Logic, Western and Indian Questions and Answers

Answer: A

The truth of an inductive argument's premises does not in an a priori way lead to the truth of its conclusion.

In a valid deductive argument the conclusion follows by logical form alone, so true premises guarantee a true conclusion necessarily and a priori. In an inductive argument the conclusion goes beyond the premises, so even true premises make it only probable; it remains logically possible that the premises are true and the conclusion false. That is Hume's problem of induction.

The other options describe the kind of support induction does give: prima facie, empirically grounded (a posteriori) and forward-looking support for the conclusion.

Answer: D

'All that glitters is not gold' is an O proposition.

Despite its universal-looking form, the proverb does not mean that nothing that glitters is gold (an E proposition, which would be false, since gold glitters). It means that not everything that glitters is gold, that is, 'Some things that glitter are not gold': a particular negative, the O form.

It is not an A ('All S is P') or an I ('Some S is P'). Sentences of the form 'All S is not P' are standardly analysed as the denial of the A proposition, which is its contradictory O.

Answer: B

The argument commits the fallacy of illicit minor.

The minor term is 'animals', the subject of the conclusion. In the conclusion 'No animals are perfect', an E proposition, it is distributed. In the premise 'All men are animals', an A proposition, it is the predicate and is undistributed. A term distributed in the conclusion must be distributed in its premise, so the minor term is illicitly distributed.

The major term 'perfect' is distributed in both its premise and the conclusion (both E), so there is no illicit major. The middle term 'men' is distributed in both premises, so there is no fallacy of the middle term. Since the error has a name, 'none of the above' is wrong.

Answer: C

The subject term of the conclusion is the minor term.

In a categorical syllogism the conclusion's subject is the minor term and its predicate the major term; the premise containing each is named after it. The middle term appears in both premises but not in the conclusion, linking the other two.

The copula is the verb 'is' or 'are' joining subject and predicate, not a term at all. For example, in 'All men are mortal; Socrates is a man; so Socrates is mortal', 'Socrates' is the minor term, 'mortal' the major and 'man' the middle.

Answer: C

Both terms are distributed in an E proposition.

'No S is P' says something about every S (each is excluded from P) and about every P (each is excluded from S), so both terms are distributed.

The other forms differ: an A proposition ('All S is P') distributes only its subject; an O proposition ('Some S is not P') distributes only its predicate; an I proposition ('Some S is P') distributes neither term.

Answer: A

A and E propositions are contraries.

On the traditional square of opposition, 'All S is P' and 'No S is P' cannot both be true, but they can both be false (when some S are P and some are not). That is the relation of contrariety.

Sub-contraries are I and O, which cannot both be false. Contradictories, A and O or E and I, have opposite truth values. Subalternation is the relation from a universal to its particular: A to I and E to O.

Answer: D

A proposition must be either true or false.

In traditional and classical logic, a proposition is what a declarative sentence asserts, and its defining feature is that it has a truth value: it is true or false, though we may not know which. Questions, commands and exclamations are not propositions because they are neither.

A proposition need not be true, since false propositions are still propositions, and it need not be tautologous, since most propositions are contingent. 'Neither true nor false' describes what is not a proposition.

Answer: A

Both 'p ⊃ q' and '~ p . q' are true.

With p false and q true:

• A material conditional is false only when its antecedent is true and its consequent false. Here the antecedent p is false, so p ⊃ q is true.
• ~ p is true (p is false), and q is true, so the conjunction ~ p . q is true.

Any option making either statement false misreads the truth tables of the horseshoe or of the dot.

Answer: A

The correct match is Tautology : ~ (p . ~ p).

~ (p . ~ p), the law of non-contradiction, is true whatever the truth value of p: p . ~ p is always false, so its negation is always true.

The others are mismatched. p ∨ ~ p, the law of excluded middle, is also a tautology, not a contingent formula. The law of double negation is p ≡ ~ ~ p, and the printed '~ (~ ∧ p)' is not even well formed. The rule shown in option D, inferring φa from (∃x) φx, is existential instantiation; existential generalisation goes the other way, from φa to (∃x) φx.

Answer: D

The correct symbolization is (x)[(Ax ∨ Px) ⊃ (Ix . Sx)].

'Authors and poets are intelligent and successful' says that anyone who is an author, and anyone who is a poet, is both intelligent and successful. So the condition is a disjunction: for every x, if x is an author or x is a poet, then x is intelligent and x is successful.

Using Ax . Px in the antecedent (options A and B) would restrict the claim to people who are both authors and poets. Option B also weakens the predicate to 'intelligent or successful', and option C, with a dot in place of the horseshoe, claims that everything is an author, a poet, intelligent and successful.