Semantic theory of truth
Encyclopedia
A semantic theory of truth is a theory of truth in the philosophy of language
which holds that truth is a property of sentences.
and deflationary
conceptions, is due to work published by Polish
logic
ian Alfred Tarski
in the 1930s. Tarski, in "On the Concept of Truth in Formal Languages", attempted to formulate a new theory of truth in order to resolve the liar paradox
. In the course of this he made several metamathematical discoveries, most notably Tarski's undefinability theorem using the same formal technique as Kurt Gödel
used in his incompleteness theorems
. Roughly, this states that a truth-predicate satisfying convention-T for the sentences of a given language cannot be defined within that language.
es like the liar paradox
, it is generally necessary to distinguish the language that one is talking about, the so-called object language
, from the language that one is using, the so-called metalanguage
. In the following, quoted sentences like "'P'" are always names of sentences whereas the unquoted sentences are the sentences in the metalanguage which are translations of sentences in the object language. Tarski demanded that the object language was 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 of a language, a sentence of the form (T):
(1) 'P' is true if, and only if, P.
(where 'P' is the name of the sentence P in the metalanguage which in turn is a translation of the corresponding sentence in the object language.)
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. But this would also be a T-sentence:
(3) 'Der Schnee ist weiß' is true (in German) 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
s. He gave a number of reasons for not extending his theory to natural languages, including the problem of there being no systematic way of deciding whether a given sentence of a natural language is well-formed, and that natural languages are 'closed'; that is, they can describe the semantic characteristics of their own elements. But Tarski's approach was extended by Davidson
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
.)
Tarski developed the theory to give an inductive definition of truth as follows.
For a language L containing ~ ("not"), & ("and"), v ("or") and quantifiers ("for all" and "there exists"), Tarski's inductive definition of truth looks like this:
These explain how the truth conditions of complex sentences (built up from connective
s and quantifiers) can be reduced to the truth conditions of their constituent
s. The simplest constituents are atomic sentence
s. A contemporary semantic definition of truth would define truth for the atomic sentences as follows:
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 terms of truth, so it would be circular were he 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 much contemporary philosophy of language
. It is a rather controversial matter whether Tarski's semantic theory should be counted as either a correspondence theory
or as a deflationary theory
.
Philosophy of language
Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for analytic philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the relationship between 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 correspondenceCorrespondence theory of truth
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 that world...
and deflationary
Deflationary theory of truth
A deflationary theory of truth is one of a family of theories which all have in common the claim that assertions that predicate truth of a statement do not attribute a property called truth to such a statement.-Redundancy theory:...
conceptions, is due to work published by Polish
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
ian Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
in the 1930s. Tarski, in "On the Concept of Truth in Formal Languages", attempted to formulate a new theory of truth in order to resolve the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
. In the course of this he made several metamathematical discoveries, most notably Tarski's undefinability theorem using the same formal technique as Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
used in his incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
. Roughly, this states that a truth-predicate satisfying convention-T for the sentences of a given language cannot be defined within that language.
Tarski's Theory
To formulate linguistic theories without semantic paradoxParadox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es like the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
, it is generally necessary to distinguish the language that one is talking about, the so-called object language
Object language
An object language is a language which is the "object" of study in various fields including logic, linguistics, mathematics and theoretical computer science. The language being used to talk about an object language is called a metalanguage...
, from the language that one is using, the so-called metalanguage
Metalanguage
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language...
. In the following, quoted sentences like "'P'" are always names of sentences whereas the unquoted sentences are the sentences in the metalanguage which are translations of sentences in the object language. Tarski demanded that the object language was 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 of a language, a sentence of the form (T):
(1) 'P' is true if, and only if, P.
(where 'P' is the name of the sentence P in the metalanguage which in turn is a translation of the corresponding sentence in the object language.)
