term must be distributed
in at least one of the premises.
Fallacy of Undistributed Middle
Middle term must be distributed in at least one of the premises
otherwise syllogism becomes invalid and has “The fallacy of
Undistributed Middle”.
Example: All cows are mammals.
All cats are mammals.
Therefore, all cats are cows.
Here, Middle term “Mammals” is not distributed in any of the premises,
Hence it’s invalid syllogism.
Rule 3. Any term distributed in the conclusion must be distributed in
the premises.
➢ If a term is distributed in the conclusion then it must be distributed
in a premise.
Fallacy of Illicit Minor and Illicit Major
➢ A syllogism becomes invalid if a term is distributed in the
conclusion but undistributed in the premises. On the basis of this
theory there are two type of fallacies:
• Fallacy of Illicit Major
• Fallacy of Illicit Minor

Pranav__k91
Fallacy of Illicit Minor
➢ If minor term is distributed in the conclusion but undistributed in
the minor premises then syllogism would be invalid because of
“Fallacy of Illicit Minor”.
Example:
All fishes are mammals.
All mammals are animals.
Therefore, all animals are fishes.
Here Minor Term “Animal” is distributed in the conclusion but
Undistributed in the minor premises.
Fallacy of Illicit Major
➢ If major term is distributed in the conclusion but undistributed in
the major premises then syllogism would be invalid because of
“Fallacy of Illicit Major”.
Example:
All horses are animals. (A)
Some dogs are not horses. (O)
Therefore, some dogs are not animals. (O)
Here Major Term “Animal” is distributed in the conclusion but
Undistributed in the major premises.

Pranav__k91
Rule 4. Avoid two negative premises.
➢ Two negative premises (E/O) are not allowed in valid categorical
syllogism.
Fallacy of Exclusive Premises:
A syllogism having both the negative premises is invalid because of
Fallacy of exclusive premises.
Example:
No fish are mammals.
Some dogs are not fish.
Therefore, some dogs are not mammals.
Here, major premise is “E” that is universal negative and minor
Premises is “O” that is particular negative.
Rule 5. If either premise is negative, the conclusion must be negative.
Fallacy of drawing affirmative conclusion from a negative premise:
➢ If one of the premises is negative (E/0) and the conclusion is
affirmative (A/l) then it makes invalid syllogism by fallacy of
drawing affirmative conclusion from a negative premise.
Example:
All monkey are birds.
Some wolves are not monkey.

Pranav__k91
Therefore, some wolves are birds.
Here, minor premise is Negative and conclusion is affirmative. In this
Way this syllogism is invalid.
Rule 6. From two universal premises no particular conclusion may be
drawn.
➢ If both premises are universal (A/E) then the conclusion cannot be
particular (E/0).
Existential Fallacy:
➢ If syllogism having both universal premises and a particular
conclusion then it would be invalid by Existential Fallacy.
Example:
All mammals are animals
All dogs are mammals.
Therefore, some dogs are animals.
Here, according to existential import universal proposition don’t say
anything about existence but particular does say something. So in this
case the conclusion contains more information than premises, thereby
making it invalid.

Pranav__k91
Flowchart for Applying the Six Syllogistic Rules
The following chart captures the process for working through
the six rules of validity for categorical syllogisms:

Pranav__k91
Example
Test the validity/Invalidity of the given syllogistic forms by the
traditional method of rules and fallacies in the figures indicated against
them.
AAA-2
Solution:
PM(A) All P is M.
SM(A) All S is
Therefore, S P (A) .: All S is P.
It is invalid by fallacy of Undistributed Middle term. Because both the
Premises are a proposition and the predicate term of a proposition
become always Undistributed. Here, Middle term is in the place of
Predicate, So Middle term is also undistributed.
Venn Diagram Technique for Testing Validity

Pranav__k91
Few steps for testing validity or invalidity of syllogism.
1. First, make Diagram for the premises.

Pranav__k91
2. Make Diagram for the conclusion.
3. look the both diagram and justify that conclusion is inherent in the
premises or not.
4. After looking the both diagrams, we can state that conclusion is
contained in the premises thus the augmented shown to be valid
otherwise invalid.
5. The argument is valid if the conclusion is already diagrammed in the
premises. The shaded part of conclusion also should be shaded in the
premises for validity.
Example 1: No doctors are professor.
All lawyers are doctors.
Therefore, no lawyers are professors.
Here both premises are universal, we can diagram either premise first.
Then draw the conclusion separately.
Here shaded area of both diagrams of premises and conclusion is
showing that conclusion is implicitly contained in the premises.

