I'm just beginning to study propositional logic myself in advance of a course I'm going to be taking on it, and I'm slightly confused by the notion of semantic equivalency (the double turnstile). What exactly do e.g. "A" and "B" in the statement A ⊨ B represent? Are they just free variables, or something else? I'm not sure how free variables can have semantic properties. Also, is semantic equivalency a relation of identity or of entailment? I've seen it referred to as "semantic entailment" but entailment and equivalency seem rather different to me. Thanks in advance!
- "A" and "B" generally represent formulas. Hence, on its own, "A ⊨ B" is a just scheme. You can fill in specific formulas to get an evaluable statement, e.g. "P, P ⊃ Q ⊨ Q", which is true (modus ponens), or "P ⊨ (P ∧ Q)", which is false.
- At least given standard terminology, "A" and "B" therefore are not free variables. Free variables are a matter in predicate logic. In propositional logic, however, there are no free variables at all.
- The double turnstile (⊨) does not mean semantic equivalence, but mere semantic entailment. So a statement of the form "A ⊨ B" means that whenever A is true, B is also true (in propositional logic, any assignment that makes A true also makes B true).
- Semantic equivalence means that (given two formulas A and B), both A ⊨ B and B ⊨ A. In other words, semantic equivalence means that A and B entail each other, or in propositional logic, that A and B are true under exactly the same assignments.