Logic, Western and Indian Questions and Answers

Answer: B

The incorrect statement is All its premises are true but its conclusion is false.

A deductive argument is valid when it is impossible for all its premises to be true and its conclusion false. That combination is exactly what validity rules out, so a valid argument can never have true premises and a false conclusion.

Every other combination is possible for a valid argument. True premises with a true conclusion is the sound case. False premises can lead validly to a false conclusion, or validly to a true one, as in 'All fish are mammals; all whales are fish; so all whales are mammals'.

Answer: A

By quantity, propositions are divided into universal and particular.

In traditional logic the quantity of a categorical proposition depends on whether it refers to the whole of its subject class ('All S is P', 'No S is P') or only part of it ('Some S is P', 'Some S is not P'). The first are universal, the second particular.

Affirmative and negative is the division by quality. Necessary and probable is a division by modality. 'Existential' is a term from modern quantification theory, contrasting with universal quantifiers, not the traditional division by quantity.

Answer: C

The answer is 'p ∨ q' is true and '~p' is false.

A disjunction is true when at least one disjunct is true. Since p is true, p ∨ q is true whatever the value of q. A negation has the opposite value of what it negates, so ~p is false.

'Both are true' and 'both are false' each get one of the two values wrong, and the last option reverses both.

Answer: D

The correct form is [~(p • q) ≡ (~p ∨ ~q)] and [~(p ∨ q) ≡ (~p • ~q)].

De Morgan's laws say that the negation of a conjunction is equivalent to the disjunction of the negations, and the negation of a disjunction is equivalent to the conjunction of the negations: 'not both' means 'at least one is not', and 'not either' means 'neither'.

The first option gives material equivalence and a related form, and the second gives the commutation laws for conjunction and disjunction. The third negates only p inside the brackets, not the whole conjunction or disjunction, so it is not De Morgan's law.

Answer: B

The fallacy is composition.

The fallacy of composition argues that what is true of each part, taken separately, must be true of the whole. Here the premise is about each individual (each person's pleasure is good for that person) and the conclusion is about the whole (the pleasure of everybody is good for everybody). This is the classic objection to J.S. Mill's proof of the principle of utility.

Division is the reverse, from whole to parts. The fallacy of accident applies a general rule to an exceptional case, and the converse fallacy of accident generalises from an exceptional case, neither of which happens here.

Answer: D

The correct symbolization is (x) {(Bx ∨ Wx) ⊃ [(Ax ∨ Fx) ⊃ Sx]}.

'Bees and wasps sting' does not mean that anything is both a bee and a wasp; it means anything that is a bee or a wasp stings, so the subject is Bx ∨ Wx. 'If they are either angry or frightened' is the condition Ax ∨ Fx, and the consequence is Sx. Nesting the condition gives: for any x, if x is a bee or a wasp, then if x is angry or frightened, x stings.

The other options use 'Vx', which is not one of the given predicates, and also join the subjects with a conjunction (or the emotions with one), making the statement apply only to things that are both bees and wasps, or both angry and frightened.

Answer: B

The Rule of Detachment is a rule of transformation.

A formal system has rules of formation, which say which strings of symbols are well-formed formulas, and rules of transformation (inference), which say how new formulas may be derived from given ones. The rule of detachment, modus ponens, lets us derive q from p and p ⊃ q, 'detaching' the consequent. It is therefore a rule of transformation.

It does not define what counts as a formula, so it is not a rule of formation, and it is not both or neither.

Answer: A

The key gives valid.

An effective decision procedure (as for truth tables in logic) must be mechanical, requiring no ingenuity; it must end in a finite number of steps; and it must be reliable, always giving a correct yes or no answer.

Validity is a property of arguments, which a decision procedure may be used to test, not a characteristic of the procedure itself.

Answer: D

The characteristics are consistency, completeness and independence.

An axiomatic system is judged by three properties: consistency (no contradiction can be derived from the axioms), completeness (every truth of the field can be derived), and independence (no axiom can be derived from the others).

'Contingent' describes a proposition that is neither necessarily true nor necessarily false; it is not a requirement for a system of axioms, so every option including it is wrong.

Answer: D

The correct symbolisation is (X) [Sx . (Px ∨ Ox) ⊃ Hx].

'All successful politicians and orators are hard-working' says of anything that if it is successful and is a politician or an orator, then it is hard-working. The 'and' joining politicians and orators lists two groups, so it becomes a disjunction: a successful orator need not also be a politician.

Option A, with 'Px . Ox', would cover only those who are both politicians and orators, and leaves Hx outside the quantifier. Option C asserts that everything is successful and hard-working, and option B separates the quantifiers.