View source for T-schema

{{Short description|Testing device for logical soundness}}
The '''T-schema''' ("truth [[schema (logic)|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 as the "Equivalence Schema", a synonym introduced by [[Michael Dummett]].<ref name=Künne2005>{{cite book |last=Künne |first=Wolfgang |year=2003 |title=Conceptions of truth |publisher=Clarendon Press |isbn=978-0-19-928019-3 |page=[https://archive.org/details/conceptionsoftru0000kunn/page/18 18] |url=https://archive.org/details/conceptionsoftru0000kunn |url-access=registration}}</ref>

The T-schema is often expressed in [[natural language]], but it can be formalized in [[Predicate logic|many-sorted predicate logic]] or [[modal logic]]; such a formalisation is called a "'''T-theory'''."{{Citation needed|reason=I was not able to find a formalisation of the T-schema in either of the two references for this article. Please add references for the formalisation of the T-schema in predicate logic or in modal logic, preferably both.|date=October 2019}} 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 abbreviated to S): 'S' is true [[if and only if]] 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 value]]s [[disquotational principle|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. Said again, a sentence of the form "A" is true if and only if A is true. 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 (rule of inference)|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.
Predicates for truth that meet all of these criteria are called "satisfaction classes", a notion often defined with respect to a fixed language (such as the language of [[Peano arithmetic]]); these classes are considered acceptable definitions for the notion of truth.<ref>H. Kotlarski, [https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093635929 Full Satisfaction Classes: A Survey] (1991, [[Notre Dame Journal of Formal Logic]], p.573). Accessed 9 September 2022.</ref>

==Natural languages==

[[Joseph Heath]] 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."<ref name=Heath2001>{{cite book |last=Heath |first=Joseph |year=2001 |title=Communicative action and rational choice |publisher=MIT Press |isbn=978-0-262-08291-4 |page=186 |url=https://books.google.com/books?id=3i-QXQQOfZQC&pg=PA186}}</ref> He gives as an "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|logical biconditional]].

==See also==
* [[Principle of bivalence]]
* [[Law of excluded middle]]

==References==
{{reflist}}

==External links==

* {{cite SEP |url-id=tarski-truth |title=Tarski's Truth Definitions}}
* {{cite SEP |url-id=self-reference/#Matt-sema |title=Consequences of the Semantic Paradoxes}}

{{Theories of truth}}
{{Mathematical logic}}
[[Category:Mathematical logic]]
[[Category:Philosophical logic]]
[[Category:Truth]]
[[Category:Logical expressions]]