Logic, Western and Indian Questions and Answers

Q: Law of contradiction states that :

The law of contradiction states that nothing can be both (A) and not (A).The law of (non-)contradiction says a thing cannot both have and not have the same property at the same time and in the same respect; in logic, ~(p . ~p).'Everything is identical to itself' is the law of identity, and 'everything is either A or not A' is the law of excluded middle. 'A proposition can be both true and false' denies the law outright.

Q: The fallacy committed in the following syllogism is .................. ."No tragic actors are idiots.Some comedians are not idiots.Therefore all comedians are not tragic actors.

The fallacy is undistributed minor.The conclusion 'All comedians are not tragic actors' is an E proposition (no comedians are tragic actors), which distributes its subject, comedians, the minor term. In the minor premise 'Some comedians are not idiots', comedians is the subject of an O proposition and is undistributed. A term distributed in the conclusion but not in its premise commits the fallacy of the illicit minor.The major term, tragic actors, is distributed in both places, so there is no illicit major. The middle term, idiots, is distributed as the predicate of the negative premises. (The syllogism also has two negative premises, but that is not among the options.)

Q: Match List I and List II and select the correct answer from the codes given below :List IList II(a) ~ (p &or; ~ p)(1) Identity(b) ~ (p • ~ p)(2) Contingency(c) ~ (p ⊃ q)(3) Tautology(d) p ⊃ p(4) Contradiction

The match is a-4, b-3, c-2, d-1.~(p &or; ~p) denies the law of excluded middle, so it is false in every row: a contradiction. ~(p • ~p) is the law of non-contradiction, true in every row: a tautology. ~(p ⊃ q) is true only when p is true and q false, and false otherwise, so it is contingent. p ⊃ p is the law of identity in propositional form.Option A calls ~(p &or; ~p) a tautology, which reverses it, and the remaining codes mismatch the contingent formula.

Q: "Sudhir had not locked the gates yesterday. Either the watchman or Sudhir locks the gates. Therefore, the watchman must have locked the gates yesterday". This argument is ..................... .

The argument is deductive and valid.It is a disjunctive syllogism: either the watchman or Sudhir locks the gates (W &or; S); Sudhir did not (~S); therefore the watchman did (W). If the premises are true the conclusion must be true, so the argument is deductive and its form is valid.An inductive argument would claim only probable support, which is not the case here, and the form has no gap that would make it invalid.

Q: If the statement 'p' is true and the statement 'q' is false, the truth value of the statements 'p &equiv; ~ q' and '~ q ⊃ ~ p' are :

The answer is that p &equiv; ~q is true and ~q ⊃ ~p is false.With p true and q false, ~q is true and ~p is false. A biconditional is true when both sides have the same value: p and ~q are both true, so p &equiv; ~q is true. A conditional is false only when its antecedent is true and its consequent false: ~q is true and ~p is false, so ~q ⊃ ~p is false.That rules out 'both true', 'both false' and the reverse assignment in option D.

Q: Correct formulation of the Rule U.I. (Universal Instantiation) is ........... .

The correct rule is (X) (φx), therefore φu, where u is any individual symbol.Universal instantiation lets us infer, from a universally quantified statement, any substitution instance of it. In Copi's formulation, from (x)(φx) we may infer φν, where ν is any individual symbol, whether an individual constant or the individual variable y used in the proof.Restricting it to individual constants (B) or to variables (D) gives too narrow a rule, and option A states the inference without saying what may be substituted.

Q: Which of the following is incorrect with reference to the law of identity ?

The incorrect statement is Everything is unequal to itself.The law of identity states that everything is identical with itself: A is A, or, in the words of Bishop Butler, everything is what it is and not another thing. In propositional logic it is expressed as \(P \supset P\), if P then P, which is a tautology.Options A, B and C are therefore all statements of the law of identity. 'Everything is unequal to itself' denies it, so it is the incorrect one.

Q: If the truth value of 'A' proposition is true, the truth values of 'E' 'I' 'O' propositions are ........... respectively.

If A is true, E, I and O are F, T, F.On the traditional square of opposition:E is the contrary of A. Contraries cannot both be true, so if A is true, E is false.I is the subaltern of A. The truth of the universal carries down to the particular, so I is true.O is the contradictory of A. Contradictories always take opposite values, so O is false.Example: if 'All men are mortal' is true, then 'No men are mortal' is false, 'Some men are mortal' is true and 'Some men are not mortal' is false. Option D reverses every value, and B and C each get E or I wrong.

