Semantics of logic
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math :: logic :: terms
The validity of an argument depends upon the meaning or semantics of the sentences that make it up.
Aristotle's "Organon", especially "On Interpretation", gives a cursory outline of semantics which the scholastic logicians, particularly in the XIII and XIV century, developed into a complex and sophisticated theory, called Supposition Theory.
This showed how the truth of simple sentences, expressed schematically, depend on how the terms "supposit" or stand for certain extra-linguistic items.
For example, in part II of his "Summa Logicae", William of presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not. Thus "every A is B' is true iff there is something for which 'A' stands, and there is nothing for which 'A' stands, for which 'B' does not also stand."
Early modern logic defined semantics purely as a relation between ideas.
Antoine Arnauld in the "Port Royal Logic", says that "after conceiving things by our ideas, we compare these ideas, and, finding that some belong together and some do not, we unite or separate them. This is called affirming or denying, and in general judging". Thus truth and falsity are no more than the agreement or disagreement of ideas.
This suggests obvious difficulties, leading Locke to distinguish between 'real' truth, when our ideas have 'real existence' and 'imaginary' or 'verbal' truth, where ideas like harpies or centaurs exist only in the mind. This view was taken to the extreme in the XIX century, and is generally held by modern logicians to signify a low point in the decline of logic before the XX century.
Modern semantics is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subject-predicate analysis that underlies medieval semantics.
The main modern approach is model-theoretic semantics, based on Alfred Tarski's semantic theory of truth. The approach assumes 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 domain of discourse: an interpretation of first-order 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".
Model-theoretic semantics is one of the fundamental concepts of model theory. Modern semantics also admits rival approaches, such as the proof-theoretic semantics that associates the meaning of propositions with the roles that they can play in inferences, an approach that ultimately derives from the work of Gerhard Gentzen on structural proof theory and is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
In logic, the semantics of logic is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment.
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.
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 subject-predicate 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:
Model-theoretic semantics is the archetype of Alfred Tarski's semantic theory of truth, based on his T-schema, 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 first-order 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". Model-theoretic semantics provides the foundations for an approach to the theory of meaning known as Truth-conditional semantics, which was pioneered by Donald Davidson. Kripke semantics introduces innovations, but is broadly in the Tarskian mold.
Proof-theoretic 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".
Truth-value 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 first-order 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 truth-value semantics).
Game-theoretical 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 truth-value semantics. Like truth-value semantics, it is also non-referential in nature.