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7 Conservativeness and Deflationism Inthischapter,weexploretherelationbetweendeflationismandthearithmetical nonconservativeness of the compositional theory of truth. In this context, we also describe the role that the concept of truth plays in specific philosophical disciplines such as epistemology, philosophy of language, and metaphysics. 7.1 Defining Conservativeness Deflationist theories hold that truth is somehow an “insubstantial” notion, that it does not carry much ontological weight. This claim is sometimes combined with the view that the purpose of the notion of truth is to express certain things (infinite conjunctions) that we could not otherwise express, but not to prove things about matters not concerning truth that we cannot otherwise prove. In other words, some deflationists hold that the notion of truth is argumentatively weak. Some believe that deflationism is wedded to conservativeness claims: A reasonable truth theory should be conservative over an underlying reasonable philosophical (metaphysical, epistemological, semantic, etc.) theory.1 Shapiro expresses this conviction in the following manner [Shapiro 1998, pp. 497–498]: I submit that in one form or other, conservativeness is essential to deflationism. Suppose, for instance, that Karl correctly holds a theory [A] in a language that cannot express truth. He adds a truth predicate to the language and extends [A] to a theory [A’] using only axioms essential to truth. Assume that [A’] is not conservative over [A]. Then there is a sentence  in the original language so that  is a consequence of [A’] but not a consequence of [A]. That is, it is logically possible for the axioms of [A] to be true yet  false, but it is not logically possible for the axioms of [A’] to be true and  false. This undermines the central deflationist theme that truth is insubstantial. Before 1. See [Horsten 1995] and [Field 1999]. The author of the former article now regrets the (cautious) remarks that he made in this direction. 80 Chapter 7 Karl moved to [A’], ¬ was possible. The move from [A] to [A’] added semantic content sufficient to rule out the falsity of . But by hypothesis, all that was added in [A’] were principles essential to truth. Thus, those principles have substantial semantic content. In fact, most attention in the literature has been devoted to the question of conservativeness of a reasonable theory of truth not over reasonable philosophical but over reasonable mathematical theories ([Ketland 1999], [Halbach 2001]). We can generalize the concept of conservativeness that we have discussed so far. Consider a theory Th formulated in a language LTh. Th may be a scientific (physical, mathematical, etc.) or a philosophical theory. Consider a theory of truth Tr that is intended to apply to the language of the theory Th. The theory Tr may be thought of as being formulated in the language LTr, which is defined as LTh ∪ {T }, where T is a truth predicate. Now we say that the theory Tr of truth is conservative over Th if and only if for every sentence φ of the language of LTh in which the truth predicate does not occur the following holds: If Th ∪ Tr ⊢ φ, then Th ⊢ φ. Some philosophers insist that the language LTh, in which the theory Th is formulated, must not already contain the truth predicate [Field 1999]. If this restriction is accepted, then it can usually be shown for reasonable choices of Th and Tr that the conservativeness property indeed holds. If this restriction is rejected, then for many reasonable choices of Th and Tr the conservativeness property will not hold. Already in the case where Th is a mathematical theory, it seems somewhat artificial to impose such a restriction on the language of Th. Suppose, for concreteness , that Th is Peano arithmetic. Then the restriction that the language of Th should not contain the truth predicate entails that in instances of the induction scheme, the truth predicate is not allowed to occur. This seems awkward because the induction scheme is intended to apply to all property-expressing predicates. So let us from now on allow that the language LTh in which the theory Th is formulated may already contain the truth predicate. Instead of the proof-theoretic characterization of conservativeness, one can opt for a semantic conception of conservativeness. We say that the theory Tr of truth is semantically conservative over Th if and only if for every sentence φ of the language of LTh in which the truth predicate does not occur the following holds: If Th ∪ Tr |= φ, then Th |= φ. [3.137.185.180] Project...

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