Q: Identify the fallacy committed in the following argument is ................."All men are mortal.All animals are mortal.&there4; "All animals are men."

The argument commits the fallacy of undistributed middle.The conclusion's terms are 'animals' (minor) and 'men' (major), so the middle term is 'mortal'. It appears as the predicate of two universal affirmative (A) propositions, and the predicate of an A proposition is undistributed. Since the middle term is not distributed in at least one premise, the premises do not connect men and animals: both may be mortal without being the same class.The major term 'men' is distributed in its premise and undistributed in the conclusion, which is allowed, so there is no illicit major (undistributed major). The minor term 'animals' is distributed in both premise and conclusion, so there is no illicit minor either.

Q: Correct symbolization of the following statement is :Statement : Only students are members. (Sx, Mx)

The correct symbolization is (X) (Mx \(\supset\) Sx).'Only students are members' means that nothing is a member unless it is a student, that is, all members are students. With Sx for 'x is a student' and Mx for 'x is a member':\[ (x)(Mx \supset Sx) \]Option B, \((x)(Sx \supset Mx)\), says all students are members, which reverses the claim; 'only' marks the predicate of the universal, not its subject. Option A says everything is both a student and a member, far too strong, and option D says only that some student is a member.

31.

Law of contradiction states that :

A.Everything is identical to itself
B.Nothing can be both (A) and not (A)
C.Proposition can be both true and false
D.Everything is either (A) or not (A)

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: B

The law of contradiction states that nothing can be both (A) and not (A).

The law of (non-)contradiction says a thing cannot both have and not have the same property at the same time and in the same respect; in logic, ~(p . ~p).

'Everything is identical to itself' is the law of identity, and 'everything is either A or not A' is the law of excluded middle. 'A proposition can be both true and false' denies the law outright.

Discuss in Forum

32.

The fallacy committed in the following syllogism is .................. .

"No tragic actors are idiots.
Some comedians are not idiots.
Therefore all comedians are not tragic actors.

A.Undistributed Middle
B.Undistributed Major
C.Undistributed Minor
D.Undistributed Major and Minor

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The fallacy is undistributed minor.

The conclusion 'All comedians are not tragic actors' is an E proposition (no comedians are tragic actors), which distributes its subject, comedians, the minor term. In the minor premise 'Some comedians are not idiots', comedians is the subject of an O proposition and is undistributed. A term distributed in the conclusion but not in its premise commits the fallacy of the illicit minor.

The major term, tragic actors, is distributed in both places, so there is no illicit major. The middle term, idiots, is distributed as the predicate of the negative premises. (The syllogism also has two negative premises, but that is not among the options.)

Discuss in Forum

33.

Match List I and List II and select the correct answer from the codes given below :

List I	List II
(a) ~ (p ∨ ~ p)	(1) Identity
(b) ~ (p • ~ p)	(2) Contingency
(c) ~ (p ⊃ q)	(3) Tautology
(d) p ⊃ p	(4) Contradiction

A.a-3, b-1, c-4, d-2
B.a-4, b-3, c-2, d-1
C.a-2, b-1, c-3, d-4
D.a-3, b-2, c-4, d-1

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: B

The match is a-4, b-3, c-2, d-1.

~(p ∨ ~p) denies the law of excluded middle, so it is false in every row: a contradiction. ~(p • ~p) is the law of non-contradiction, true in every row: a tautology. ~(p ⊃ q) is true only when p is true and q false, and false otherwise, so it is contingent. p ⊃ p is the law of identity in propositional form.

Option A calls ~(p ∨ ~p) a tautology, which reverses it, and the remaining codes mismatch the contingent formula.

Discuss in Forum

34.

"Sudhir had not locked the gates yesterday. Either the watchman or Sudhir locks the gates. Therefore, the watchman must have locked the gates yesterday". This argument is ..................... .

A.Deductive and valid
B.Deductive and invalid
C.Inductive and invalid
D.Inductive and good

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

The argument is deductive and valid.

It is a disjunctive syllogism: either the watchman or Sudhir locks the gates (W ∨ S); Sudhir did not (~S); therefore the watchman did (W). If the premises are true the conclusion must be true, so the argument is deductive and its form is valid.

An inductive argument would claim only probable support, which is not the case here, and the form has no gap that would make it invalid.

Discuss in Forum

35.

