Abstract
We say that a sentence A is a permissive consequence of a set of premises Γ whenever, if all the premises of Γ hold up to some standard, then A holds to some weaker standard. In this paper, we focus on a three-valued version of this notion, which we call strict-to-tolerant consequence, and discuss its fruitfulness toward a unified treatment of the paradoxes of vagueness and self-referential truth. For vagueness, st-consequence supports the principle of tolerance; for truth, it supports the requisit of transparency. Permissive consequence is non-transitive, however, but this feature is argued to be an essential component to the understanding of paradoxical reasoning in cases involving vagueness or self-reference.
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Notes
- 1.
(Smith 2008), for instance, defines a notion of permissive consequence for fuzzy logic with continuum many truth values linearly ordered (the real interval [0,1]), whereby a sentence A is said to follow from a set of premises Γ provided in all models in which all formulae in Γ have value strictly greater than half, A gets a value greater or equal than 1/2. This notion of permissive consequence, as fine-grained though it appears, can in fact be shown to be representable without loss of generality in a three-valued framework, and indeed, both Smith’s notion and its three-valued version coincide with classical consequence for first-order logic. We refer to (Cobreros et al. Ms.) for details and more ample discussion of this point.
- 2.
We state the definition in terms of a multi-premise and multi-conclusion setting, although, for most of the applications we are interested in here, we can limit our perspective to the multi-premise-single-conclusion case.
- 3.
Malinowski’s notion of q-consequence is defined exactly in terms of the basic attitudes of acceptance and rejection. A sentence A is a q-consequence of a set of premises Γ whenever A is accepted when all of the premises in Γ are not rejected.
- 4.
The generalization to n-ary predicates presents no special difficulty, and only involves the definition of appropriate similarity relations between n-tuples.
- 5.
For more rigor, we might have chosen to break the proviso 21.1 into two separate constraints: first, the crispness constraint that \(M\models^{s} aI_{P}b\) iff \(M\models^{t} aI_{P}b\) (see (Cobreros et al. 2012b)), and secondly, the closeness constraint proper that if \(M\models^{s/t} aI_{P}b\), then \(| I(Pa)-I(Pb)|<1\) (not assuming the “only if” part). That way of doing things actually appears preferable to us in general, but we collapse both constraints in 21.1 for the sake of simplicity.
- 6.
- 7.
We are indebted to an anonymous reviewer for this important clarification. As pointed out by the reviewer, in Kripke’s construction the minimal fixed point V for the supervaluation schema satisfies the fixed point property, but not transparency, since as a consequence of the lack of value-functionality in the supervaluation schema, \(V(\lambda \vee{T}{\langle{\lambda}\rangle})=V(\neg{T}{\langle{\lambda}\rangle} \vee{T}{\langle{\lambda}\rangle})=1\), but \(V({T}{\langle{\lambda}\rangle} \vee{T}{\langle{\lambda}\rangle})=1/2\). Conversely, a transparent theory of truth may fail identity, for example if you start from a Kripke-Kleene model M and generate a new model \(M'\) that assigns to each sentence A the pair \(<M(A), A>\) as a value, and then simply ignore the second coordinate of its values when defining validity. This sort of model will yield the same logic as the original models, but without ever assigning the same value to any two distinct sentences, so it will exhibit transparency without identity.
- 8.
(Smith 2008) presents closeness as an explicit weakening of tolerance in his fuzzy approach, and means to endorse closeness without endorsing tolerance. See (Cobreros et al. (Ms.)) however for a more thorough discussion of the status of both principles in relation to Smith’s notion of consequence. Conversely, the theory of vagueness presented in (Cobreros et al. 2012b), which involves the notion of classical extension for vague predicates, is one in which tolerance is st-valid, but without involving the notion of closeness in truth values. Tolerance t-holds in all models despite the existence of elements a and b for which \(aI_{P}b\) holds (strictly or tolerantly), but such that \(|I(Pa)-I(Pb)|=1\) in the two-valued models used.
- 9.
See also (Cobreros et al. 2013) for a discussion of structural motivations for the admission of non-transitive consequence.
- 10.
On the comparison between norms of assertion with regard to theories of truth, see especially (Wintein 2012). Chapter 7 of (Wintein 2012), in particular, presents a theory of truth based on the strict-tolerant distinction, but taking a different perspective on Kripke-Kleene models for truth as well as on assertibility proper.
- 11.
See, again, (Zardini 2008).
- 12.
Our view on this should be compared to Priest’s original view on the status of modus ponens, a rule that is not LP-valid, but that Priest calls a “quasi-validity”, still applicable to sentences that are not paradoxical. In our system, modus ponens is a validity, but wherever Priest talks of quasi-validities that are lost in relation to the conditional, we can speak of corresponding classical metainferences that are lost for consequence in the vicinity of paradoxes.
- 13.
We are indebted to an anonymous reviewer for drawing attention to this point.
- 14.
- 15.
Rogers’ emphasis on the use of partial, as opposed to total, functions as a way of blocking diagonalization arguments bears some analogy with the idea of limiting the expressiveness of our language to block the strengthened Liar.
- 16.
See (Ripley 2013a) for an application of this strategy to deal with a particular version of higher-order vagueness.
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Acknowledgments
We are grateful to two anonymous referees for detailed comments, and to the editors for their assistance in the preparation of this paper. We also thank audiences in Amsterdam, Barcelona and Paris. We thank the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n 229 441-CCC, the ANR project BTAFDOC (“Beyond Truth and Falsity: Degrees of Confidence”), and the program “Non-Transitive Logics” (Ministerio de Economía y Competitividad, Government of Spain, FFI2013-46451-P). We also thank grants ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL*.
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Cobreros, P., Egré, P., Ripley, D., Rooij, R. (2015). Vagueness, Truth and Permissive Consequence. 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_21
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