A Paraconsistent and Substructural Conditional Logic

I introduce and motivate a conditional logic based on the substructural system HL from Paoli (Substructural logics: a primer, Kluwer, Dordrecht, 2002). Its hallmark is the presence of three logical levels (each one of which contains its own conditional connective), linked to one another by means of appropriate distribution principles. Such a theory brings about a twofold benefit: on the one hand, it yields a new classification of conditionals where the traditional dichotomies (indicative vs subjunctive, factual vs counterfactual) do not play a decisive role; on the other hand, it allows to retain suitable versions of both substitution of provable equivalents and simplification of disjunctive antecedents, while still keeping out such debatable principles as transitivity, monotonicity, and contraposition.

1. 1.

   Observe that Weatherson is defending a substantially different theory in his more recent Weatherson (2009).

2. 2.

   Of course we must rule out such pseudoconditionals as Austin’s “There are biscuits on the sideboard if you want some”. These sentences will not be considered further in this paper.

3. 3.

   What about associativity? I do not have univocal intuitions either way. In my formal theory below, I will not assume that this property holds.

4. 4.

   And, if the order of the disjuncts in the original disjunction had been reversed, the resulting conditional would instead have been acceptable simply in virtue of the fact that its main clause was taken for granted.

5. 5.

   Do these conditionals actually occur in everyday speech? People often use conditionals like “If Pete is a good barber, then I’m a monkey’s uncle”. Nonetheless, they are often dismissed as mere rhetorical devices. It is remarked in Anderson and Belnap (1975), p. 163: “It is of course sometimes said that the ‘if-then’ we use admits that false or contradictory propositions imply anything you like, and we are given the example ‘If Hitler was a military genius, then I’m a monkey’s uncle’. But it seems to us unsatisfactory to dignify as a principle of logic what is obviously no more than rhetorical figure of speech, and a facetious one at that”. In my opinion, however, the decisive issue is not whether to dignify or not the ex absurdo quodlibet as a principle of logic: it is only that we must correctly determine for which logical constants it holds and for which ones it fails. “Monkey’s uncle” conditionals are commonly used by ordinary English speakers and have an “if…then” grammatical form; moreover, people freely use the ex absurdo quodlibet in dealing with these sentences. If it is thought that they are not proper hypotheticals, independent reasons should be given to justify this claim. I believe that the linguistic data do not lead to dispose of them offhand as pseudoconditionals—but, as long as they are conveniently kept distinct from conditionals which do not abide by that law, there is little to worry about. This issue, by the way, marks a first difference between my perspective and the relevant one (more will emerge soon): since the implicational paradoxes plainly fail for the relevant conditional, Anderson and Belnap conclude that they are fallacious altogether. On the contrary, I think that they are false of other kinds of conditionals but true of this one.

6. 6.

   Advocates of the Gricean conversational account of paradoxes of material implication, like Lewis, will disagree: however, Read (1988) and others have convincingly shown that the recourse to implicature yields no advantage in the case of nested conditionals and wherever the conditional is not the principal connective of the sentence at issue. Therefore, it does not offer a viable solution to the paradoxes.

7. 7.

   In the framework of standard possible world semantics, in fact, A > B would be true at w just in case f _A (w) ⊆ [B]], while ¬B >  ¬A would be true at w just in case f ¬B (w) ⊆ [ ¬A]. Likewise, A⋎B would be true at w just in case f ¬A (w) ⊆ [B], while B⋎A would be true at w just in case f ¬B (w) ⊆ [A]. Obviously, these conditions need not be equivalent.

8. 8.

   Caution: Anderson and Belnap use the horseshoe to denote both the classical conditional, which obeys modus ponens in any arbitrary theory, and the extensional conditional of their relevant logic, which does not. My chosen notation avoids any possible misunderstanding.

9. 9.

   See again Paoli (2007) for a more detailed discussion of this point.

10. 10.

    Anderson and Belnap are not the sole authors in the relevant tradition who seem committed to such an equivocation. Hunter (1993), for one, followed their lead. A happy exception is the paper Mares and Fuhrmann (1995), where the arrow is carefully distinguished from the corner, although the squiggle is assigned no special status.

11. 11.

    The last statement is not always correct, though. Consider the perfectly grammatical disjunctive paraphrase of a subjunctive conditional: “I had to jot that down or I would have forgotten it” (Dowing 1975, p. 86).

12. 12.

    Cp. the remark in Smiley (1984): “When a conditional conveys temporal succession […] it becomes an understatement to say that contraposition fails. Either the contraposed conditional conveys a message unrelated to the original (compare ‘If the surgeon didn’t operate the patient would die’ with ‘If the patient didn’t die the surgeon would operate’) or it fails to convey a coherent message at all (try contraposing ‘If the surgeon didn’t operate tonight the patient would die tomorrow’)”.

13. 13.

    It suffices to consider any subjunctive conditional with the opt-out property: “If Oswald hadn’t killed Kennedy no one else would have”, indeed, does not seem to imply “If Oswald didn’t kill Kennedy no one else did”.

14. 14.

    In most relevant logics, to be sure, distribution for extensional connectives is available, whereas it fails in linear logic and in Meyer (1966) LR, whose sequent version is obtained from the classical sequent calculus by dropping the rules of weakening. I argued in Paoli (2007) that nondistributive logics are the best motivated relevant logics.

15. 15.

    I tried to keep my preference for a proof-conditional (over a truth-conditional) semantics for logical constants in the background but, as you see, I failed. Observe, however, that what I said so far is independent from such a personal liking. For a defence of this view, see Paoli (2007).

16. 16.

    As it is customary to do in the tradition of abstract algebraic logic, I will not distinguish between logical languages and algebraic similarity types; therefore, I will use the same symbols for logical connectives and the corresponding operation symbols. Fm(£) will denote both the set of all formulas of £ (seen as a logical language) and the set of all terms of £ (seen as an algebraic similarity type), and likewise for Fm(£ ^′).

A version of the present paper was presented at the 4th World Congress on Paraconsistency (Melbourne, Australia, July 2008). I thank the audience of my talk for their stimulating questions and comments. I also thank Charlie Donauhe and an anonymous referee for their precious suggestions.

Authors and Affiliations

1. Department of Pedagogy, Psychology, Philosophy, University of Cagliari, Cagliari, Italy

   Francesco Paoli

Corresponding author

Correspondence to Francesco Paoli .

Editors and Affiliations

1. , Department of Philosophy, University of Auckland, Arts 2, 18 Symonds Street, Auckland, 1142, New Zealand

   Koji Tanaka

2. , Department of Philosophy, University of Aberdeen, King's College, High Street, Aberdeen, AB24 3UB, United Kingdom

   Francesco Berto

3. , Department of Philosophy, Victoria University of Wellington, Murphy 518, Kelburn Parade, Wellington, New Zealand

   Edwin Mares

4. , Department of Education, University of Cagliari, Via Is Mirrionis 1, Cagliari, 09123, Italy

   Francesco Paoli

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