If the statement 'p' is true and the statement 'q' is false, the truth value of the statements 'p ≡ ~ q' and '~ q ⊃ ~ p' are :

A.Both the statements are true
B.Both the statements are false
C.The statement 'p ≡ ~ q' is true and the statement '~ q ⊃ ~ p' is false
D.The statement 'p ≡ ~ q' is false and the statement '~ q ⊃ ~ p' is true

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The answer is that p ≡ ~q is true and ~q ⊃ ~p is false.

With p true and q false, ~q is true and ~p is false. A biconditional is true when both sides have the same value: p and ~q are both true, so p ≡ ~q is true. A conditional is false only when its antecedent is true and its consequent false: ~q is true and ~p is false, so ~q ⊃ ~p is false.

That rules out 'both true', 'both false' and the reverse assignment in option D.

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36.

Correct formulation of the Rule U.I. (Universal Instantiation) is ........... .

A.(X) φx / ∴ φy
B.(X) (φx) / ∴ φu (where u is any individual constant)
C.(X) (φx) / ∴ φu (where u is any individual symbol)
D.(X) φx / ∴ φy (where y is any variable)

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The correct rule is (X) (φx), therefore φu, where u is any individual symbol.

Universal instantiation lets us infer, from a universally quantified statement, any substitution instance of it. In Copi's formulation, from (x)(φx) we may infer φν, where ν is any individual symbol, whether an individual constant or the individual variable y used in the proof.

Restricting it to individual constants (B) or to variables (D) gives too narrow a rule, and option A states the inference without saying what may be substituted.

Discuss in Forum

37.

Which of the following is incorrect with reference to the law of identity ?

A.Everything is identical with itself
B.Everything is what it is
C.\(P \supset P\)
D.Everything is unequal to itself

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: D

The incorrect statement is Everything is unequal to itself.

The law of identity states that everything is identical with itself: A is A, or, in the words of Bishop Butler, everything is what it is and not another thing. In propositional logic it is expressed as \(P \supset P\), if P then P, which is a tautology.

Options A, B and C are therefore all statements of the law of identity. 'Everything is unequal to itself' denies it, so it is the incorrect one.

Discuss in Forum

38.

If the truth value of 'A' proposition is true, the truth values of 'E' 'I' 'O' propositions are ........... respectively.

A.F, T, F
B.T, T, F
C.F, F, F
D.T, F, T

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

If A is true, E, I and O are F, T, F.

On the traditional square of opposition:

E is the contrary of A. Contraries cannot both be true, so if A is true, E is false.
I is the subaltern of A. The truth of the universal carries down to the particular, so I is true.
O is the contradictory of A. Contradictories always take opposite values, so O is false.

Example: if 'All men are mortal' is true, then 'No men are mortal' is false, 'Some men are mortal' is true and 'Some men are not mortal' is false. Option D reverses every value, and B and C each get E or I wrong.

Discuss in Forum

39.

Identify the fallacy committed in the following argument is .................

"All men are mortal.
All animals are mortal.
∴ "All animals are men."

A.Fallacy of undistributed middle
B.Fallacy of undistributed major
C.Fallacy of undistributed minor
D.Fallacy of illicit major

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

The argument commits the fallacy of undistributed middle.

The conclusion's terms are 'animals' (minor) and 'men' (major), so the middle term is 'mortal'. It appears as the predicate of two universal affirmative (A) propositions, and the predicate of an A proposition is undistributed. Since the middle term is not distributed in at least one premise, the premises do not connect men and animals: both may be mortal without being the same class.

The major term 'men' is distributed in its premise and undistributed in the conclusion, which is allowed, so there is no illicit major (undistributed major). The minor term 'animals' is distributed in both premise and conclusion, so there is no illicit minor either.

Discuss in Forum

40.

Correct symbolization of the following statement is :

Statement : Only students are members. (Sx, Mx)

A.(X) (Sx . Mx)
B.(X) (Sx \(\supset\) Mx)
C.(X) (Mx \(\supset\) Sx)
D.(\(\exists\)x (Sx . Mx)

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: C

The correct symbolization is (X) (Mx \(\supset\) Sx).

'Only students are members' means that nothing is a member unless it is a student, that is, all members are students. With Sx for 'x is a student' and Mx for 'x is a member':

\[ (x)(Mx \supset Sx) \]

Option B, \((x)(Sx \supset Mx)\), says all students are members, which reverses the claim; 'only' marks the predicate of the universal, not its subject. Option A says everything is both a student and a member, far too strong, and option D says only that some student is a member.

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