Questions: Truth Functions and Interpretation

Question 1

A student claims: 'The formula (P → Q) is false.' What is the correct response?

• A. The student is right — (P → Q) is always false because it depends on unknown variables
• B. The formula (P → Q) has no inherent truth value; it is true or false only relative to a specific interpretation that assigns values to P and Q
• C. The student is wrong because (P → Q) is a tautology — it is always true
• D. The claim is acceptable shorthand for 'P → Q seems false given what we know'

Answer: (B)

Formulas do not have inherent truth values — they are evaluated relative to interpretations. Under the interpretation P=T, Q=F, the formula (P → Q) evaluates to F. Under P=T, Q=T, it evaluates to T. Under P=F, Q=anything, it evaluates to T. Without specifying an interpretation, 'the formula is true/false' is a category error. Only tautologies (true under all interpretations) and contradictions (false under all interpretations) have truth values independent of interpretation.

Question 2

Under the interpretation A = F, B = T, what is the truth value of ¬(A ∧ ¬B)?

• A. False — because A is false, the conjunction fails
• B. True — computed step by step: ¬B = F, A ∧ F = F, ¬F = T
• C. True — because ¬B = F and negating a false conjunction gives true
• D. Indeterminate — without knowing what A and B mean, we cannot evaluate

Answer: (B)

Apply truth functions bottom-up through the parse tree. Step 1: ¬B = ¬T = F. Step 2: A ∧ ¬B = F ∧ F = F. Step 3: ¬(A ∧ ¬B) = ¬F = T. The result is T. The key skill here is mechanical application of truth functions — no interpretation of 'what A means' is required or useful. Option D (indeterminate) represents the misconception that formulas need semantic content to evaluate; they only need an interpretation assigning T/F to each atom.

Question 3

Two formulas are logically equivalent if and only if they produce the same truth value under every possible interpretation.

• T. True
• F. False

Answer: True

True — this is the definition of logical equivalence (φ ≡ ψ). Two formulas define the same truth function if and only if they agree on every row of the truth table. For example, (P → Q) ≡ (¬P ∨ Q) because both evaluate to F only when P=T and Q=F, and T otherwise. Logical equivalence is verified exhaustively over all 2^n interpretations for n atoms. This makes propositional logic decidable: you can always determine equivalence by truth table.

Question 4

The formula (A ∨ ¬A) is true because nearly every proposition A refers to something that is either true or false in the real world.

• T. True
• F. False

Answer: False

False — and this gets the explanation backwards. (A ∨ ¬A) is a tautology because of its logical structure: under any interpretation, either A = T (making A ∨ ¬A = T ∨ F = T) or A = F (making A ∨ ¬A = F ∨ T = T). The formula's tautologous status follows purely from how the truth functions for ∨ and ¬ interact — no facts about the real world are needed. The claim in the question reverses the logic: the tautology is not true because of empirical facts; it would remain a tautology even in a world where nothing is true or false in any meaningful sense.

Question 5

Why do formulas in propositional logic have no inherent truth value, and what determines their truth value instead?

Model answer: A formula is a syntactic object — a string of symbols. It has no inherent truth value because its atomic propositions (P, Q, A, etc.) are uninterpreted variables, not statements about a fixed state of affairs. What determines a formula's truth value is an interpretation: a function that assigns T or F to each atomic proposition. Given an interpretation, the truth value of the whole formula is computed mechanically by applying truth functions bottom-up through the formula's structure. The same formula evaluates differently under different interpretations unless it is a tautology (always T) or contradiction (always F).

This separation between syntax (the formula) and semantics (its truth value under an interpretation) is foundational to logic. It means we can study the logical relationships between formulas — entailment, equivalence, satisfiability — without committing to what the variables 'really mean.' It also makes logic computationally tractable: to check whether a formula is a tautology, enumerate all 2^n interpretations and verify T on each. This would be impossible if formulas had meaning-dependent truth values that couldn't be enumerated.