The T-schema ("truth
schema
Schema may refer to:
Science and technology
* SCHEMA (bioinformatics), an algorithm used in protein engineering
* Schema (genetic algorithms), a set of programs or bit strings that have some genotypic similarity
* Schema.org, a web markup vocab ...
", 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
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 ...
's
semantic theory of truth. Some authors refer to it as the "Equivalence Schema", a synonym introduced by
Michael Dummett.
The T-schema is often expressed in
natural language
A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...
, but it can be formalized in
many-sorted predicate logic or
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 ...
; 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
Analytic philosophy is a broad movement within Western philosophy, especially English-speaking world, anglophone philosophy, focused on analysis as a philosophical method; clarity of prose; rigor in arguments; and making use of formal logic, mat ...
.
As expressed in semi-natural language (where 'S' is the name of the sentence abbreviated to S): 'S' is true
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
s
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.
* 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.
[H. Kotlarski]
Full Satisfaction Classes: A Survey
(1991, Notre Dame Journal of Formal Logic, p.573). Accessed 9 September 2022.
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."
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
logical biconditional
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment, is the logical connective used to conjoin two statements P and Q to form th ...
.
See also
*
Principle of bivalence
*
Law of excluded middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...
References
External links
*
*
{{Mathematical logic
Mathematical logic
Philosophical logic
Truth
Logical expressions