Questions: Truth Assignments and Valuations

Question 1

Under the truth assignment P = T, Q = F, what is the truth value of the formula (P → Q)?

• A. True — because P is true, the implication is validated
• B. False — because the antecedent is true but the consequent is false
• C. True — because Q being false makes the implication vacuously true
• D. Undefined — implications require both sides to have the same truth value

Answer: (B)

The material conditional (P → Q) is false only when the antecedent (P) is true and the consequent (Q) is false — exactly this assignment. P = T and Q = F gives T → F = F. Options A and C both make the same error: expecting that a true antecedent 'validates' the implication. The truth-functional definition of → says the opposite — a true antecedent with a false consequent is the one case that makes the implication false.

Question 2

A formula has 2 atomic propositions and is true under 3 of the 4 possible truth assignments. Which category describes this formula?

• A. Tautology — it is true under the majority of assignments
• B. Contradiction — it is false under at least one assignment
• C. Contingency — it is true under some assignments and false under others
• D. Valid — it is true more often than false

Answer: (C)

A tautology is true under ALL assignments; a contradiction is false under ALL assignments; a contingency is true under SOME and false under SOME. This formula is true under 3/4 assignments and false under 1/4 — so it is a contingency. 'Majority' has no semantic significance. Note that option B confuses 'contradiction' (false under all assignments) with 'not a tautology' (false under some). The single false assignment is exactly what makes this a contingency rather than a tautology.

Question 3

The truth value of a compound formula is determined entirely by the truth values of its atomic propositions and the connectives used — the subject matter or meaning of those propositions is irrelevant.

• T. True
• F. False

Answer: True

This is the compositionality principle of truth-functional semantics. Once you have a truth assignment for the atomic propositions, the valuation function recursively computes truth values bottom-up through the formula tree. Whether P means 'It is raining' or 'The number is prime' plays no role — only whether it is assigned T or F. This is what makes propositional logic formal and mechanical: the computation is purely syntactic once you have the assignment.

Question 4

A formula that is true under nearly every truth assignment you have tested so far is therefore a tautology.

• T. True
• F. False

Answer: False

A tautology must be true under ALL 2ⁿ possible truth assignments (for n atomic propositions). Testing a subset — even a large one — does not establish tautologyhood, because untested assignments might falsify the formula. The only complete method is a truth table enumerating all 2ⁿ assignments, or a proof in a formal system. Tautology is a universal quantifier over all assignments, not a statistical claim.

Question 5

What does it mean for a formula to be a contingency, and why does the concept of a truth assignment make this possible?

Model answer: A contingency is a formula that is true under some truth assignments and false under others — neither always true (tautology) nor always false (contradiction). The concept of a truth assignment makes this possible because it decouples a formula's truth value from any fixed meaning: the same formula can evaluate to T under one assignment and F under another. Without truth assignments, a formula would have a single fixed truth value, and the three-way distinction among tautology, contradiction, and contingency would not exist.

This is the crucial conceptual shift from natural language to formal logic. In natural language, 'It is raining' has a truth value determined by the world. In propositional logic, the atomic proposition P has no fixed truth value — it takes whatever value the assignment gives it. This freedom is what makes formal logic a tool for studying the structure of reasoning rather than specific empirical claims. A tautology is true regardless of what the propositions mean; a contingency's truth depends on what they happen to mean in a given context.