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Disjunction and Disjunctive Syllogism

Published online by Cambridge University Press:  01 January 2020

Peter Milne
Peter Milne*
Affiliation:
University of Edinburgh, David Hume Tower-George Square, Edinburgh, United KingdomEH8 9JX
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Extract

The validity of argument by disjunctive syllogism (henceforth, DS) has been denied by proponents of relevant and paraconsistent logic (who are sometimes one and the same). DS is stigmatised for its role in inferences — most notably C.I. Lewis's derivation of that fallacy of irrelevance ex falso quodlibet (EFQ) — that involve both it and other rules of inference governing disjunction, or, to speak more precisely, other rules of inference taken to apply to the very same disjunction that obeys DS. In avoiding these inferences the road less travelled is to deny the identity rather than to deny DS: what follows is, then, an exercise in disjoining disjunctions.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 28 , Issue 1 , March 1998 , pp. 21 - 32
Copyright
Copyright © The Authors 1998

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References

1 The Lewis argument might better be attributed to Albert of Saxony, who had it in all essentials some six centuries before Lewis, or perhaps to William of Soissons in the twelfth century. For the former see, e.g., Bochenski, J.M. Formale Logik, second edition (Freiburg and Munich: Karl Alber 1962), 237-8Google Scholar; for the latter, Martin, Christopher ‘William's Machine,’ Journal of Philosophy 83 (1986) 564-72CrossRefGoogle Scholar (I owe this reference to Stephen Read).

2 Allen, Colin and Hand, Michael Logic Primer (Cambridge, MA: The MIT Press 1992), xii and 18–19Google Scholar

3 See Prawitz, Dag ‘Meaning and Proofs: On the Conflict between Classical and Intuitionist Logic,’ Theoria 43 (1977) 2–40CrossRefGoogle Scholar, and ‘Proofs and the Meaning and Completeness of the Logical Constants,’ in Hintikka, J. Niiniluoto, I. and Saarinen, E. eds., Essays on Mathematical and Philosophical Logic (Dordrecht: Reidel 1979) 25–40CrossRefGoogle Scholar.

4 In order to read OS as a rule not just in which negation is eliminated but giving the meaning of negation one has to take it that not a single inference but the totality of instances of OS in which it occurs go to fix the meaning of a negated proposition: i.e., what fixes the meaning of -.A is that,for every sentence B, -.A and AvB suffice to yield B (as do -.A and BvA).

5 Belnap, Nuel ‘Tonk, Plonk and Plink,’ Analysis 22 (1962) 130-4CrossRefGoogle Scholar; reprinted in Strawson, P.F. ed., Philosophical Logic (Oxford: Oxford University Press 1967) 132-7Google Scholar

6 Read, Stephen Relevant Logic (Oxford: Blackwell 1988), 31-4Google Scholar, and Thinking About Logic (Oxford: Oxford University Press 1994), 60, 160, 163

7 Dummett, Michael ‘The Source of the Concept of Truth,’ in Boolos, George ed., Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge: Cambridge University Press 1990) 1-15, at 7–8Google Scholar

8 Dummett, Michael The Logical Basis of Metaphysics (London: Duckworth 1991), 256Google Scholar. It should perhaps be noted that Dummett does describe DS as a fundamental form of argument, The Logical Basis, 293.

9 Dummett, ‘The Source,’ 7–8Google Scholar

10 See Milne, Peter ‘Classical Harmony: Rules of Inference and the Meaning of the Logical Constants,’ Synthese 100 (1994) 49–94, at 90, n. 31CrossRefGoogle Scholar, for a proof that a conditional containing the Sasaki hook is locally weakest in the sense given in the text. As is well known, adding an intuitionist conditional to quantum logic collapses it to classical logic.

11 Tennant, Neil Natural Logic, second edition (Edinburgh: Edinburgh University Press 1990)Google Scholar.

12 Milne, ‘Classical Harmony,’ 76Google Scholar

13 I mean this comparison of strength quite strictly, so that, for example, logics that contain DNE but lack other principles of intuitionist logic and may therefore intuitively be regarded as quite weak are neither stronger nor weaker than intuitionist logic. One is, of course, entitled to view the absence of DNE as a weakness of intuitionist logic. Discussion with Stephen Read prompted this footnote.

14 Read, Thinking about Logic, 60Google Scholar

15 The non-classicallogic in question is called MC+ in Parsons, Charles ‘A Propositional Calculus Intermediate between the Minimal Calculus and the Classical,’ Notre Dame Journal of Formal Logic 7 (1966) 353-8CrossRefGoogle Scholar, and JX in Segerberg, Krister ‘Propositional Logics Related to Heyting's and Johansson's,’ Theoria 34 (1968) 26-61.Google Scholar From the perspective of the present paper, in which we have distinguished two disjunctions, Allen and Hand take the introduction rules of one disjunction and the elimination rules of the other, something that can evidently be done in two ways. Parsons’ MC+, Segerberg's JX, rather than intuitionist logic, would have played a much more prominent role here had Allen and Hand chosen +-introduction and orthodox v-elimination as the rules governing their disjunction, admittedly a most unlikely event.