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. But this would also be a T-sentence:
(3) 'Der Schnee ist weiß' is true (in German) 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
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s. He gave a number of reasons for not extending his theory to natural languages, including the problem of there being no systematic way of deciding whether a given sentence of a natural language is well-formed, and that natural languages are 'closed'; that is, they can describe the semantic characteristics of their own elements. But Tarski's approach was extended by Davidson
Donald Davidson (philosopher)
Donald Herbert Davidson was an American philosopher born in Springfield, Massachusetts, who served as Slusser Professor of Philosophy at the University of California, Berkeley from 1981 to 2003 after having also held teaching appointments at Stanford University, Rockefeller University, Princeton...
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
Truth-conditional semantics is an approach to semantics of natural language that sees the meaning of assertions as being the same as, or reducible to, their truth conditions...
.)
Tarski developed the theory to give an inductive definition of truth as follows.
For a language L containing ~ ("not"), & ("and"), v ("or") and quantifiers ("for all" and "there exists"), Tarski's inductive definition of truth looks like this:
- (i) A negation ~A is true if and only if A is not true.
- (ii) A conjunction A&B is true if and only if A is true and B is true.
- (iii) A disjunction A v B is true if and only if A is true, B is true, or both are true.
- (iv) A universal statement "for all x A(x)" is true if and only if each object satisfies "A(x)".
- (v) An existential statement "there exists x A(x)" is true if and only if there is an object which satisfies "A(x)".
These explain how the truth conditions of complex sentences (built up from connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s and quantifiers) can be reduced to the truth conditions of their constituent
Constituent (linguistics)
In syntactic analysis, a constituent is a word or a group of words that functions as a single unit within a hierarchical structure. The analysis of constituent structure is associated mainly with phrase structure grammars, although dependency grammars also allow sentence structure to be broken down...
s. The simplest constituents are atomic sentence
Atomic sentence
In logic, an atomic sentence is a type of declarative sentence which is either true or false 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:
- (vi) An atomic sentence F(x1,...,xn) is true (relative to an assignment of values to the variables x1, ..., xn)) if the corresponding valueValue (mathematics)In mathematics, value commonly refers to the 'output' of a function. In the most basic case, that of unary, single-valued functions, there is one input and one output .The function f of the example is real-valued, since each and every possible function value is real...
s of variablesVariable (mathematics)In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
bear the relationRelation (mathematics)In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
expressed by the predicate 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 terms of truth, so it would be circular were he to use one of them in the definition of truth itself. Tarski's semantic conception of truth plays an important role in modern logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
and also in much contemporary philosophy of language
Philosophy of language
Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for analytic philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the relationship between language...
. It is a rather controversial matter whether Tarski's semantic theory should be counted as either a correspondence theory
Correspondence theory of truth
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 that world...
or as a deflationary theory
Deflationary theory of truth
A deflationary theory of truth is one of a family of theories which all have in common the claim that assertions that predicate truth of a statement do not attribute a property called truth to such a statement.-Redundancy theory:...
.
Further reading
- Simon BlackburnSimon BlackburnSimon Blackburn is a British academic philosopher known for his work in quasi-realism and his efforts to popularise philosophy. He recently retired as professor of philosophy at the University of Cambridge, but remains a distinguished research professor of philosophy at the University of North...
and Keith Simmons, eds., 1999. Truth. Oxford University Press, ISBN 0-19-875250-4. - Wilfrid HodgesWilfrid HodgesWilfrid Augustine Hodges is a British mathematician, known for his work in model theory. He was Professor of Mathematics at Queen Mary, University of London from 1987 to 2006, and is the author of numerous books on logic....
, 2001. Tarski's truth definitions. In the Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
. - Richard KirkhamRichard KirkhamRichard Ladd Kirkham is an American philosopher. Among his published works are the much-cited Theories of Truth , "Does the Gettier Problem Rest on a Mistake?" Mind , and "On Paradoxes and a Surprise Exam" Philosophia . Kirkham graduated from Cornell College in 1977 and received his Ph.D...
, 1992. Theories of Truth. Bradford Books, ISBN 0-262-61108-2. - Alfred TarskiAlfred TarskiAlfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, 1944. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy and Phenomenological Research 4.
External links
- Tarski's Truth Definitions (an entry of Stanford Encyclopedia of Philosophy)
- Alfred TarskiAlfred TarskiAlfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, 1944. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy and Phenomenological Research 4.