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When reading some SEP articles, this is a phrase I commonly came across, "this provides a semantics for this logic". But what does it mean?

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    Maybe a ref to the relevant SEP article may help... Jun 15, 2023 at 8:11
  • Having said that, see simple example regarding Propositional Logic: for classical logic the usual truth table interpretation of the propositional connectives. Jun 15, 2023 at 8:12
  • Unfortunately I didn't take note of where I saw it, and I can only use phone for next few hours so it'll be a while till I can update . Requesting patience Jun 15, 2023 at 8:13
  • See the entry on Classical Logic for details. Jun 15, 2023 at 8:14
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    I would guess it means "Define a mapping from a (pure) syntax to a model"
    – Rushi
    Jun 15, 2023 at 8:36

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Formal logic which often appears to be and is mathematical in nature (consider mathematical logic for instance) is a formal system in which the emphasis is on syntax. This means that the logic is both abstract and uses artificial symbols in contrast to logic expressed in natural language.

P->Q is a simple example where P and Q represent clauses and -> represents some form of logical consequence so that P->Q can represent any number of meaningful expressions. It might mean 'If Bob goes into the house, then he will find his keys' or it might mean 'If the keys are on the table, then Bob left them there'. Without providing an explanation of P->Q, it simply is unknown what the various symbols refer to. This is what it means to give a logic a semantics in the simple sense.

In the more technical, broader sense, to give a logic a semantics isn't to give a specific series of meanings to the symbols, but to define the rules of the symbols in the abstract. For instance, in Hoare Logic {P}C{Q}, {P} and {Q} are again variables, but this time instead of providing specific natural language, one restricts or binds the variables to a specific domain of discourse where the former are preconditions of state, and the latter are postconditions of state and C is the command or operation performed in the system of computation which may be among a subset of operations provided by a a programming language specified in BNF. Such a logic is used in formal verification of the correctness of a computer program.

Either way, when using abstract symbols, the symbols themselves may simplify and clarify logic, but unless they are meaningful, they are not useful.

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The simple answer is, the symbols in formal logic refer to something, like the variable x refers to some number, in math. “Semantics” is meaning: what the symbols refer to.

What a symbol refers to can be called its extension. The relationship a logical symbol has to other symbols within the logical system as it has been defined, can be called its intension. These can correspond roughly to “external reference”, and “internal meaning”, respectively.

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