Pranav__k91
In other way we can that shaded area in the conclusion was shaded
already when we diagrammed the premises.
Thus, the argument is valid.
UNIT-III : Propositional Logic, Basic Truth-functions of the
Propositional Calculus, Testing arguments by truth table method.
Relation between Truth functions,
Propositional Logic
What is a proposition? A proposition is the basic building block of logic. It
is defined as a declarative sentence that is either True or False, but not
both. The Truth Value of a proposition is True(denoted as T) if it is a true
statement, and False(denoted as F) if it is a false statement. For Example,
1. The sun rises in the East and sets in the West.
2. 1 + 1 = 2
3. ‘b’ is a vowel.
All of the above sentences are propositions, where the first two are
Valid(True) and the third one is Invalid(False). Some sentences that do
not have a truth value or may have more than one truth value are not
propositions. For Example,
1. What time is it?

Pranav__k91
2. Go out and play.
3. x + 1 = 2.
The above sentences are not propositions as the first two do not have a
truth value, and the third one may be true or false. To represent
propositions, propositional variables are used. By Convention, these
variables are represented by small alphabets such as p,q,r,s . The area of
logic which deals with propositions is called propositional calculus or
propositional logic. It also includes producing new propositions using
existing ones. Propositions constructed using one or more propositions
are called compound propositions. The propositions are combined
together using Logical Connectives or Logical Operators.
Truth Table
Since we need to know the truth value of a proposition in all possible
scenarios, we consider all the possible combinations of the propositions
which are joined together by Logical Connectives to form the given
compound proposition. This compilation of all possible scenarios in a
tabular format is called a truth table. Most Common Logical connectives-
1. Negation – If p is a proposition, then the negation of p is denoted
by “~” or “¬”, which when translated to simple English means- “It
is not the case that p ” or simply “not p “. The truth value of
¬p is the opposite of the truth value of p . The truth table of
¬p is -

Pranav__k91
Example, The negation of “It is raining today”, is “It is not the case that is
raining today” or simply “It is not raining today”.
2. Conjunction – For any two propositions p and q , their
conjunction is denoted by p∧ q , which means “p and q “. The
conjunction p ∧ q is True when both p and q are True,
otherwise False. The truth table of p ∧ q is-
Example, The conjunction of the propositions p – “Today is Friday” and
q – “It is raining today”, p ∧ q is “Today is Friday and it is raining

Pranav__k91
today”. This proposition is true only on rainy Fridays and is false on any
other rainy day or on Fridays when it does not rain.
3. Disjunction – For any two propositions p and q , their disjunction
is denoted by p V q , which means “p or q “. The disjunction p V
q is True when either p or q is True, otherwise False. The truth
table of p V q is-
Example, The disjunction of the propositions p – “Today is Friday” and q
– “It is raining today”, p V q is “Today is Friday or it is raining today”. This
proposition is true on any day that is a Friday or a rainy day(including
rainy Fridays) and is false on any day other than Friday when it also does
not rain.
4. Exclusive Or – For any two propositions p and q , their exclusive
or is denoted by p ⊕ q , which means “either p or q but not
both”. The exclusive or p ⊕ q is True when either p or q is True,
and False when both are true or both are false. The truth table of p
⊕ q is-

Pranav__k91
Example, The exclusive or of the propositions p – “Today is Friday” and
q – “It is raining today”, p ⊕ q is “Either today is Friday or it is raining
today, but not both”. This proposition is true on any day that is a Friday
or a rainy day(not including rainy Fridays) and is false on any day other
than Friday when it does not rain or rainy Fridays.
5. Implication – For any two propositions p and q , the statement “if
p then q ” is called an implication and it is denoted by p→ q . In
the implication p → q , p is called the hypothesis or antecedent
or premise and q is called the conclusion or consequence. The
implication is p → q is also called a conditional statement. The
implication is false when p is true and q is false otherwise it is
true. The truth table of p → q is-

Pranav__k91
Example, “If it is Friday then it is raining today” is a proposition which is
of the form p → q . The above proposition is true if it is not
Friday(premise is false) or if it is Friday and it is raining, and it is false
when it is Friday but it is not raining.
6. Biconditional or Double Implication – For any two propositions p
and q , the statement “p if and only if(iff) q ” is called a
biconditional and it is denoted by p ⇔ q . The statement p ⇔ q
is also called a bi-implication. p ⇔ q has the same truth value as
(p ⇔ q) ∧(q ⇔ p) The implication is true when p and q have
same truth values, and is false otherwise. The truth table of p ⇔ q
is-
Example, “It is raining today if and only if it is Friday today.” is a
proposition which is of the form p ⇔ q . The above proposition is true if
it is not Friday and it is not raining or if it is Friday and it is raining, and it
is false when it is not Friday or it is not raining.

