In linguistics, formal semantics seeks to understand linguistic meaning by constructing precise mathematical models of the principles that speakers use to define relations between expressions in a natural language and the world that supports meaningful discourse. The mathematical tools used are the confluence of formal logic and formal language theory, especially typed lambda calculi.
Linguists rarely employed formal semantics until Richard Montague showed how English (or any natural language) could be treated like a formal language. His contribution to linguistic semantics, which is now known as Montague grammar, was the basis for further developments, like the categorial grammar of Bar-Hillel and colleagues, and the more recent type-logical semantics (or grammar) based on Lambek calculus.
There is some disagreement concerning the explanatory roles attributed to formal semantics. Several theorists ground semantics on facts about communication, convention and truth, whereas others tend to see it as a syntactically-driven project primarily concerned with explaining productivity and systematicity in natural language, and thus part of a larger linguistic enterprise such as Chomskyan linguistics or any other modular view of the human linguistic ability.
Most current approaches to formal semantics fall within the paradigm of the so-called truth-conditional semantics, which attempts to explain the meaning of a sentence by providing the conditions under which it would be true. However, several adherents to the truth-conditional program have also argued that there is more to meaning than truth-conditions. Alternative approaches include more cognitive-oriented proposals such as Pietroski's treatment of meanings as instructions to build concepts, sentences being devoid of truth-conditions. Another line of inquiry, using linear logic, is glue semantics, which is based on the idea of "interpretation as deduction", closely related to the "parsing as deduction" paradigm of categorial grammar.
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