logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, the semantics of logic or formal semantics is the study of the meaning and interpretation of

formal languages In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a set called "#Definition, alphabet". The alphabet of a formal language consists of symbol ...

, formal systems, and (idealizations of) natural languages. This field seeks to provide precise mathematical models that capture the pre-theoretic notions of

truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...

, validity, and

logical consequence Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...

. While logical syntax concerns the formal rules for constructing well-formed expressions, logical semantics establishes frameworks for determining when these expressions are true and what follows from them. The development of formal semantics has led to several influential approaches, including model-theoretic semantics (pioneered by

Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...

), proof-theoretic semantics (associated with Gerhard Gentzen and Michael Dummett), possible worlds semantics (developed by Saul Kripke and others for modal logic and related systems), algebraic semantics (connecting logic to

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

), and game semantics (interpreting logical validity through game-theoretic concepts). These diverse approaches reflect different philosophical perspectives on the nature of meaning and truth in logical systems.

Overview

The truth conditions of various sentences we may encounter in

argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...

s 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 A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

, an idealised sentence suitable for logical manipulation. Until the advent of modern logic,

Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

'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 syllogisms, but with the generality of modern logics based on the quantifier. The main modern approaches to semantics for formal languages are the following: * The archetype of ''model-theoretic semantics'' is

's semantic theory of truth, based on his T-schema, and is one of the founding concepts of

model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

. 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 Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...

'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 J. Michael Dunn, Nuel Belnap, and Hugues 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 semantics'' or ''game-theoretical semantics'' made a resurgence 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 Hartry 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.

References

* Jaakko Hintikka (2007),
Socratic Epistemology: Explorations of Knowledge-Seeking 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. {{Philosophy of language Mathematical logic Model theory Philosophy of language Semantics Theories of deduction Formal semantics (natural language)

Overview

See also

References