Non-classical Metatheory for Non-classical Logics

Bacon, Andrew

doi:10.1007/s10992-012-9223-9

Cited by 50 publications

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“…respect of A as ∼∼(Valid(A) → Prov(A)), lifting this terminology to a logic (in particular to Heyting's) when the corresponding property is possessed in respect of all formulas. (Note incidentally that this has nothing to do with the usual 'derivability' vs. 'outright provability' contrast between strong and weak completeness or between the similarly named properties from (Bacon 2013), touched on in the longer notes on Section 2 in the Appendix, headed 'Bacon and others'. ) What the adherent of a weaker logic regards as the apparatus for making finer discriminations in the subject matter than are available to the adherent of the stronger logic, who is faced instead with a proliferation problem: how on earth to respond the requirement to 'deconflate'.…”

Section: Weaken Your Metalogic: Easier Said Than Donementioning

confidence: 99%

Semantics without Toil? Brady and Rush Meet Hallden

Humberstone

2019

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The present discussion takes up an issue raised in Section 5 of Ross Brady and Penelope Rush's paper 'Four Basic Logical Issues' concerning the (claimed) triviality-in the sense of automatic availability-of soundness and completeness results for a logic in a metalanguage employing at least as much logical vocabulary as the object logic, where the metalogical behaviour of the common logical vocabulary is as in the object logic. We shall see-in Propositions 4.5-4.7-that this triviality claim faces difficulties in the face of Halldén incompleteness, for essentially the same reasons that Halldén thought this phenomenon raised semantic difficulties for the modal logics of C. I. Lewis exhibiting it. To counter any inclination to dismiss the phenomenon as providing at best a marginal range of counterexamples to the triviality claim, a Postscript assembles some reminders of the extent of-and the varied considerations favouring-Halldén incompleteness.

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Section: Weaken Your Metalogic: Easier Said Than Donementioning

confidence: 99%

Semantics without Toil? Brady and Rush Meet Hallden

Humberstone

2019

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“…So the criticism would apply not just to the nontransitive approach. There are, of course, exceptions; some people do their metatheory in a nonclassical logic (Bacon ; Weber, Badia, and Girard ; Zardini , ). Let us assume that this is the way to go.…”

Section: A Problem With Expressing Admissibilitymentioning

confidence: 99%

The Cut-Free Approach and the Admissibility-Curry

Hlobil¹

2018

Thought: A Journal of Philosophy

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The perhaps most important criticism of the nontransitive approach to semantic paradoxes is that it cannot truthfully express exactly which metarules preserve validity. I argue that this criticism overlooks that the admissibility of metarules cannot be expressed in any logic that allows us to formulate validity-Curry sentences and that is formulated in a classical metalanguage. Hence, the criticism applies to all approaches that do their metatheory in classical logic. If we do the metatheory of nontransitive logics in a nontransitive logic, however, there is no reason to think that the argument behind the criticism goes through. In general, asking a logic to express its own admissible metarules may not be a good idea.Until recently, philosophical common sense had it that, on pain of triviality, you cannot accept all of classical logic, allow for self-reference and let your language contain its own truth predicate (satisfying all Tarski biconditionals). The nontransitive approach to semantic paradoxes has proved common sense wrong. It offers a supraclassical logic, called STT, with a transparent truth predicate (i.e., a truth predicate, T, such that A and T⟨A⟩ are everywhere intersubstitutable) and self-reference (Cobreros et al. 2012(Cobreros et al. , 2013Ripley 2012Ripley , 2013. Transitivity, that is, Cut, fails in STT. But it only fails for paradoxical sentences, like the liar sentence or Curry sentences. That seems like a small price to pay for the full strength of classical logic together with a transparent truth predicate. With that much to recommend itself, I think we should try to hold on to the nontransitive approach as long as we can. My goal in this paper is to show that, contrary to recent criticism (Barrio, Rosenblatt, and Tajer 2016; Rosenblatt 2017), we shouldn't reject the nontransitive approach because it cannot express the admissibility of its own metarules.Along the way, we can observe some general facts about the possibility of expressing, in the object language, which metarules are admissible. In particular, I point out that no one who does her metatheory in classical logic and allows for self-reference can allow her logic to truthfully express which metarules are admissible, according to her own logic.

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“…However no definition of equivalence in terms of provability will ensure this due to the incompleteness theorems (one can always construct a Gödel sentence which ought to be intersubstitutable with 0 = 0.) 2 A better way to at least characterise the extension of the relation of logical equivalence would be to select a suitable class of models and define logical truth as truth in all models in that class. 3 Of course this helps us little unless we know which models are suitable, and this depends in turn on the prior conception of logical truth we are trying to characterise.…”

Section: Substitutability Of Logical Equivalentsmentioning

confidence: 99%

“…Alternatively one could run the model theory within a non-classical theory of sets as suggested in [2]. In the former case one can expect 'S is logically true' to behave classically (i.e.…”

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Paradoxes of Logical Equivalence and Identity

Bacon

2013

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Assuming the unrestricted application of classical logic, the paradoxes of truth, sets and properties make trouble for naïve intersubstitutivity principles, such as the principle that allows one to substitute, in non-intentional contexts, the claim that φ for the claim that 'φ' is true, or the claim that x is F for the claim that x has the property of being F . One way to respond to these paradoxes is to reject the logical assumptions the paradoxes rest on, allowing one to instead accept naïve intersubstitutivity principles governing sets, truth and properties.In this paper I show that even in these weakened logics intersubstitutivity principles can wreak havoc. In these discussions intersubstitutivity principles are normally formulated using relatively weak metalinguistic rules; the following paradoxes arise when a stronger intersubstitutivity axiom, formulated entirely in the object language, is assumed.In section 1 I outline a few different applications of these paradoxes: truth theorists, for example, might want to endorse a principle stating that logically equivalent sentences are substitutable salve veritate. Property and set theorists might want to endorse a version of Leibniz's law. In section 2 I present two paradoxes that show that in either case they cannot endorse the principle in question, given background assumptions. The second of these paradoxes uses very little in the way of logical machinery, and thus applies to most logics developed to deal with the semantic and set theoretic paradoxes (see for example Priest [14].) In the second half of section 2 I note that both paradoxes, when interpreted in terms of the notion of logical equivalence, are similar in spirit to recent versions of Curry's paradox that employ the notion of a valid argument (see, for example, [19] and [4].) I then show that the present paradoxes can be formulated so as not to depend on any distinctive structural rules and, as a result, they are problematic for recent approaches to the validity Curry paradox that relax the rule of structural contraction (Zardini [20], Priest [15], Murzi & Shapiro [12].)

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Non-classical Metatheory for Non-classical Logics

Cited by 50 publications

References 20 publications

Semantics without Toil? Brady and Rush Meet Hallden

Semantics without Toil? Brady and Rush Meet Hallden

The Cut-Free Approach and the Admissibility-Curry

Paradoxes of Logical Equivalence and Identity

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