Questions: Tautologies and Contradictions

Question 1

In a proof by contradiction, you assume ¬Q and derive a contradiction. What exactly does deriving a contradiction establish?

• A. That ¬Q is false in some cases, so Q is probably true
• B. That the logical system itself contains an inconsistency
• C. That ¬Q cannot possibly be true under any truth assignment, so Q must be true
• D. That Q is a tautology — true under all interpretations of its component variables

Answer: (C)

A contradiction is false under every possible truth assignment — it cannot be true. If valid logical steps from assumption ¬Q lead to a contradiction, then ¬Q cannot be true in any world where logic holds. The inference is: if ¬Q were true, the contradiction would have to be true (since each step was valid); but the contradiction cannot be true; therefore ¬Q cannot be true. This is a deductive certainty, not a probabilistic claim. Option D is wrong: Q need not be a tautology — it just must be true in the specific context at hand.

Question 2

Which of the following is a tautology?

• A. p → q
• B. p ∨ q
• C. p → (q → p)
• D. ¬p → q

Answer: (C)

p → (q → p) is true under every truth assignment. If p is true, then q → p is true (p is the consequent and it's true), so the whole statement is true. If p is false, the antecedent of the outer conditional is false, making the whole statement true regardless of q. Options A and D are contingencies — false when antecedent is true and consequent false. Option B is false when both p and q are false. Only C is always true.

Question 3

The statement 'The sun is either currently shining or it is not currently shining' is a tautology.

• T. True
• F. False

Answer: True

This is an instance of p ∨ ¬p — the law of excluded middle — which is always true regardless of the actual weather. A tautology is true by its logical structure alone, not because of contingent facts. Whether it's sunny or cloudy today is irrelevant; the statement's truth is guaranteed by its form.

Question 4

Any statement that has been observed to be true in nearly every case examined so far is a tautology.

• T. True
• F. False

Answer: False

A tautology must be true under every logically possible truth assignment, not just every empirically observed case. 'All swans are white' was observed to be true in Europe for centuries — but it was a contingency, not a tautology, because non-white swans were possible (and turned out to exist). Tautologies are true by virtue of logical structure; a contingency can be consistently false under some assignment even if you've never witnessed that assignment.

Question 5

Why does deriving a contradiction from an assumption prove that the assumption is false? What property of contradictions makes proof by contradiction work?

Model answer: A contradiction is false under every possible truth assignment — there is no possible world where it is true. If valid logical steps from assumption A lead to a contradiction, then A cannot be true in any world where logic holds: if A were true, the contradiction would have to be true (each step was valid); but contradictions are impossible; therefore A is impossible. The contradiction functions as a logical impossibility — a destination that proves the journey to it was impossible.

This distinguishes proof by contradiction from merely finding a false or surprising conclusion. A surprising conclusion is still possible; a contradiction (R ∧ ¬R) violates the basic law of non-contradiction. Reaching it under valid inference condemns the premise that set you on that path — which is why the method is a deductive proof, not an argument by implausibility.