Questions: Logical Consistency and Contradiction

Question 1

A legal brief relies on two premises: 'A contract requires consideration to be binding' and 'This contract is binding even though no consideration was exchanged.' What is the logical consequence of using these premises together?

• A. The argument is weakened but can still support a narrow conclusion
• B. The argument fails only if the conclusion is also false
• C. From these contradictory premises, any conclusion whatsoever can be derived in classical logic
• D. The argument is invalid but not necessarily unsound

Answer: (C)

The two premises directly contradict each other: the first says consideration is required; the second says the contract is binding without it. In classical logic, a contradictory set of premises entails everything — this principle is called ex contradictione quodlibet ('from contradiction, anything'). An argument 'proving' everything proves nothing; its conclusion carries no evidential weight. The inconsistency must be resolved before any valid inference can be drawn.

Question 2

Which of the following sets of statements is logically consistent?

• A. 'No mammals can fly.' 'Bats are mammals.' 'Bats can fly.'
• B. 'The defendant was in Paris.' 'The defendant was in Chicago.' 'The clocks in both cities were accurate.' (same moment in time)
• C. 'She arrived before noon.' 'She arrived at 2 PM.' 'The clock was accurate.'
• D. 'All prime numbers greater than 2 are odd.' '17 is prime.' '17 is odd.'

Answer: (D)

Options A, B, and C all contain contradictions — no possible situation makes all their claims simultaneously true. Option D is consistent: all three statements are true simultaneously (17 is prime, all primes > 2 are odd, so 17 is odd — the claims reinforce each other). Consistency requires only that the claims *can* all be true at once, not that they are mutually supportive or even interesting together.

Question 3

A consistent set of premises guarantees that an argument is sound.

• T. True
• F. False

Answer: False

Consistency means the premises can all be true simultaneously — but they might all be consistently false. Soundness requires both that the argument is valid (conclusion follows from premises) AND that the premises are actually true. A consistent set of false premises produces an unsound argument even if it's valid. Consistency is a necessary condition to avoid ex contradictione quodlibet, but it is nowhere near sufficient for soundness.

Question 4

If a set of premises is inconsistent, then every conclusion that can be derived from them using valid inference rules is technically entailed by those premises.

• T. True
• F. False

Answer: True

This is the principle of explosion (ex contradictione quodlibet): in classical logic, if you accept P and not-P, you can derive any proposition Q. The proof uses disjunction introduction (from P, derive P ∨ Q) and disjunctive syllogism (from P ∨ Q and not-P, derive Q). Because every proposition is entailed by a contradiction, a contradictory argument cannot distinguish true conclusions from false ones — it 'proves' everything and therefore nothing of use.

Question 5

Why does an argument with contradictory premises 'prove' nothing, even when each individual inference step is logically valid?

Model answer: Because in classical logic, a contradiction entails every proposition. If premises include both P and not-P, any conclusion Q can be derived using valid steps (disjunction introduction and disjunctive syllogism). Since the argument can equally 'prove' Q and not-Q, it provides no evidence for either — the proof is technically valid but epistemically worthless.

This exposes why consistency is a prerequisite, not just a preference. Valid reasoning preserves truth: if the premises are true and the reasoning is valid, the conclusion must be true. But contradictory premises can never all be true simultaneously, so the truth-preservation guarantee breaks down entirely. The argument form becomes a machine that outputs anything, which means it outputs nothing meaningful. Consistency is the minimum standard a set of premises must meet before logical analysis of the argument can proceed.