Substructural heresies

ABSTRACTThis paper discusses two revisionary views about substructurality. The first attempts to reduce the structural features of a logic to properties of its logical vocabulary. It will be found to be untenable. The second aims to separate the structural features of a logic from the properties of logical consequence and to reinterpreted them as sui generis proof resources. I will argue that it is a viable path for a renewed understanding of substructurality.KEYWORDS: Substructural logiclogical consequencesequents Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 These ‘collections’ can be sequences (lists), multisets (lists with repetitions but without order), sets simpliciter, etc. A full characterisation of a sequent (in a specific calculus) requires the specification of their precise mathematical nature. But here we allow, in principle, variations of the components of the sequents. Therefore generality is preferable to precision hence the use of the term ‘collection’ – with the caveat that the default choice will be to treat collections as sets.2 I follow the usual notational conventions and use minuscules from the second half of the Latin alphabet as sentential variables. Majuscules, with or without superscripts, from the beginning of the alphabet are metavariables ranging over sentential variables while those from its end range over collections (see the previous note) of formulae.3 An anonymous referee points out that the same view appears in Sambin, Battilotti, and Faggian (Citation2000). I am not convinced that this is the position advocated in that paper. The matter deserves more attention than I can give it here, but the following passage from Sambin, Battilotti, and Faggian (Citation2000) seems to be decisive for the overall understanding of their position: ‘It is an ambition of basic logic to offer a new perspective and new tools to the search for unity in logic. …[O]ur plan is to look for the basic principles and structures common to many different logics. So one aim is to obtain each specific logic by the addition of rules concerning exclusively the structure (i.e. structural rules dealing only with assertions), while keeping the logic of propositions (i.e. operational rules dealing with logical constants) absolutely fixed’. I take this to indicate a commitment to the priority of the structural level over the operational one which is quite antithetical to A-heresy.4 Henceforth I will use ‘structural property’ and ‘structural rule’ interchangeably. Each structural rules generates a structural property and each structural property correlates in some way with a structural rule.5 By ‘set-theoretic aggregate’ I mean any kind of collection that can be represented within set-theory; that includes sequences, multisets, sequences of set-theoretic aggregates, etc. For the most part, however, I will represent these using the set-theoretic accolades.6 See note 16 for details and examples.7 A model is a quadruple ⟨W,v0,R,⋆⟩, with W a non-empty set of points (possible worlds/situations) and v0∈W a special world, R a ternary relation on W and ⋆ a function from W to itself needed for negation about which we won't care much here. As in the more familiar case of normal modal logics, R is subject to particular conditions, depending on the logics at hand. Then the clause for fusion would be something like v⊩A∘B if (w⊩A and u⊩B) and Rwuv.8 Should one use classical logic in the metatheory of non-classical logics (assuming that the aforementioned extensional conjunctive operation is indeed classical)? I have no ready answer to this question. The status quo is to provide an affirmative answer; it has been challenged in, e.g. (Tanaka and Girard Citation2023).9 In the usual preservationist or propagationist accounts of consequence, a certain property shared by the premisses is transmitted to (at least one) conclusion. However, there are accounts of consequence that are not preservationist and in which its obtaining is marked by the conclusion having some property that is not necessarily the one had by the premisses (Cobreros et al. Citation2012), just like there are accounts that take consequence as primitive and make no attempt to explain it in terms of properties of the premisses or conclusions (Schroeder-Heister Citation2011). (I am indebted to an anonymous referee for reminding me of the latter.)10 Recall that this conjunction, although symbolised here by ∧, need not be the extensional conjunction of, say, classical or intuitionistic logic. Rather, it is some appropriate conjunctive operation in the object language of the logic