Abstract
We present an overview of typed and untyped disquotational truth theories with the emphasis on their (non)conservativity over the base theory of syntax. Two types of conservativity are discussed: syntactic and semantic. We observe in particular that TB—one of the most basic disquotational theories—is not semantically conservative over its base; we show also that an untyped disquotational theory PTB is a syntactically conservative extension of Peano Arithmetic.
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- 1.
It’s worth mentioning that Horwich attributes truth to propositions, not sentences or utterances.
- 2.
Admittedly, it is not the only possible option. Cf. Beall (2009), where disquotationalism is understood as a view that truth is a “fully transparent device” (p. 3); more exactly: it’s “a device introduced via rules of intersubstitution: that \(Tr(ulcorner \alpha urcorner)\) and α are intersubstitutable in all (nonopaque) contexts” (p. 1). On this approach, adopting T-biconditionals as axioms might be just one of the possible ways to give justice to the disquotationalist’s intuitions.
- 3.
See e.g. Halbach (2001), p. 188: “As far as I can see, neither have deflationists subscribed to conservativeness explicitly nor does it follow from one of their other doctrines. (…) But if the deflationist understands his claim that truth is not a substantial notion as implying that his truth theory has no substantial consequences, he commits a mistake.”
- 4.
- 5.
Cf. Halbach (2011), p. 55, where the proof is given that UTB 1 is conservative over PA.
- 6.
On partial truth predicates, see Kaye (1991), p. 119 ff.
- 7.
Cf. (Kaye 1991), p. 228, Proposition 15.4.
- 8.
Although we worked in M 2, the transition to M 1 is made possible by the fact that all formulas in our type belong to the language L n , i.e. they do not contain “T n ”, so if they are satisfied in M 2, they are also satisfied in M 1.
- 9.
After obtaining Theorem 7, I found out that the result was proved earlier by Fredrik Engström. Engström’s work is unpublished.
- 10.
For more about coded sets, see e.g. Kaye (1991) p. 141 ff.
- 11.
More information about prime models can be found in Kaye (1991), p. 91 ff.
- 12.
One path could consist in considering maximal conservative sets of substitutions of a T-schema. It has been shown however, that there are uncountably many such sets and none of them is axiomatizable. See Cieśliński (2007).
- 13.
In Curry’s paradox we consider a sentence ψ satisfying the condition: \(\psi \equiv [T(ulcorner \psi urcorner) \rightarrow 0 = 1]\). It turns out then that adopting a T-biconditional for ψ results in a contradiction. However, a Curry sentence ψ constructed by diagonalization contains an occurrence of the truth predicate which is not negated.
- 14.
A presentation of KF can be found in Halbach (2011), starting on p. 195.
- 15.
The proof of Theorem 13 was presented on the “Truth be told” conference in Amsterdam (2011), and also in (Cieśliński 2011).
- 16.
Strictly speaking, T 0 will contain not only (codes of) arithmetical sentences true in M, but also these nonstandard numbers a, for which the formula “\(a \in d_0\)” is satisfied in M.
- 17.
In this respect the situation of an adherent of a typed disquotational theory is quite comfortable; his problems lie elsewhere: in the deductive weakness of his theory.
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Acknowledgements
This work was supported by a grant number N N101 170438 by the Polish Ministry of Science and Education (MNiSW).
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Cieśliński, C. (2015). Typed and Untyped Disquotational Truth. In: Achourioti, T., Galinon, H., Martínez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_15
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