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4. The Disquotational Theory
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4 The Disquotational Theory Tarski wanted to define truth. He saw that truth could not be defined in the object language: It has to be done in an essentially stronger metalanguage. In his definition , Tarski assigned a pivotal role to the Tarski-biconditionals. Nowadays, many a deflationist philosopher takes a suitable class of Tarski-biconditionals as the axioms of her theory of truth. 4.1 Tarski on Defining Truth Tarski’s theory, as set out in [Tarski 1935], is the point of departure for most, if not all, recent publications on truth. Rather than giving a historically precise account ofTarski’s account, I elaborate on those features that are most important for later developments. Tarski’s truth predicate pertains to a specific formal language, the so-called objectlanguage. In our case, the objectlanguage is LPA. The truth predicate does not belong to the objectlanguage. It forms part of the metalanguage. In the simplest case, the objectlanguage is a sublanguage of the metalanguage (i.e., all formulae of the objectlanguage are also formulae of the metalanguage). If the metalanguage does not include the objectlanguage, then a translation of the object- into the metalanguage is required. Whether a translation is correct depends of course on the meaning of the sentences of the objectlanguage, but— surely Pilate has asked himself this question—what is “meaning”? Thus, the need for a translation has caused much anxiety. We stick to the simple case and assume that LPA is a sublanguage of the metalanguage. Tarski develops his theory at first in natural language enriched by some mathematical symbols. However, Tarski is aware of the need for a precisely defined metalanguage that allows for strict formal proofs in it. For the metalanguage, Tarski needs some rules governing the use of the expressions of the objectlanguage. Tarski assumed that we have axioms and 48 Chapter 4 rules, stated in the metalanguage, that allow for derivations within the metalanguage . He called the resulting theory “Metawissenschaft,” which is usually translated as metatheory. Tarski stated some conditions that should be satis- fied by the metatheory. It is a deductive system with axioms and rules, which contains the theory of the objectlanguage (if there is one) and is able to prove certain facts about expressions. It is assumed that the metalanguage has a name for each sentence of the objectlanguage. However, not just any kind of name will do. It must be possible to read off the syntactic shape of the sentence from its name. For this purpose, Tarski introduced his structural-descriptive names. We do not discuss Tarski’s original approach, but we present two examples of naming systems that comply with the requirement that one must be able to recover the shape of an expression from its name. In natural language, the quotational name of a sentence satisfies the condition that the name has to reveal the structure of the sentence: The singular term “Snow is white” designates the sentence within the quotation marks, and thus the name displays the exact shape of the sentence. Because the metalanguage contains the objectlanguage LPA and therefore also all standard numerals, the metalanguage has names for all sentences of the objectlanguage. Indeed, a gödel code implicitly contains the structure of the expression that it codes. Tarski aimed at a definition of truth. A (potential) definition of truth in the objectlanguage is given as an explicit definition of the primitive predicate symbol T in the metalanguage. It takes the following form: ∀x ∈ LPA : T (x) ↔ TDef (x), where TDef (x) is a complex formula in the metalanguage, containing one free variable x. The resulting theory is a definitional extension of the metatheory. We are sloppy and call the formula TDef (x) itself a truth definition. Of course, the sentence ∀x ∈ LPA : T (x) ↔ TDef (x) is the general pattern for introducing new unary predicate predicates by an explicit definition. Thus, obviously not every choice of TDef (x) is acceptable as a definition of truth. However, there may be different acceptable choices.What are the distinguishing features of an adequate definition of truth? Tarski’s answer to this question is contained in his material adequacy condition . Whether a definition is materially adequate depends on the metatheory. The metatheory must prove certain things about the predicate TDef (x) that is supposed to define truth. [18.188.175.182] Project MUSE (2024-04-17 04:37 GMT) The Disquotational Theory...