Semantics of logic
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Part of Formal languages 
In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal languages and (idealizations of) natural languages usually trying to capture the pretheoretic notion of logical consequence.
Overview
The truth conditions of various sentences we may encounter in arguments 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 the proposition, an idealised sentence suitable for logical manipulation.^{[citation needed]}
Until the advent of modern logic,
The main modern approaches to semantics for formal languages are the following:
 The archetype of modeltheoretic semantics is firstorder predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false". Modeltheoretic semantics provides the foundations for an approach to the theory of meaning known as truthconditional semantics, which was pioneered by Donald Davidson. Kripke semanticsintroduces innovations, but is broadly in the Tarskian mold.
 Prooftheoretic semantics associates the meaning of propositions with the roles that they can play in inferences. Gerhard Gentzen, Dag Prawitz and Michael Dummett are generally seen as the founders of this approach; it is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
 Truthvalue semantics (also commonly referred to as substitutional quantification) was advocated by Ruth Barcan Marcus for modal logics in the early 1960s and later championed by J. Michael Dunn, Nuel Belnap, and Hugues Leblanc for standard firstorder logic. James Garson has given some results in the areas of adequacy for intensional logics outfitted with such a semantics. The truth conditions for quantified formulas are given purely in terms of truth with no appeal to domains whatsoever (and hence its name truthvalue semantics).
 Henkin quantifiers.
 Probabilistic semantics originated from Hartry Field and has been shown equivalent to and a natural generalization of truthvalue semantics. Like truthvalue semantics, it is also nonreferential in nature.
See also
References
 Jaakko Hintikka (2007), Socratic Epistemology: Explorations of KnowledgeSeeking by Questioning, Cambridge: Cambridge University Press.
 Ilkka Niiniluoto (1999), Critical Scientific Realism, Oxford: Oxford University Press.
 John N. Martin (2019), The Cartesian Semantics of the Port Royal Logic , Routledge.