Abstract
Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic has not been made sense of, despite its well-developed mathematics. In this paper, I examine the nature of logic in order to understand what it means to make sense of logic. I then show that, just as one can make sense of non-normal modal logics (as Cresswell demonstrates), we can make ‘sense’ of paraconsistent logic. Finally, I turn the tables on classical logicians and ask what sense can be made of explosive reasoning. While I acknowledge a bias on this issue, it is not clear that even classical logicians can answer this question.
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- 1.
In this paper, I often use the term ‘paraconsistent logic’ as a mass noun.
- 2.
Putnam is concerned with the sense of a statement or a question rather than of a formal system. My statement is an application of his claim rather than his claim itself.
- 3.
A similar charge against paraconsistent logic was put to me by my former colleague, Max Cresswell (though his charge was not to do with the conception of logic but with a coherent interpretation of logical connectives). This paper is, in part, a response to him.
- 4.
See particularly Cresswell (1967), pp. 202–203. Note that Cresswell doesn’t hold the view that the laws of logic are laws of thought. The purpose of his article is to show that different modal logics reflect different interpretations of the necessity operator, just as I am trying to show, in the first half of this paper, that paraconsistent logic (some relevant logics at least) reflects a different interpretation of the conditional operator. Thanks go to Max Cresswell for clarifying his position in personal communication.
- 5.
I don’t take this statement to be uncontroversial. However, I am concerned with attaching a ‘sense’ to non-normal worlds and to non-normal modal logics and I take it that there is a sense in which a law of logic is expressed by a logical truth as was held by, for example, Frege, Russell and Hilbert.
- 6.
- 7.
Exactly who came up with the first semantics for the Anderson and Belnap systems is a matter of dispute, just like who came up with the first semantics for Lewis modal logics is, I believe, a matter of dispute. I let historians settle the issue.
- 8.
Note that there can be more than one world which has this privileged status. However, completeness doesn’t force one to assume that there is more than one.
- 9.
Priest (1992) provides a different analysis of the relevant non-normal worlds. His analysis does not seem to be a generalisation of Kriple’s non-normal worlds, despite his claim that the Routleys and Meyer generalised Kripke’s non-normal worlds.
- 10.
Strictly speaking, there are no logical truths in FDE. Even though FDE may not be the best paraconsistent logic to be used for the current purpose, it is the easiest to understand the nature of paraconsistency with and, hence, I have used FDE in my discussion. One can replace FDE with LP of Priest (1979) which imposes the exhaustion principle: for all p, either ⟨p, 1⟩ ∈ μ or ⟨p, 0⟩ ∈ μ.
- 11.
Putnam in fact declines to attribute this view to Frege (p. 247). See also Goldfarb (2001) who presents Frege as holding a different conception of logic.
- 12.
Notable paraconsistent logics in which ¬(p ∧ ¬p) is not a logical truth are the Logics of Formal Inconsistency (LFIs). See, for example, Carnielli et al. (2007). My defence of paraconsistent logic doesn’t extend to LFIs. I let the advocates of LFIs provide their own defence.
- 13.
I note that ¬(p ∧ ¬p), in fact, fails to be a logical truth in FDE. But this is because of truth-value gap rather than truth-gap glut that FDE allows: p may be assigned no truth value in which case ¬(p ∧ ¬p) lacks truth value too. I take it that the concern of Putnam is with contradiction rather than indeterminacy of truth value.
- 14.
This is also the case in LP.
- 15.
Some paraconsistent logicians define this principle for someA. It does not make any difference for the purpose of this paper whether it is defined for some or anyA.
- 16.
As far as I know, no one has explicitly formulated this claim on paper. However, the claim is often put to me by my colleagues, for example.
References
Anderson, A.R., and N.D. Belnap. 1975. Entailment: The logic of relevance and necessity, vol. I. Princeton: Princeton University Press.
Anderson, A.R., N.D. Belnap, and J.M. Dunn. 1992. Entailment: The logic of relevance and necessity, vol. II. Princeton: Princeton University Press.
Carnielli, W.A., M.E. Coniglio, and J. Marcos. 2007. Logics of formal inconsistency. In Handbook of philosophical logic, vol. 14, 2nd ed., ed. D. Gabbay, and F. Guenthner, 15–107. Berlin: Springer.
Cresswell, M.J. 1967. The interpretation of some lewis systems of modal logic. Australasian Journal of Philosophy 45: 198–206.
Dunn, J.M. 1976. Intuitive semantics for first-degree entailment and ‘Coupled Trees’. Philosophical Studies 29: 149–168. Republished in Anderson et al. (1992).
Goldfarb, W.D. 2001. Frege’s conception of logic. In Future pasts: The analytic tradition in twentieth-century philosophy, ed. J. Floyd and S. Shieh, 25–41. Oxford: Oxford University Press.
Haack, S. 1978. Philosophy of logic. Cambridge: Cambridge University Press.
Hughes, G.E., and M.J. Cresswell. 1996. A new introduction to modal logic. London: Routledge.
Kripke, S.A. 1963. Semantical analysis of modal logic I. Normal modal propositional calculi. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9: 67–96.
Kripke, S.A. 1965. Semantical analysis of modal logic II. Non-normal modal propositional calculi. In The theory of models: Proceedings of the 1963 international symposium at Berkeley, ed. J.W. Addison, L. Henkin, and A. Tarski, 206–220. Amsterdam: North-Holland.
Lemmon, E.J. 1957. New foundations for lewis modal systems. The Journal of Symbolic Logic 22(2): 176–186.
Lemmon, E.J. 1965. Beginning logic. London: Thomas Nelson and Sons.
MacFarlane, J. 2000. What does it mean to say that Logic is Formal? Ph.D. Dissertation, University of Pittsburgh.
MacFarlane, J. 2002. Frege, Kant, and the logic in logicism. The Philosophical Review 111: 25–65.
Priest, G. 1979. Logic of paradox. Journal of Philosophical Logic, 8: 219–241.
Priest, G. 1992. What is a non-normal World? Logique et Analyse 35: 219–302.
Priest, G., and K. Tanaka. 2009. Paraconsistent logic. In Stanford encyclopaedia of philosophy, Summer 2009 edn., ed. E.N. Zalta. Stanford: Stanford University.
Putnam, H. 1994. Rethinking mathematical necessity. In Words and Life, ed. J. Conant, 245–263. Cambridge: Harvard University Press.
Routley, R. and Meyer, R.K. 1972. The semantics of entailment – II. Journal of Philosophical Logic 1: 53–73.
Routley, R. and Meyer, R.K. 1973. The semantics of entailment – I. In Truth, syntax and modality, ed. H. Leblanc, 199–243. Amsterdam: North-Holland.
Routley, R. and Routley, V. 1972. The semantics of first degree entailment. Noûs 6: 335–359.
Tanaka, K. 2000. The labyrinth of trees and the sound of silence: Topics in dialetheism. Ph.D. thesis, University of Queensland.
Urquhart, A. 1972. Semantics for relevant logics. The Journal of Symbolic Logic 37: 159–169.
Weber, Z. 2010a. Extensionality and restriction in naive set theory. Studia Logica 94: 109–126.
Weber, Z. (2010b). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic 3: 1–22.
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Tanaka, K. (2013). Making Sense of Paraconsistent Logic: The Nature of Logic, Classical Logic and Paraconsistent Logic. 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_2
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