A semantic theory of truth is a

theory of truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...

in the

philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...

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 deflationary conceptions, is due to work by

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...

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 premise ...

ian

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 ...

. Tarski, in "On the Concept of Truth in Formal Languages" (1935), attempted to formulate a new theory of truth in order to resolve the

liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth ...

. In the course of this he made several metamathematical discoveries, most notably

Tarski's undefinability theorem Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that ''arithmetical truth ...

using the same formal technique Kurt Gödel used in his

incompleteness theorems Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

. Roughly, this states that a truth-predicate satisfying

Convention T 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 ...

for the sentences of a given language cannot be defined ''within'' that language.

Tarski's theory of truth

To formulate linguistic theories without semantic

paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

es such as the

, it is generally necessary to distinguish the language that one is talking about (the ''object language'') from the language that one is using to do the talking (the '' metalanguage''). In the following, quoted text is use of the object language, while unquoted text is use of the metalanguage; a quoted sentence (such as "''P''") is always the metalanguage's ''name'' for a sentence, such that this name is simply the sentence ''P'' rendered in the object language. In this way, the metalanguage can be used to talk about the object language; Tarski's theory of truth (

1935) demanded that the object language be contained in the metalanguage. Tarski's material adequacy condition, also known as Convention T, holds that any viable theory of truth must entail, for every sentence "''P''", a sentence of the following form (known as "form (T)"): (1) "P" is true

if, and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...

, P. For example, (2) 'snow is white' is true if and only if snow is white. These sentences (1 and 2, etc.) have come to be called the "T-sentences". The reason they look trivial is that the object language and the metalanguage are both English; here is an example where the object language is German and the metalanguage is English: (3) 'Schnee ist weiß' is true if and only if snow is white. It is important to note that as Tarski originally formulated it, this theory applies only to

formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sy ...

s. He gave a number of reasons for not extending his theory to natural languages, including the problem that there is no systematic way of deciding whether a given sentence of a natural language is well-formed, and that a natural language is ''closed'' (that is, it can describe the semantic characteristics of its own elements). But Tarski's approach was extended by

Davidson Davidson may refer to: * Davidson (name) * Clan Davidson, a Highland Scottish clan * Davidson Media Group * Davidson Seamount, undersea mountain southwest of Monterey, California, USA * Tyler Davidson Fountain, monument in Cincinnati, Ohio, USA * ...

into an approach to theories of '' meaning'' for natural languages, which involves treating "truth" as a primitive, rather than a defined, concept. (See

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 ...

.) Tarski developed the theory to give an

inductive definition In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include fact ...

of truth as follows. (See

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 ...

) For a language ''L'' containing ¬ ("not"), ∧ ("and"), ∨ ("or"), ∀ ("for all"), and ∃ ("there exists"), Tarski's inductive definition of truth looks like this: * (1) A primitive statement "''A''" is true if, and only if, ''A''. * (2) "¬''A''" is true if, and only if, "''A"'' is not true. * (3) "''A''∧''B''" is true if, and only if, "''A" is true'' and "''B" is true''. * (4) "''A''∨''B''" is true if, and only if, "''A" is true'' or "''B" is true'' or ("''A" is true'' and "''B" is true''). * (5) "∀''x''(''Fx'')" is true if, and only if, for all objects x; "Fx" is true. * (6) "∃''x''(''Fx'')" is true if, and only if, there is an object ''x'' for which "Fx" is true. These explain how the truth conditions of ''complex'' sentences (built up from connectives and quantifiers) can be reduced to the truth conditions of their

constituent Constituent or constituency may refer to: Politics * An individual voter within an electoral district, state, community, or organization * Advocacy group or constituency * Constituent assembly * Constituencies of Namibia Other meanings * Const ...

s. The simplest constituents are

atomic sentence In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences ...

s. A contemporary semantic definition of truth would define truth for the atomic sentences as follows: * An atomic sentence ''F''(''x''₁,...,''x''_''n'') is true (relative to an

assignment Assignment, assign or The Assignment may refer to: * Homework * Sex assignment * The process of sending National Basketball Association players to its development league; see Computing * Assignment (computer science), a type of modification to ...

of values to the variables ''x''₁, ..., ''x''_''n'')) if the corresponding

value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...

s of variables bear the relation expressed by the

predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...

''F''. Tarski himself defined truth for atomic sentences in a variant way that does not use any technical terms from semantics, such as the "expressed by" above. This is because he wanted to define these semantic terms in the context of truth. Therefore it would be circular to use one of them in the definition of truth itself. Tarski's semantic conception of truth plays an important role in modern logic and also in contemporary

. It is a rather controversial point whether Tarski's semantic theory should be counted either as a

correspondence theory In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world ...

or as a deflationary theory.

Kripke's theory of truth

Kripke's theory of truth (

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

1975) is based on partial logic (a logic of partially defined

truth predicate In formal theories of truth, a truth predicate is a fundamental concept based on the sentences of a formal language as interpreted logically. That is, it formalizes the concept that is normally expressed by saying that a sentence, statement or idea ...

s instead of Tarski's logic of totally defined truth predicates) with the strong Kleene evaluation scheme.Axiomatic Theories of Truth (Stanford Encyclopedia of Philosophy)
/ref>

References

External links

Semantic Theory of Truth
''Internet Encyclopedia of Philosophy''
Tarski's Truth Definitions
(an entry o
Stanford Encyclopedia of Philosophy
*

, 1944.
The Semantic Conception of Truth and the Foundations of Semantics
''Philosophy and Phenomenological Research'' 4. {{Mathematical logic Mathematical logic Semantics Theories of truth Theories of deduction

Origin

Tarski's theory of truth

Kripke's theory of truth

See also

References

Further reading

External links