Pranav__k91
UNIT-IV: Inductive Reasoning and Probability, Simple
Enumeration and Analogy.
Mill’s Methods of Experimental Enquiry.
Simple Enumeration
Simple Enumeration is a common man’s method of arriving at a
generalization. Generalization is a statement of the type, ‘All A is B’.It is
the simplest kind of induction. The generalization of a common man
differs from that of a scientist. Common man uses simple enumeration
whereas scientist use scientific Induction for establishing generalizations.
Simple enumeration is the process of establishing a generalization on the
basis of the observation of some cases or instances of a kind.
Generalization in simple enumeration is supportedby direct evidence. In
induction by simple enumeration We generalize by going beyond what
has been experienced. Induction by simple enumeration can be defined
as “what is true of several cases of a kind is true of all the cases of that
kind”. It establishes a generalization on the basis of uniform and
uncontradicted experience. For example:
First observed crow is black
Second observed crow is black.
Third observed crow is black.
One lakh observed crows are black.
Therefore All crows are black

Pranav__k91
Generalizations established by simple enumeration have the followings
characteristics :-
• Uniform and Uncontradicted Experience :
Generalization in simple enumeration is based on uniform and
uncontradicted Experience.
For example :Ice is cold, fire is hot etc. We have never come across any
contrardictory experience of ice being hot and fire being cold. In these
examples the scope of generalizations are unlimited hence it is larger
than the scope of evidence.
• Absence / Lack of analysis of property:
Simple enumeration is the process of simply counting the instances
(cases) to find that all these cases share a common property, However
it does not involve analysis : for example- why crows are black, or why
roses have thorns. Here one is not concerned in finding out why
blackness goes with crows or why thorns are associated with roses.
• Unrestricted generality :
The generalization established by simple Enumeration is not
about a class with limited number of members, for example :
Some students in this class are smart
Therefore, All students in this class are smart.
In the given example a generalization is established but it is of
restricted generality. Therefore such kind of arguments are not induction
by simple enumeration. In SimpleEnumeration the conclusion i.e.
generalization is about unrestricted number of members. Forexample:
Some polar bears are white

Pranav__k91
Therefore, All polar bearsare white
In case of simpleEnumeration there is an inductive leap or jump from
observed cases to unobserved or known cases to unknown cases The
scope of our generalization is unlimited and hence larger than the scope
of evidence.
• Low degree of Probability :
As the generalization of Simple Enumeration are based on uniform
experience of some cases, we cannot be sure of the unobserved
cases/instances possessing the same characteristics as the observed
ones.Generalization such as - ‘All crows are black’ is accepted as true
on the basis of observation, i.e.Direct evidence. But we cannot rule
out the possibility of a Contrardictory instance. Therefore it is saidto be
probable.
• Value of Induction by Simple Enumeration :
The generalizations established by simple enumeration are not equally
good that is to say some generalizations are good and some are bad
For example: “All crows are black” is a good one but “All swans are
white” js a bad one. Mill and Bacon considers the process of simple
enumeration as childish and unreliable. According to them the value of
Simple Enumeration depends upon the number of instances observed.
However they were wrong in saying because valueof generalizations
depends upon some more conditions. They are as follows:
1. Wider Experience : The generalizations of Simple Enumeration are
based on wider experience. For example : All crows are black, is

