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Meta-Inferences and Supervaluationism

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Meta-Inferences and Supervaluationism Journal of Philosophical Logic https://doi.org/10.1007/s10992-021-09618-4 Meta-inferences and Supervaluationism Luca Incurvati1 · Julian J. Schloder¨ 1 Received: 27 May 2020 / Accepted: 1 July 2021 / © The Author(s) 2021 Abstract Many classically valid meta-inferences fail in a standard supervaluationist frame- work. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissi- bility of inference rules. Thus, supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the conse- quence relation that preserves truth in a model of the non-normal modal logic NT. Finally, we extend our approach to a first-order setting and show that supervaluation- ism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented. Keywords Meta-inferences · Supervaluationism · Global consequence · Multilateral logic · Higher-order vagueness 1 Introduction Necessary truth is often identified with truth in all possible worlds. The super- valuationist approach to vagueness identifies truth simpliciter with truth on all Luca Incurvati [email protected] Julian J. Schloder¨ [email protected] 1 Department of Philosophy & ILLC, University of Amsterdam, Amsterdam, Netherlands L. Incurvati, J.J. Schloder¨ precisifications. David Lewis [17] famously said that possible worlds are ways the world could have been. Precisifications are ways language could be made precise. In standard modal logic, consequence is defined locally, as the preservation of truth at a world. In a supervaluationist framework, it is natural to define con- sequence globally, as the preservation of truth on all precisifications. But if the supervaluationist object language contains a definitely operator Δ, certain classi- cal meta-inferences fail for global consequence, notably conditional proof, reductio ad absurdum, proof by cases, contraposition and, in a first-order setting, existen- tial instantiation. This calls into question whether supervaluationism can provide an account of good deductive reasoning [10, 11, 28]. If certain classically valid argument patterns fail, supervaluationists must indicate which argument patterns may be employed in legitimate inference. This is the proof-theoretic problem for supervaluationism. To solve the proof-theoretic problem, we provide a proof system for superval- uationist logic which includes supervaluationistically acceptable versions of the problematic meta-inferences. The proof system is a multilateral calculus in the style we introduced in [13]and[14]: inferences may involve asserted sentences as well as rejected and weakly asserted ones. The proof system emerges naturally by think- ing of truth simpliciter as licensing assertion, falsity simpliciter as licensing negative assertion, and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. The supervaluationist can specify which argument patterns govern good deductive reasoning when vague terms are involved. According to supervaluationism, the vagueness of bald lies in the fact that it admits borderline cases—it may be that it is neither definitely the case that Harry is bald nor is it definitely the case that he is not bald. Now, the logic of assertion of our proof system is sound and complete with respect to the global consequence relation of the modal logic S5. It follows that it is either definitely the case that Harry is definitely bald or it is definitely the case that he is not definitely bald. However, higher-order predicates such as definitely bald would seem to admit of borderline cases, just as their first-order counterparts. This is the basic problem of higher-order vagueness for supervaluationism. To solve it, it is natural for supervaluationists to abandon the modal logic Axiom 4 and retreat to a modal logic in which it no longer follows that it is either definitely the case that Harry is definitely bald or it is definitely the case that he is not definitely bald. However, the broader problem of higher-order vagueness for supervaluationism cuts deeper than this. Delia Graff Fara [10] argued that to account for higher-order vagueness, the supervaluationist must accept gap principles stating that a definitely bald person cannot become not definitely bald by gaining one hair, and similarly for any finite iteration of definitely. The gap principles give rise to paradoxes of higher- order vagueness similar to the standard (first-order) sorites paradox that are not as easily avoided as the basic problem. According to an emerging consensus on the problem of higher-order vagueness, the culprit is global consequence, which should be relinquished at the expense of its regional counterpart [3]. In particular, Graff Fara’s argument employs the argu- ment pattern allowing one to infer definitely A from A, which fails under a regional Meta-inferences and Supervaluationism consequence relation. However, this argument pattern seems central to the logic of definitely. What is more, Elia Zardini [32] has recently shown that the pattern is not needed to derive a higher-order sorites paradox. To address the problem of higher-order vagueness, we again turn to the multilateral framework. Proof analysis within this framework reveals that the fault lies with some of the rules for the Δ operator. Model-theoretically, this means that the culprit is not global consequence, but the modal properties of Δ. By removing the problematic rules, we obtain a logic in which Δ is non-normal—that is, it does not satisfy Δ(A → B) → (ΔA → ΔB), the modal logic axiom K. In particular, the asserted fragment of the resulting logic coincides with the global consequence relation of the non-normal modal logic NT. The non-normality of Δ will turn out to be a crucial ingredient in avoiding Zardini’s higher-order sorites paradox. The modal logic NT may appear to be too weak to represent the supervaluation- ist canons of good deductive reasoning. However, its global consequence relation is strong enough to account for the validity of several argument patterns that are beyond reach of its local consequence relation. For instance, although Axiom K fails in NT, the global definition of validity means that the assertion of ΔB can be inferred from the assertion of ΔA and the assertion of Δ(A → B). Similarly, although A → ΔA fails, the assertion of ΔA can be inferred from the assertion of A, thus vindicat- ing a central argument pattern of the logic of definitely which fails under a regional consequence relation. The difference between an argument and the corresponding material conditional will be a recurring theme of our discussion. In many cases, supervaluationism val- idates the argument pattern but not the material conditional. As we shall see, the supertruth of a material conditional corresponds to the relevant argument pattern being locally valid. Supervaluationists should therefore stress the primacy of global validity qua preservation of truth simpliciter. The demand to validate material condi- tionals that correspond to valid argument patterns may stem from the prevalence of standard modal logic, in which truth simpliciter is truth at a world and consequence is therefore local. Proof-theoretically, the failure of principles such as Axiom K and A → ΔA and the validity of the corresponding inferences can coexist thanks to the failure of the classical meta-inferences, in particular conditional proof. Supervaluationists should therefore embrace this failure as an integral part of their approach to vagueness, rather than excuse it as an embarrassing but negligible outcome thereof. The plan is as follows. We begin by explaining the proof-theoretic problem for supervaluationism, paying special attention to the role of the problematic meta- inferences (Section 2). We then present supervaluationist multilateral logic and show how it addresses the proof-theoretic problem (Section 3). We go on to introduce the problem of higher-order vagueness (Section 4) and note that the basic version of the problem already forces us to revise supervaluationist multilateral logic. We prove that the resulting logic is sound and complete with respect to NT global consequence (Section 5). We explain how these revisions put the supervaluationist in a position to account for vagueness in the same way at all orders (Section 6). We formally execute this plan by extending our logic to the first-order domain. The result is a system which naturally blocks the derivation of Zardini’s higher-order sorites paradox (Section 7). L. Incurvati, J.J. Schloder¨ We conclude by commenting on the different roles played by the proof theory and the model theory in our approach (Section 8). 2 Supervaluationism and the Proof-Theoretic Problem According to a standard way
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