Abstract
Substructural pluralism about the meaning of logical connectives is best understood as the view that natural language connectives have all (and only) the properties conferred by classical logic, but that particular occurrences of these connectives cannot simultaneously exhibit all these properties. This is just a more sophisticated way of saying that while natural language connectives are ambiguous, they are not so in the way classical logic intends them to be. Since this view is usually framed as a means to resolve paradoxes, little attention is paid to the logical properties of the ambiguous connectives themselves. The present paper sets out to fill this gap. First, I argue that substructural logicians should care about these connectives; next, I describe a consequence relation between a set of ambiguous premises and an ambiguous conclusion, and review the logical properties of ambiguous connectives; finally, I highlight how ambiguous connectives might explain our intuitions about logical rivalry.
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Notes
- 1.
Unfortunately, due to the usage of Prawitz-style ND, the introduction-rule for the lattice-conjunction is based on a not so natural way of expressing the shared context used to derive both conjuncts. In particular, one should be attentive to the fact that both occurrences of [A]i in ( ⊓I) count as a single assumption. Also, rather than to specify the rules for the negation, we only need to note that we use a De Morgan negation which satisfies ∼ ∼ A ⊣ ⊢ A, as well as all usual De Morgan equivalences for both the lattice and the group-theoretical connectives.
- 2.
Basically, a situation where a first agent uses addition to obtain a disjunctive formula which he then passes on to a second agent who uses the disjunctive syllogism to recover the original disjunct the first agent started from, see Burgess (1981).
- 3.
The main complication I ignore here bears on the fact that the L-information of a message is defined as the non-tautological deductive yield of that message. Since classical tautologies can have a tautological and a non-tautological disambiguation, the diagnosis should in fact be that the channel is both equivocal and noisy (cfr. the ‘p { or} { not}–p’ example from Allo (2007)). This more refined diagnosis does not affect my contention that the channel is not merely equivocal, but also noisy.
- 4.
Two remarks: (1) Such failures of transitivity are, as far as I can see, only superficially related to what we find in Tennant (2004); (2) As remarked by Elia Zardini (pc), the suggestion that explosion is a fallacy of equivocation is entirely compatible with the suggestion that transitivity is the real culprit. As a consequence, the proposal set out in the next section can only be motivated if the Xerox-principle is already presupposed.
- 5.
Given the usage of a sequent-calculus in van Eijck and Jaspars (1996), this surfaces as a failure of the principal cuts.
- 6.
The usage of ⊔ for the disjunctive analysis does not block this argument, whereas the usage of ⊕ reduces to the former in the presence of contraction (which is exactly what is required for A 1 ⊢ A 2).
- 7.
The naming convention is inherited from combinatory logic, see Mares and Meyer (2001).
- 8.
Of course, the use of classical disjunctions as a means to analyse ambiguity would restore the required distribution principles. Given the implicit aim to recover ambiguity within the substructural language, this is not an option I’m inclined to investigate.
References
Allo, P. 2007. Logical pluralism and semantic information. Journal of Philosophical Logic 36(6): 659–694.
Anderson, A.R., and N.D. Belnap. 1975. Entailment: The logic of relevance and necessity, vol. 1. Princeton: Princeton University Press.
Azzouni, J., and B. Armour-Garb. 2005. Standing on common ground. The Journal of Philosophy 102(10): 532–544.
Batens, D. 1997. Inconsistencies and beyond: A logical-philosophical discussion. Revue Internationale de Philosophie 51(2): 259–273.
Beall, J.C., and G. Restall. 2006. Logical pluralism. Oxford: Oxford University Press.
Burgess, J.P. 1981. Relevance: A fallacy? Notre Dame Journal of Formal Logic 22(2): 97–104.
Corcoran, J. 1998. Information-theoretic logic. In Truth in Perspective, ed. C. Martnez, U. Rivas, and L. Villegas-Forero, 113–135. Aldershot: Ashgate.
da Costa, N.C.A. 1997. Logiques classiques et non classiques: Essai sur les fondements de la logique. Paris: Masson.
Dos̆en, K. 1993. A historical introduction to substructural logic. In Substructural logics, ed. K. Dos̆en and P. Schroeder Heister, 1–30. Oxford: Oxford University Press.
Dretske, F. (1999). Knowledge and the flow of information. Stanford: CSLI.
Fine, K. 1975. Vagueness, truth and logic. Synthese 30(3): 265–300.
Haack, S. 1974. Deviant logic. Some philosophical issues. Cambridge: Cambridge University Press.
Humberstone, I.L. 2005. Logical discrimination. In Logica universalis, ed. J.-Y. Béziau, 207–228. Basel: Birkhäuser Verlag.
Lewis, D. (1982). Logic for equivocators. Noûs 16: 431–441.
Mares, E. 1997. Relevant logic and the theory of information. Synthese 109(3): 345–360.
Mares, E. 2004. Relevant logic: A philosophical interpretation. Cambridge: Cambridge University Press.
Mares, E., and R.K. Meyer. 2001. Relevant logics. In The blackwell guide to philosophical logic, ed. L. Goble, 280–308. Oxford: Blackwell.
Paoli, F. 2007. Implicational paradoxes and the meaning of logical constants. Australasian Journal of Philosophy 85(4): 553–579.
Prawitz, D. 2006. Meaning approached via proofs. Synthese 148(3): 507–524.
Priest, G. 2001. Logic: One or many? In Logical consequence: Rival approaches, ed. J. Woods and B. Brown, 23–38. Stanmore: Hermes.
Read, S. 1981. What is wrong with disjunctive syllogism? Analysis 41: 66–70.
Read, S. 1988. Relevant logic: A philosophical examination of inference. Oxford: Basil Blackwell.
Read, S. 2000. Harmony and autonomy in classical logic. Journal of Philosophical Logic 29(2): 123–154.
Tennant, N. 2004. Relevance in reasoning. In Handbook of philosophy of logic and mathematics, ed. S. Shapiro, 696–726. Oxford: Oxford University Press.
van Deemter, K. 1996. Towards a logic of ambiguous expressions. In Semantic ambiguity and underspecification, ed. K. van Deemter and S. Peters, 203–237. Stanford: CSLI.
van Eijck, D.J.N., and J.O.M. Jaspars. 1996. Ambiguity and reasoning. Amsterdam: CWI, CS-9616. http://db.cwi.nl/rapporten/abstract.php?abstractnr=568
Acknowledgements
The author wishes to thank the audiences and organisers of the World Congress on Paraconsistency (Melbourne) and the Foundations of Logical Consequence Workshop at Arché (Saint-Andrews), as well as Francesco Paoli and Sebastian Sequoiah-Grayson for correspondence on the material in this paper.
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Allo, P. (2013). Noisy vs. Merely Equivocal Logics. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_5
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