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 A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathe ...

, and (idealizations of)

natural languages A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and Language change, change. It can take different forms, typically either a spoken language or a sign language. Na ...

. 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 Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the prop ...

(associated with

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 ...

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." H ...

possible worlds semantics A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...

(developed by

Saul Kripke Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American analytic philosophy, analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emer ...

and others for

modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...

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 Game semantics is an approach to Formal semantics (logic), formal semantics that grounds the concepts of truth or Validity (logic), validity on Game theory, game-theoretic concepts, such as the existence of a winning strategy for a player. In this ...

(interpreting logical validity through

game-theoretic Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

concepts). These diverse approaches reflect different philosophical perspectives on the nature of meaning and truth in logical systems.

Overview

The

truth condition In semantics and pragmatics, a truth condition is the condition under which a sentence is true. For example, "It is snowing in Nebraska" is true precisely when it is snowing in Nebraska. Truth conditions of a sentence do not necessarily reflect c ...

s 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 The ''Organon'' (, 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, who maintained against the ...

'', especially ''

De Interpretatione ''On Interpretation'' (Greek: , ) is the second text from Aristotle's ''Organon'' and is among the earliest surviving philosophical works in the Western tradition to deal with the relationship between language and logic in a comprehensive, explic ...

'', provided the basis for understanding the significance of logic. The introduction of quantification, needed to solve the

problem of multiple generality The problem of multiple generality names a failure in traditional logic to describe valid inferences that involves multiple quantifiers. For example, it is intuitively clear that if: :''Some cat is feared by every mouse'' then it follows logical ...

, rendered impossible the kind of subject–predicate analysis that governed Aristotle's account, although there is a renewed interest in

term logic In logic and formal semantics, 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 ...

, attempting to find calculi in the spirit of Aristotle's

syllogism A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defin ...

s, 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

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 deflat ...

, 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 (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 First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

is given by a mapping from terms to a universe of

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

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 associ ...

, which was pioneered by Donald Davidson.

Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...

introduces innovations, but is broadly in the Tarskian mold. * ''

Proof-theoretic semantics Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the prop ...

'' associates the meaning of propositions with the roles that they can play in inferences.

Dag Prawitz Dag Prawitz (born 1936, Stockholm) is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction, and for his contributions to proof-theoretic semantics. Prawitz is a member of the ...

and

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 In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and J. Michael Dunn and Nuel Belnap. It is also called the ''substitution interpretation'' ...

'' (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

s in the early 1960s and later championed by J. Michael Dunn,

Nuel Belnap Nuel Dinsmore Belnap Jr. (; May 1, 1930 – June 12, 2024) was 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 ...

, and Hugues Leblanc for standard first-order logic. James Garson 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 individu ...

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 is an approach to Formal semantics (logic), formal semantics that grounds the concepts of truth or Validity (logic), validity on Game theory, game-theoretic concepts, such as the existence of a winning strategy for a player. In this ...

'' 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. Hintikka is regarded as the founder of formal epistemic logic and of game semantics for logic. Life and career Hintikka was born in ...

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 Type theory, theory of types. He was an ...

, who studied Henkin quantifiers. * '' Probabilistic semantics'' originated from

Hartry Field Hartry Hamlin Field (born November 30, 1946) is an American philosopher. He is Silver Professor of Philosophy at New York University; he is a notable contributor to philosophy of science, philosophy of mathematics, epistemology, and philosophy of ...

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 12 March 1946) is a Finnish philosopher and mathematician, serving as a professor of philosophy at the University of Helsinki since 1981. He was appointed as rector of the University of Helsinki on 1 August 20 ...

(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