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!
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They are formulas; see Logical consequence.– Mauro ALLEGRANZACommented May 2, 2017 at 6:41
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Sorry about beating this dead horse, but I don't think they can be formulas, they are names of formulas, I'm actually quite certain that this is the case, I assume it's common knowledge. Take an actual example: P, P ⊃ Q ⊨ Q. That is a shorthand for saying: "P","P⊃ Q" ⊨ "Q". It's just that most books never mention it, they simply are too lazy to use the quoting devises.– JohannesCommented May 2, 2017 at 10:27
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1 Answer
Some remarks:
- "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.
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One should be careful here "P, P ⊃ Q ⊨ Q" doesn't actually make sense, if the formulas flanking "⊨" are not names but are object language sentences.– JohannesCommented May 1, 2017 at 13:56
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@Johannes Sorry, I don't get what you mean, can you please explain?– user26652Commented May 1, 2017 at 14:35
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"P, P ⊃ Q ⊨ Q" is a sentence in the metalanguage, usually interpreted as expressing that the object language sentence "Q" is a consequence of the sentences "P" and "P ⊃ Q", so it has to talk about formulas hence it has to use the names of those formulas. Note that therefore in the schema "A ⊨ B" (assuming it is a schema), the schematic letters "A" and "B" take meta-language names as instances. Just like in the T-schema: "X is true iff p", this schema is expressed in the meta-meta-language, so the "X" in this schema takes as instance meta-language names of object language sentences....– JohannesCommented May 1, 2017 at 15:56
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@Johannes Okay, so you interpret "P" as a name for a sentence in another language (perhaps a natural language)? I think this is at least not standard; in propositional logic, "P" is already taken to be in object language. However, even if you take "P" to be in meta language, why would "P, P ⊃ Q ⊨ Q" not actually make sense? Of course, you will have to define "⊨" for the then-object language, but this may (for instance) be done in terms of possible worlds.– user26652Commented May 1, 2017 at 16:16
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@Johannes By the way, I don't think that in the T-schema you've given, "X" is in meta-meta-language, but I see it rather as a shorthand for a name of the sentence p. This is because we predicate truth here, so in subject position you want to have the name of a sentence, not the name of a name of a sentence.– user26652Commented May 1, 2017 at 16:17