Pranav__k91
based on observation. When a large numbers of instances are
observed, it is possible to come across contradictory instance if
any.
Example : we do not come across any non black crows we
conclude ‘All crows are black.’
2. Variety of experience : Instead of observing maximum number of
crows from one part of the world, if we observe some crows from
different parts of the world then the generalization becomes more
probable or reliable because we all are aware that sometimes the colour
of the animals depends upon the climate or other conditions of that
region.
E.g. Some bears are black.
Therefore , All bears are black.
Here this argument is bad because in polar region we find white
bears due to climatic condition.
3. Resemblances : Value of simple enumeration is also affected by the
nature of resemblances. For example- crows a part from being black,
resemble each other in other physical characteristics also like pointed
beak,clawed feet, etc. which are equally important characteristics
of a crow.
Analogy : Analogy is a type of inductive reasoning. Analogy is a
man’s inference in which the conclusion is drawn on the basis of

Pranav__k91
observed resemblances(similarities). In analogy we proceed form
particular to particular instance.
It may be defined as an argument
from known resemblance to further resemblance, that is to say, if two (or
more) things resemble each other in certain characteristics and if one of
them have further / additional characteristics, the other is also likely to
have that characteristics.
The form of analogical argument is as follows
A - is observed to have the properties P1, P2,P3,….Pn
B - is observed to have the properties P1, P2, P3,….Pn
A possess additional property ‘q’
Therefore , B also has the property ‘q’.
Example : On the basis of the observed similarities between Earthand
Mars, Lowell putforward ananalogical argument.
Both Earthand Mars are planets.
They revolve round the Sun.
Both have water, moderatetemperature and are surrounded by an
atmosphere.
There is life on Earth
Therefore there is life on Mars.
The logical basis of the analogical argument is that the characteristics
found together are likely to be connected with one another and

Pranav__k91
therefore from the presence of one characterstics we infer the presence
of another.
Value of Analogy : Some analogical arguments are good whereas some
are bad. The soundness of analogical depends upon the following
factors:
• Relevant and important resemblances : When the resemblance is in
important and relevant characteristics,the analogical argument is good.
For example : Lowell’s analogy about life on Mars is good example
because they both resemble each other in important characteristics and
are also relevantto the characteristic infered i.e. existence of life as we
all know water, temperature and atmosphere is necessary for existence
of life.
• Important differences : If the differences are in important aspects,
then the analogical argument is bad.
• For example : Man and monkey, both have two legs, two eyes, two
hands, one nose, two ears. Man can read and write. Therefore,
monkeys can also read and write.
In this example, there are many similarities between both man
and monkey but the difference is very important, i.e. man is rational
whereas monkey is not as rational as man and therefore it is a bad
analogical argument. It is important to note that the conclusion
established by analogical reasoning is always probable and never
certain.

Pranav__k91
MILL’S METHODS OF EXPERIMENTAL ENQUIRY
Introduction: John Stuart Mill (20 May 1806 – 7 May 1873) was an
English philosopher, political economist, Member of Parliament (MP),
and civil servant. One of the most influential thinkers in the history of
classical liberalism, he contributed widely to social theory, political
theory, and political economy. Dubbed “the most influential English-
speaking philosopher of the nineteenth century”, he conceived of liberty
as justifying the freedom of the individual in opposition to unlimited
state and social control. John Stuart Mill joined the debate over the
scientific method which followed John Herschel’s 1830 publication of A
Preliminary Discourse on the study of Natural Philosophy, which
incorporated inductive reasoning from the known to the unknown,
discovering general laws in specific facts and verifying these laws
empirically. Mill countered this in 1843 in A System of Logic (fully titled A
System of Logic, Ratiocinative and Inductive, Being a Connected View of
the Principles of Evidence, and the Methods of Scientific Investigation).
In “Mill’s Methods” (of induction), as in Herschel’s, laws were discovered
through observation and induction and required empirical verification.
In his book, System of Logic (Book III), Mill set forth five methods of
experimental inquiry, calling them the Method of Agreement, the
Method of Difference , the Joint Method of Agreement and Difference,
the Method of Residues, and the Method of Concomitant Variation

