Formal semantics (logic)
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In logic, formal semantics OR logical semantics,^{1}^{2}^{3} is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pretheoretic notion of entailment. (Although both linguistics and logic lay claim to providing theories of natural language, according to Geach, logic generally ignores the "idiotism of idiom", and sees natural languages as cluttered with idioms of no logical interest.)^{4}
A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols (also called an alphabet) and a set of formation rules (also called a formal grammar) which determine which strings of symbols are wellformed formulas. When transformation rules (also called rules of inference) are added, and certain sentences are accepted as axioms (together called a deductive system or a deductive apparatus) a logical system is formed. An interpretation of a formal language is (roughly) an assignment of meanings to its symbols and truthconditions to its sentences.^{5}
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, Aristotle's Organon, especially De Interpretatione, provided the basis for understanding the significance of logic. The introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subjectpredicate analysis that governed Aristotle's account, although there is a renewed interest in term logic, attempting to find calculi in the spirit of Aristotle's syllogistic but with the generality of modern logics based on the quantifier.
The main modern approaches to semantics for formal languages are the following:
 Modeltheoretic semantics is the archetype of Alfred Tarski's semantic theory of truth, based on his Tschema, and is one of the founding concepts of model theory. This is the most widespread approach, and is based on the idea that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined mathematical domains: an interpretation of 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 semantics introduces 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 Dunn, Belnap, and 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).
 Gametheoretical semantics has made a resurgence lately mainly due to Jaakko Hintikka for logics of (finite) partially ordered quantification which were originally investigated by Leon Henkin, who studied Henkin quantifiers.
 Probabilistic semantics originated from H. Field and has been shown equivalent to and a natural generalization of truthvalue semantics. Like truthvalue semantics, it is also nonreferential in nature.
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