under consideration.11 One may take issue with the qualifier ‘independently’, and not only in the way considered by Zardini, which is mentioned below. In the literature there are arguments that premiss-combination is not constituted independently of conjunction just like conjunction is not constituted independently of premiss-combination (Dicher Citation2016, Citation2020). According to these arguments, the structural properties of a logic and that logic's operators stand in a relation of co-determination. However, because these arguments draw on resources too particular to be presumed acceptable to Zardini, and their defence would be too involved to pursue here, I do not discuss them further.12 Repeated application of the deduction theorem would transform X,A⊢B into ⊢A1→(A2→…(A→B)…), for Ai∈X(i=1,2,3,…,n). The astute reader may then wonder why is it that multiple premisses are to be interpreted as connected conjunctively rather than as connected conditionally?13 Although I don't know Zardini's exact target. He is talking about ‘entailments’ and seems unconcerned with finer points such as distinguishing between an inference as an act and an inference as an object (=consequence claim).14 This point is credited to Gareth Evans in Shoesmith and Smiley (Citation1978), and it occurs in various other places in the literature, cf. Dummett (Citation1991); Steinberger (Citation2010).15 There may be a way to have the cake and eat it too, but that will happen in a rather different restaurant, cf. Dicher (Citation2019).16 Their absence has ‘lexical’ effects, as already mentioned supra on p. 5. Operational rules that are interderivable (= equivalent) in LK cease to be so in linear logic. For instance, the following two rules A,X:YA∧B,X:YB,X:YA∧B,X:Yare interderivable with the rule A,B,X:YA∧B,X:Yusing Weakening and Contraction. (An application of Weakening to the premiss of either rule in the top row produces A, B, X : Y to which the rule in the bottom row can be applied to get A∧B,X:Y. For the converse, apply the top rules to the premiss of the bottom rule and contract on A∧B.) Without Weakening and Contraction this is no longer the case and indeed in linear logic one can distinguish two different conjunctive connectives: an additive conjunction governed by the rules X:Y,AX:Y,BX:Y,A∧B∧RA,X:YA∧B,X:Y∧L1B,X:YA∧B,X:Y∧L2and a multiplicative conjunction governed by the rules X:Y,AW:Z,BX,W:Y,Z,A⊗BA,B,X:YA⊗B,X:YIn general, every binary connective of LK gets duplicated (some would say: disambiguated (Paoli Citation2007)) so that there is an additive and a multiplicative version of it.17 Whether Girard uses ‘consequence’ in the same sense in which I have used it here or, on the contrary, he uses it to mean ‘inference’ in this paper's sense is a moot point.18 I am indebted to an anonymous referee for suggesting these last two possible interpretations of Weakening and Contraction as sui generis proof resources.19 See the discussion below, including note 20, and also, for different explorations of the same possibility, Yagisawa (Citation1993) or Russell (Citation2018).20 To be sure, this was known/embraced in full generality for some time now. Logics that are equipped to handle demonstratives/indexicals even when they change reference mid-argument have been explored by Georgi (Citation2015), Zardini (Citation2014).21 The (general) idea that certain structural rules can be used to control the synonymy of formulae occurring in a sequent has its origins in Girard (Citation1976); see also Girard, Lafont, and Taylor (Citation1989) and Dicher (Citation2019).22 I am grateful to audiences at the Alef Seminar (Babes-Bolyai University, Romania) and at the LanCog Seminar (Centre of Philosophy of the University of Lisbon, Portugal) for comments and suggestions on previous versions of this paper. I am also indebted to an anonymous referee for this journal for many suggestions and corrections which have greatly improved the paper. I owe special thanks to Bruno Jacinto and Elia Zardini for many conversations on the topics broached here.Additional informationFundingThis work was financially supported by the Fundação para a Ciência e a Tecnologia (FCT), Portugal, through grants CEECIND/02877/2018 (‘Metainferences: New perspectives on logic’) and 2022.03194.PTDC (‘On the Objects and Grounds of Substructural Rules’). Further financial support was received from the European Union, through grant 101086295 (‘Philosophical, Logical, and Experimental Perspectives on Substructurality–PLEXUS’, a Marie Sklodowska-Curie action funded by the EU under the Horizon Europe Research and Innovation Programme).