Pranav__k91
The Method of Agreement
According to the method of agreement, if two or more examples of a
phenomenon only share in a single antecedent condition, that single
condition is the cause of the examples of the phenomena (Hung, 1997).
By such reasoning, if, in all cases of tree rot, I identify a type of tick to be
present, I conclude the tick to have caused the tree rot. The tick,
however, may be ubiquitous and found in trees that are not rotting;
there could be an unobservable virus. As Hung points out, the method is
open to difficulties in interpretation. There may be a relationship that is
merely coincidental between the antecedent and the consequent, as
was the case in the preceding example. Second, the cause may not have
been included in the antecedent conditions, e.g., the miniscule virus.
Furthermore, there could be multiple causes, such as the presence of
the virus and a certain temperature range. On top of that, the cause
could be non-uniform; the same end result may be caused by a number
of factors acting separately. With tree rot, the rot may be the same but
one tree may rot due to soil that is too damp, another because of
woodpeckers, and another because of insect infestation, with our
ubiquitous tick still present in every instance. Finally, it is possible that
two events may occur in succession, invariably, without a causal relation
between them, such as with thunder and lightning, both of which share
in a cause but differ in the time their effects take to reach us.
We might assume the first experience (seeing lightning) to be the cause
of the second experience (hearing thunder). The differences in rates of
wave travel produce the temporal differences in receipt of stimulation at
the receptors. One has to conclude that the above problems make the
generalization questionable.

Pranav__k91
The Method of Difference
With the method of difference, there are cases where the phenomenon
under scrutiny is sometimes present and sometimes absent, and in
which all other elements, but one, remain constant (over those instances
of the “phenomenon-present” and the “phenomenon-absent”). Given
those conditions, the element that is present when the other is present,
and absent when the other is absent, is causally related to the other. In
this case, given that all other aspects within the circumstances being
considered have to be the same, we have an example of what, in
modern terminology, is called the control of confounding variables.
Joint Method of Agreement and Difference
With the joint method of agreement and difference, one gathers a
number of instances of positive and negative cases (without everything
the same being constant, as with the method of difference above). The
multiplication of the number of instances observed is intended to make
the method more reliable. The approach expands upon the method of
agreement since both positive and negative instances are drawn upon
rather than just positive instances.
Through the use of this method one concludes that if, due to a process
of elimination, the antecedent condition is always present when the

Pranav__k91
consequent is present, but is never present when the consequent is not
present, that the antecedent condition is the cause.
Method of Concomitant Variation
With the method of concomitant variation one is looking for
circumstances where one of the elements, say the antecedent, is varying
in magnitude and cases where some other variable varies in a similar
manner (Day, 1964). Under such circumstances one determines the first
to be causally related to the second if an increase (or decrease) is
accompanied by an increase (or a decrease) in the second. For instance,
the amount of time that the sun is present during the day is associated
with the average temperature. When the sun is briefly present
temperature is low and vice versa. (This is what we would now refer to
as a positive correlation.)
Method of Residues
With the method of residues, one commences with the knowledge that
something (A), the antecedent, causes an effect ( E ) , and, furthermore,
that the antecedent contains within it an element (A:1) that also has a
known effect (E:1). Now, given these conditions, it can be concluded that
the first cause (A) minus its component cause (A:1) will equal the first

Pranav__k91
effect ( E ) less the second, componential, effect (E:1) or (A – A:1 = E –
E:1). A person weighing her or himself, for instance, who is rather shy
around doctors, can get on a scale fully clothed and then go behind a
screen and remove all clothing. The weight of the clothing subtracted
from the weight of the clothed person will give the person’s weight
unclothed. The method, as Hung (1997) pointed out, has its own
problems since some causes may not be additive. For Instance, water is
composed of both hydrogen and oxygen. Both are gases that are
flammable but, added together, they cause a dousing of flame;
conversely, water douses fire, but, if it has onHane of its components
removed, it will result in unwanted consequences when applied to
flame.
Conclusion
According to Hung (1997), the methods developed by Francis Bacon and
Mill may have very limited applications. First of all, the conclusions that
result from them may often be in error.
Second, they only apply to tidy cases where the requisite conditions are
in place. Lastly, they are essentially correlational methods in nature and
that confines scientific practice to fortuitous observations and will not
yield novel concepts (as true experiments could do). In a nutshell, Mill
has falsely confused correlational research with experimental research.
Nonetheless, his was a further expansion upon earlier methods in
teasing out consistent relations between phenomena. More than that,
he was attempting to codify practices that would support the exploration
of nature and its lawful regularities.

Pranav__k91
------------------------------------××××××××××××××××××------------------------------
Thank You
Pranav Kumar

Logic 3rd Semester Notes.pdf. Bhu 3rd sem logic

More Related Content

Similar to Logic 3rd Semester Notes.pdf. Bhu 3rd sem logic (19)

Recently uploaded (20)

Logic 3rd Semester Notes.pdf. Bhu 3rd sem logic