Abstract
The paper is concerned with the fine boundary between expressive power and reducibility of semantic and intensional notions in the context of arithmetical theories. I will consider three notions of reduction of a theory characterizing a semantic or a modal notion to the underlying arithmetical base theory – relative interpretability, speed up, conservativeness – and highlight a series of cases where moving between equally satisfactory base theories and keeping the semantic or modal principles fixed yields incompatible results. I then consider the impact of the non-uniform behaviour of these reducibility relations on the philosophical significance we usually attribute to them.
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Notes
- 1.
- 2.
Note on terminology: In what follows, we employ the adjective ‘intensional’ and ‘modal’ in an inclusive sense, familiar to medieval authors such as Ockham, which comprehend also truth and propositional attitudes.
- 3.
- 4.
For details on the subexponential theories \(\mathsf {S}^1_2\), I Δ 0 + Ω 1 the standard reference is still (Hájek and Pudlák 1993, §V).
- 5.
It should be noticed that Halbach’s (1999) contains a cut-elimination argument that was later shown to contain a gap. The problem there was exactly this absence of a suitable machinery to control the complexity of boxed formulas in proofs of \(\mathbb {L}_B\)-formulas.
- 6.
For the reader not familiar with subsystems of first-order arithmetic, it may be useful to know that it’s easy to find theories that are proper sub-theories of E A but still sequential. One example is Buss’ theory \(\mathsf {S}^1_2\) mentioned above.
- 7.
The argument is folklore, see for instance Halbach (2014, §8).
- 8.
For a formally precise definition, see for instance (Visser 2013).
- 9.
Here’s a proof: Let’s assume that U is locally interpretable in V . In V , either C o n(U) or ¬C o n(U). If the former, an interpretation can be found via the Henkin-Feferman arithmetized completeness theorem. If the latter, then there is a finite V 0 ⊆ V such that V ⊢C o n(V 0) →C o n(U 0) for all finite U 0 ⊂ U. Therefore V proves C o n(U 0). But then, by the well-known argument due to Feferman (1960), one can find an intensional consistency statement C o n ∗(U) such that V ⊢C o n ∗(U). We can then employ the Henkin-Feferman construction again.
- 10.
Personal communication.
- 11.
For a full argument, see Nicolai (2016a).
- 12.
Notice for instance, that a form of full commutation with negation is needed even for the principle
(K)that is required to close provability under
. - 13.
Where \(2^x_0=x\) and \(2^x_{y+1}=2^{2^x_y}\).
- 14.
See Fischer (2014, 4.11) for a proof in the case of PA. The method can however be generalized to all B.
- 15.
This desideratum was clearly emphasized in Halbach’s influential monograph:
…a base theory must contain at least a theory about the objects to which truth can be ascribed. […] The axioms of the truth theory can serve their purpose only if the base theory allows one to express certain facts about syntactic operations; […] it should be provable in the base theory that a conjunction is different from its conjuncts and different from any disjunction.[…] The decisive advantage of using Peano Arithmetic is that I do not have to develop a formal theory that can be used as base theory…(Halbach 2014, §2)
- 16.
A suggestion along these lines has been recently made by Fujimoto in Deflationism beyond arithmetic. Unpublished manuscript.
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Acknowledgements
This research has been supported by the European Commission, grant no. 658285 FOREMOTIONS. I would like to thank Martin Fischer, Albert Visser, and an anonymous referee for useful comments and insights.
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Nicolai, C. (2018). On Expressive Power Over Arithmetic. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_3
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