logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

, the semantics of logic or formal semantics is the study of the semantics, or

interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...

s, of formal and (idealizations of)

natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

s usually trying to capture the pre-theoretic notion of

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

Overview

The truth conditions of various sentences we may encounter in

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...

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 In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

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

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

's ''

Organon The ''Organon'' ( grc, Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logical analysis and dialectic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics. The six ...

'', especially ''

De Interpretatione ''De Interpretatione'' or ''On Interpretation'' ( Greek: Περὶ Ἑρμηνείας, ''Peri Hermeneias'') is the second text from Aristotle's '' Organon'' and is among the earliest surviving philosophical works in the Western tradition to dea ...

'', 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 In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...

, 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

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 a ...

semantic theory of truth A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences. Origin The semantic conception of truth, which is related in different ways to both the correspondence and deflati ...

, based on his

T-schema The T-schema ("truth schema", not to be confused with " Convention T") is used to check if an inductive definition of truth is valid, which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it ...

, 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 (math ...

. 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

first-order predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

is given by a mapping from terms to a universe of

individual An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one's own need ...

s, 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 Truth-conditional semantics is an approach to semantics of natural language that sees meaning (or at least the meaning of assertions) as being the same as, or reducible to, their truth conditions. This approach to semantics is principally associate ...

, 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 Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...

, Dag Prawitz and

Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He ...

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- British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is consi ...

'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 Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quant ...

for modal logics in the early 1960s and later championed by

J. Michael Dunn J. Michael Dunn (June 19, 1941 – April 5, 2021) was Oscar Ewing Professor Emeritus of Philosophy, Professor Emeritus of Informatics and Computer Science, was twice chair of the Philosophy Department, was Executive Associate Dean of the College o ...

Nuel Belnap Nuel Dinsmore Belnap Jr. (; born 1930) is an American logician and philosopher who has made contributions to the philosophy of logic, temporal logic, and structural proof theory. He taught at the University of Pittsburgh from 1963 until his reti ...

, and Hugues Leblanc for standard first-order logic.

James Garson James Garson is an American philosopher and logician. He has made significant contributions in the study of modal logic and formal semantics. He is author of ''Modal Logic for Philosophers'' and ''What Logics Mean'' by Cambridge University Press ...

has given some results in the areas of adequacy for

intensional logic Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (''extensions''), by additional quantifiers that range over terms that may have such individuals ...

s 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 Game semantics (german: dialogische Logik, translated as '' dialogical logic'') is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a play ...

'' or ''game-theoretical semantics'' made a resurgence mainly due to

Jaakko Hintikka Kaarlo Jaakko Juhani Hintikka (12 January 1929 – 12 August 2015) was a Finnish philosopher and logician. Life and career Hintikka was born in Helsingin maalaiskunta (now Vantaa). In 1953, he received his doctorate from the University of Hel ...

for logics of (finite) partially ordered quantification, which were originally investigated by

Leon Henkin Leon Albert Henkin (April 19, 1921, Brooklyn, New York - November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active schola ...

, who studied Henkin quantifiers. * ''

Probabilistic semantics One of the most severe limitations of the Semantic Web is its inability to deal with uncertain knowledge. Probabilistic semanticsSalvatore F. Pileggi, Probabilistic Semantics, International Conference on Computational Science (ICCS 2016), Proced ...

'' 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

(2007),
Socratic Epistemology: Explorations of Knowledge-Seeking by Questioning
', Cambridge: Cambridge University Press. *

Ilkka Niiniluoto Ilkka Maunu Olavi Niiniluoto (born March 12, 1946) is a Finnish philosopher and mathematician, serving as a professor of philosophy at the University of Helsinki since 1981. He is currently on leave from his position, having been appointed as ...

(1999), ''Critical Scientific Realism'', Oxford: Oxford University Press. {{Philosophy of language Mathematical logic Model theory Philosophy of language Semantics Theories of deduction Theories of truth

Overview

See also

References