A student argues that 'If it is raining, then the ground is wet' is a tautology because it holds true in every real-world situation they can think of. What is wrong with this reasoning?
ANothing — a statement that is true in all real cases qualifies as a tautology
BThe statement is a tautology only in rainy climates, not universally
CA tautology must be true under all possible truth-value assignments, including the logically possible case where rain does not wet the ground — empirical reliability is not enough
DThe statement is actually a contradiction because it can be falsified in principle
Tautologies are logical, not empirical. A tautology is true in every row of its truth table — for every possible combination of truth values its components could have. 'If it rains, the ground is wet' is contingently true: its truth table has a row where 'it rains' is T and 'ground is wet' is F, giving the conditional F. That row makes it non-tautological. The student is confusing 'this has always been true in experience' with 'this cannot possibly be false' — a fundamental confusion between empirical and logical necessity.
Question 2 Multiple Choice
Which of the following compound statements is a tautology?
AP → Q
BP ∧ ¬P
CP ∨ ¬P
DP ↔ Q
P ∨ ¬P ('P or not-P') is true in every row of its truth table: when P is T, the left disjunct is T; when P is F, the right disjunct is T. So the whole statement is always T — a tautology. P ∧ ¬P is a contradiction (always F). P → Q and P ↔ Q are contingencies — their truth depends on the values of P and Q. For instance, P → Q is F when P is T and Q is F.
Question 3 True / False
A contradiction can play a useful role in mathematical proof — detecting one proves that an assumption must be false.
TTrue
FFalse
Answer: True
This is exactly the structure of proof by contradiction. You assume the negation of what you want to prove (¬P) and derive a contradiction — a statement of the form Q ∧ ¬Q, which is always false. Since a contradiction is impossible, and you derived it through valid steps from ¬P, the assumption ¬P must be false. Therefore P is true. Contradictions are not just logical dead ends — they are the diagnostic tool that makes this entire proof strategy work.
Question 4 True / False
If a statement has been verified true for a large number of specific cases, it has been shown to be a tautology.
TTrue
FFalse
Answer: False
A tautology requires truth across ALL possible truth-value assignments — infinitely many in general. Verifying many specific cases only shows the statement is frequently true, not always true. This is the same mistake as thinking many supporting examples prove a mathematical claim: no finite number of confirming instances suffices. A single counterexample — one row of the truth table where the statement is false — disqualifies it as a tautology. Tautologies must be verified by exhaustive truth-table analysis (or formal proof), not case-checking.
Question 5 Short Answer
P ∨ ¬P is a tautology, but 'the sky is always blue' is not — even if the sky really is always blue. Why does the difference matter for logical inference?
Think about your answer, then reveal below.
Model answer: P ∨ ¬P is true under every possible truth-value assignment by logic alone — it cannot be false. 'The sky is always blue' is an empirical claim that could be false in some possible world (at night, during a storm). Logical inference requires tautologies as its foundation because a valid rule must hold necessarily, not just contingently. If we built inference rules on empirically true claims, those rules would break down whenever the world changed. Tautologies guarantee that inference steps are truth-preserving in every possible case, which is what 'valid' means.
The distinction tracks the difference between logical necessity and empirical contingency. Tautologies are validated by the structure of the statement itself — no knowledge of the world is needed. This is why they can serve as the basis of inference: 'from P and P→Q, conclude Q' is valid because (P ∧ (P→Q))→Q is a tautology, not because of anything contingent about reality. Grounding inference in tautologies gives logic its generality and reliability across all domains.