Consider a vocabulary with only four propositions A, B, C and D. How many…

Concept: For a propositional vocabulary of n atomic propositions, each proposition can independently be assigned true or false, so there are 2n distinct truth assignments (models) in total. For a disjunction P ∨ Q, the sentence is false only in the single case where both P and Q are false, so its models = (total assignments) − (assignments where both P and Q are false).

Application:

1. This vocabulary has four propositions A, B, C, D, so the total number of truth assignments is 24 = 16.

2. The sentence B ∨ C is false only when B = false and C = false; A and D are then unconstrained, giving 22 = 4 assignments where the sentence is false.

3. So the number of models where B ∨ C is true = 16 − 4 = 12.

Cross-check: Split by the value of B. If B is true (8 of the 16 assignments, since A, C, D vary freely), the sentence is automatically true. If B is false, the sentence needs C to be true, which happens in 4 of the remaining 8 assignments. Total = 8 + 4 = 12, confirming the result.