Abstract
The Revision theory of truth is known, in its full (transfinite) form, as one way of dealing with circular concepts (see Gupta and Belnap 1993). The restriction of this approach which is obtained by limiting it to arbitrary, but finite steps of revision is less known, and less studied instead. In this paper we try to assess it, both from the point of view of its motivations, and of those properties which are relevant for establishing a connection with the logical investigation. Finally, we try to see how much of this approach can we make use of in the case of truth.
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Notes
- 1.
Notice that it is irrelevant that Alice knows Bob’s payoffs and vice versa. This depends upon their distribution in this game, and the fact that one and the same option for both players (not investing), is strictly “dominated” by the other alternative, as one would say it in game–theoretic terms.
- 2.
For the sake of this account, we only consider hypotheses which are different from the empty set (i.e., hypotheses that appear in a diagram representing the given situation as a game). It is clear that if one relaxes this condition then for \(H=\emptyset\) one has \(\delta_R(H)=H\) as well.
- 3.
This feature of the given game depends upon the payoffs distribution, as we said. Games like the one we have considered in the introduction are indeed regular, as Gupta (2000) calls them (as they refer to circular definitions which can be said to be regular in the sense of (Gupta and Belnap 1993, Def. 5A.8, p. 149). As it is made clear by Gupta himself, regularity plays an important role in the revision–theoretic account of finite games, though it is too much a special feature to become a standard (see also Bruni and Sillari 2011 on this).
- 4.
As a matter of fact, this impression is illusory: the reason why FR needs to be formulated in this way is stated in § 8.3, and is related to the uniform way in which the original concept of n–validity was defined.
- 5.
The material below follows a comment and a suggestion in this sense due to Philip Welch.
- 6.
See (Welch 2003). There it is shown that sets which are revision–theoretically definable with respect to the transfinite process are of a complexity that is at least \(\Updelta^1_2\) (\(\Pi^1_2\) in certain cases), independently of what limit rule is used.
- 7.
Gupta and Belnap’s book aims at dealing with a more general situation, where a base language \(\mathcal{L}\) is inflated to a language \(\mathcal{L}^+\) which contains a countable set \(\{G_i\}_{i\in\mathbb{N}}\) of circular concepts defined by formulas A i of \(\mathcal{L}^+\) (hence, a situation where any of these G i depends upon all G j ’s which occur in A i for \(i,j\in\mathbb{N}\)). The result we are referring to, that is (Gupta and Belnap 1993, Thrm. 5B.1, pp. 162 ff.), is proved to hold with respect to this expanded setting.
- 8.
The result is stated, without proof, by Gupta in (2000).
- 9.
It would be natural to ask to what an extent one can represent derivability in systems HC \(_n\) themselves by the new implication connective. Despite its interest, this topic would take us afield. Hence, we refer the interested reader to (Bruni 2012), where the issue is dealt with at length.
- 10.
The process of indexing a standard logic calculus hides some subtleties, which is nonetheless unnecessary to go into here. We refer the interested reader to (Bruni 2012) for details.
- 11.
We use the symbol ⊃ for the standard implication connective.
- 12.
Also, the reader is invited to see Gupta and Standefer (2011) for an actual elaboration of this view.
- 13.
Here we use a standard sequent notation, with Γ and ∆ indicating multisets of indexed formulas of \(\mathcal{L}^+\).
- 14.
We assume that the formulas of \(\mathcal{L}^+\) have been assigned a Gödel number according to the application of a standard arithmetization tecnique, and write \(\ulcorner A\urcorner\) for the code of the formula A.
- 15.
We use here the symbol ≡ for logical equivalence based on material implication ⊃.
- 16.
Non–standard models of theories \(\textsf{FS}\) \(_n\) could be also provided. For instance, any model of \(\textsf{FS}\) is also a model of \(\textsf{FS}\) \(_n\). Among other things, this was pointed out to the author by an anonymous referee. The author would like to thank the referee, whose comments helped to improve the previous version of the paper.
- 17.
To be more precise: let i(A) be defined for every formula A of \(\mathcal{L}^+\) by
$$i(A)=\min k\in\mathbb{N}.\mathsf{FS}_k\vdash A$$Then, the previous theorem ensures that if \(\mathsf{FS}\vdash A\), then \((\mathbb{N},\delta_T^{i(A)}(Y))\models A\) holds for every \(Y\subseteq\mathbb{N}\). Hence, A is 0–valid.
- 18.
Since the system \(\textsf{FS}\) we are trying to embed has a logic base system in axiomatic form, the required indexed version needs not to feature fully nested occurrences of the new → connective. For the sake of Theorem 4 below, the reader can think of → as occurring solely in the truth–definition axioms from point 3 of the present list. The reader should consult (Bruni 2012) for related remarks.
- 19.
This requires that the indexed modus ponens MP\(_\rightarrow\) from § 8.3.1 be among the logical inference rules of \(\textsf{T}\).
- 20.
Notice that, among the axioms we have spoken of in § 8.3.1 we have not included in \(\textsf{T}\) the schema of index shift (IS). Hence, the need of assuming A i as axiom for every formula A which is valid at every stage i. Anyway, due to the possibility that the induction schema be instantiated by formulas of \(\mathcal{L}^+\), the inclusion of (IS) in the above list would not have been sufficient to obtain an equivalent, and maybe more elegant system.
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Bruni, R. (2015). Some Remarks on the Finite Theory of Revision. 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_8
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