Semantics of logic

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.

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 are the following:

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.

Notes

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory	Set Empty set Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Range Map Function Relation Ordered pair

Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Philosophy of language

Philosophers	Plato (Cratylus) Gorgias Confucius Xunzi Aristotle Stoics Pyrrhonists Scholasticism Ibn Rushd Ibn Khaldun Thomas Hobbes Gottfried Wilhelm Leibniz Johann Herder Ludwig Noiré Wilhelm von Humboldt Fritz Mauthner Paul Ricœur Ferdinand de Saussure Gottlob Frege Franz Boas Paul Tillich Edward Sapir Leonard Bloomfield Zhuangzi Henri Bergson Lev Vygotsky Ludwig Wittgenstein Philosophical Investigations Tractatus Logico-Philosophicus Bertrand Russell Rudolf Carnap Jacques Derrida Of Grammatology Limited Inc Benjamin Lee Whorf Gustav Bergmann J. L. Austin Noam Chomsky Hans-Georg Gadamer Saul Kripke A. J. Ayer G. E. M. Anscombe Jaakko Hintikka Michael Dummett Donald Davidson Paul Grice Gilbert Ryle P. F. Strawson Willard Van Orman Quine Hilary Putnam David Lewis John Searle Joxe Azurmendi Scott Soames Stephen Yablo John Hawthorne Stephen Neale

Theories	Causal theory of reference Contrast theory of meaning Contrastivism Conventionalism Cratylism Deconstruction Descriptivist theory of names Direct reference theory Dramatism Expressivism Linguistic determinism Logical atomism Logical positivism Mediated reference theory Nominalism Non-cognitivism Phallogocentrism Quietism Relevance theory Semantic externalism Semantic holism Structuralism Supposition theory Symbiosism Theological noncognitivism Theory of descriptions Verification theory

Concepts	Ambiguity Linguistic relativity Meaning Language Truth-bearer Proposition Use–mention distinction Concept Categories Set Class Intension Logical form Metalanguage Mental representation Principle of compositionality Property Sign Sense and reference Speech act Symbol Entity Sentence Statement more...

Related articles	Analytic philosophy Philosophy of information Philosophical logic Linguistics Pragmatics Rhetoric Semantics Formal semantics Semiotics

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