T-schema
Encyclopedia
The T-schema or truth schema (not to be confused with 'Convention T') is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski
's semantic theory of truth
. Some authors refer to it as the "Equivalence Schema", a synonym introduced by Dummett.
The T-schema is often expressed in natural language
, but it can be formalized in many-sorted predicate logic or modal logic
; such a formalisation is called a T-theory. T-theories form the basis of much fundamental work in philosophical logic
, where they are applied in several important controversies in analytic philosophy
.
As expressed in semi-natural language (where 'S' is the name of the sentence abbrivated to S):
'S' is true if and only if
S
Example: 'snow is white' is true if and only if snow is white.
points out that "The analysis of the truth predicate
provided by Tarski's Schema T is not capable of handling all occurrences of the truth predicate in natural language. In particular, Schema T treats only “freestanding” uses of the predicate—cases when it is applied to complete sentences." He gives as "obvious problem" the sentence:
Heath argues that analyzing this sentence using T-schema generates the sentence fragment—“everything that Bill believes”—on the righthand side of the biconditional.
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...
's semantic theory of truth
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 deflationary conceptions, is due to work published by Polish...
. Some authors refer to it as the "Equivalence Schema", a synonym introduced by Dummett.
The T-schema is often expressed in natural language
Natural language
In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...
, but it can be formalized in many-sorted predicate logic or modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
; such a formalisation is called a T-theory. T-theories form the basis of much fundamental work in philosophical logic
Philosophical logic
Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
, where they are applied in several important controversies in analytic philosophy
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
.
As expressed in semi-natural language (where 'S' is the name of the sentence abbrivated to S):
'S' is true if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
S
Example: 'snow is white' is true if and only if snow is white.
The inductive definition
By using the schema one can give an inductive definition for the truth of compound sentences. Atomic sentences are assigned truth values disquotationally. For example, the sentence "'Snow is white' is true" becomes materially equivalent with the sentence "snow is white", i.e. 'snow is white' is true if and only if snow is white. The truth of more complex sentences is defined in terms of the components of the sentence:- A sentence of the form "A and B" is true if and only if A is true and B is true
- A sentence of the form "A or B" is true if and only if A is true or B is true
- A sentence of the form "if A then B" is true if and only if A is false or B is true; see material implication.
- A sentence of the form "not A" is true if and only if A is false
- A sentence of the form "for all x, A(x)" is true if and only if, for every possible value of x, A(x) is true.
- A sentence of the form "for some x, A(x)" is true if and only if, for some possible value of x, A(x) is true.
Natural languages
Joseph HeathJoseph Heath
Joseph Heath is a philosophy professor at the University of Toronto. He also teaches at the School of Public Policy and Governance. He received his BA from McGill University, where his teachers included Charles Taylor, and his MA and PhD degrees are from Northwestern University, where he studied...
points out that "The analysis of the truth predicate
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 "is true."...
provided by Tarski's Schema T is not capable of handling all occurrences of the truth predicate in natural language. In particular, Schema T treats only “freestanding” uses of the predicate—cases when it is applied to complete sentences." He gives as "obvious problem" the sentence:
- Everything that Bill believes is true.
Heath argues that analyzing this sentence using T-schema generates the sentence fragment—“everything that Bill believes”—on the righthand side of the biconditional.