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(I can’t get no) antisatisfaction

Abstract

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named \({\mathsf {ST}}\) and \({\mathsf {TS}}\), along with two structural cousins, \({\mathsf {LP}}\) and \({\mathsf {K3}}\). It is well known that \({\mathsf {LP}}\) and \({\mathsf {K3}}\) are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, \({\mathsf {ST}}\) and \({\mathsf {TS}}\) are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning \({\mathsf {ST}}\) which are being intensively discussed in the current literature: whether \({\mathsf {ST}}\) preserves classical logic and whether it is \({\mathsf {LP}}\) in sheep’s clothing. Explaining in what way \({\mathsf {ST}}\) and \({\mathsf {TS}}\) are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference.

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Notes

  1. See Petersen (2000), Shapiro (2010), Zardini (2011), Beall and Murzi (2013) and Rosenblatt (2019) for non-contractive approaches; Weir (2005), Cobreros et al. (2013) and Ripley (2013) for non-transitive approaches; French (2016) and Nicolai and Rossi (2018) for non-reflexive approaches and Da Ré (2020) for non-monotonic.

  2. Although we introduce here \(\top \) and \(\bot \) as logical constants their only role in this paper will be marking an empty position in a sequent.

  3. The idea that, in addition to preservation of a designated value, logical consequence can mix different notions of satisfaction was used in the 60s by Schütte and Tait [who gave a semantic proof of Cutelimination for second-order arithmetic essentially using the failure of transitivity of ST, see Girard (1987, Ch. 3) for references] and it has been applied in different areas of philosophical logic atleast in the works of Malinowski (1990), Nait-Abdallah (1995), Bennett (1998), Frankowski (2004), Zardini (2008), Cobreros et al. (2012), Cobreros et al. (2013) and Ripley (2013).

  4. Where ‘\(\Rightarrow \)’ stands for the sequent arrow. We take, as usual, a sequent as expressing an inference.

  5. Under the assumption that the notion of satisfaction applies equally to sequents at any metainferential level, see Scambler (2019) Section 3.3 for a clarification of this point. Following Barrio et al. (2020), Chris Scambler shows that there are uncountably many logics differing perhaps only at some metainferential level. A discussion of these results is beyond the scope of this paper.

  6. For simplicity we will talk about sequents involving only one premise and only one conclusion. In the four logics, the comma in the premises works as a conjunction and the comma in the conclusions as a disjunction; this is the reason why we find this simplification unproblematic.

  7. See, in particular, Cobreros et al. (2013, 853)

  8. See also Priest (2008) specially sections 8.4.8 and 8.4.11, although the trees there for \({\mathsf {LP}}\) and \({\mathsf {K3}}\) are presented in a slightly different way. The exact kind of trees used here are described in Cobreros et al. (2012).

  9. Soundness and completeness proofs are adaptations of the corresponding proofs in Priest (2008).

  10. These definitions come from Dicher and Paoli (2019) and are also used in Barrio et al. (2020). For a matter of uniformity we define metainferences with possibly multiple inferences as conclusions, although the examples we will consider below involve all a single inference in the conclusion side. The idea of validity for a metainference is also present in Barrio et al. (2015) although in a slightly different (“global-substitutional”) sense. In the paper Cobreros et al. (2013) the validity of a metainference is used in a third different way (“simply global”). Which of these notions is the appropriate one, if any, is an interesting question beyond the scope of this paper. The notion of a metainference can be generalised to cover metainferences of any order, where a metainference of order n is an arrow with metainferences of order \(n-1\) at each side. Barrio et al. (2020) use this generalisation to prove some intriguing results. See Scambler (2019) for a rejoinder.

  11. In a companion paper in preparation we show how the trees can be “turned upside down” to obtain a sequent calculus that covers the four 3-valued logic under consideration as well the hierarchy of meta-inferential logics considered by Barrio et al. (2020). As it is implicit in the construction of the trees (in particular, from the fact that to test the xy-validity of a sequent we start a tree by simply taking the formulas in the antecedent labelled by x together with the negation of those in the antecedent marked by the dual of y), the corresponding sequent calculus is two-sided and labelled, where the labels are used to reflect syntactically the key features of the semantics. The calculus (like the trees presented here) therefore fulfils the elegance conditions introduced in Fjellstad (2017) (in particular, if one forgets the labels, the rules are just those for classical logic; and one has a straightforward connection between derivability of sequents and validity), and it stands to the st semantics as the dual-two-sided calculus of Fjellstad stands to his dual valuation semantics. The duality issues we investigate in the present paper could in principle be reformulated in terms of Fjellstad’s dual valuation semantics and hence also in his sequent calculus, but they can be more straightforwardly formulated using the strict-tolerant semantic settings, which is the reason why we preferred introducing these trees rather than working with Fjellstad’s calculus.

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Acknowledgements

Previous versions of this paper were presented in the work in progress seminar of the Munich Center for Mathematical Philosophy, the “LanCog Workshop on Substructural Logics” (Lisbon), the “IX Conference of the Spanish Society for Analytic Philosophy” (Valencia), and the “The Logic of Paradox, after 40 years” (Bochum). We thank the audiences of these conferences, particularly to Hannes Leitgeb, Bogdan Dicher and Hitoshi Omori. The work of Luca Tranchini has been supported by the Deutsche Forschungsgemeinschaft as part of the project “Falsity and refutation. On the negative side of logic” (TR1112/4-1). Pablo Cobreros enjoyed a Humboldt Research Fellowship for experienced researchers (February to July 2019 and February to July 2020). This research was also supported by the Ministry of Science, Innovation and Universities of the Government of Spain with the project “Logic and Substructurality (FFI2017-84805-P)”.

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Cobreros, P., La Rosa, E. & Tranchini, L. (I can’t get no) antisatisfaction. Synthese 198, 8251–8265 (2021). https://doi.org/10.1007/s11229-020-02570-x

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Keywords

  • Substructural approaches to paradox
  • Non-transitive logic
  • Non-reflexive logic
  • Strong Kleene