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Inlogic, thesemantics of logic orformal semantics is the study of themeaning andinterpretation offormal languages,formal systems, and (idealizations of)natural languages. This field seeks to provide precise mathematical models that capture the pre-theoretic notions oftruth,validity, andlogical consequence. Whilelogical syntax concerns the formal rules for constructing well-formed expressions, logical semantics establishes frameworks for determining when these expressions are true and what follows from them.
The development of formal semantics has led to several influential approaches, includingmodel-theoretic semantics (pioneered byAlfred Tarski),proof-theoretic semantics (associated withGerhard Gentzen andMichael Dummett),possible worlds semantics (developed bySaul Kripke and others formodal logic and related systems),algebraic semantics (connecting logic toabstract algebra), andgame semantics (interpreting logical validity throughgame-theoretic concepts). These diverse approaches reflect different philosophical perspectives on the nature of meaning and truth in logical systems.
Thetruth conditions of various sentences we may encounter inarguments will depend upon their meaning, and so logicians cannot completely avoid the need to provide some treatment of the meaning of these sentences. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician traditionally is not interested in the sentence as uttered but in theproposition, an idealised sentence suitable for logical manipulation.[citation needed]
Until the advent of modern logic,Aristotle'sOrganon, especiallyDe Interpretatione, provided the basis for understanding the significance of logic. The introduction ofquantification, needed to solve theproblem of multiple generality, rendered impossible the kind of subject–predicate analysis that governed Aristotle's account, although there is a renewed interest interm logic, attempting to findcalculi in the spirit of Aristotle'ssyllogisms, but with the generality of modern logics based on the quantifier.
The main modern approaches to semantics for formal languages are the following:
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