Woods, John

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University of WindsorScholarship at UWindsorOSSA Conference Archive OSSA 2May 15th, 9:00 AM - May 17th, 5:00 PMCommentary on GirleJohn WoodsFollow this and additional works at: http://scholar.uwindsor.ca/ossaarchivePart of the Philosophy CommonsThis Commentary is brought to you for free and open access by the Faculty of Arts, Humanities and Social Sciences at Scholarship at UWindsor. It hasbeen accepted for inclusion in OSSA Conference Archive by an authorized conference organizer of Scholarship at UWindsor. For more information,please contact scholarship@uwindsor.ca.Woods, John, "Commentary on Girle" (1997). OSSA Conference Archive. 40.http://scholar.uwindsor.ca/ossaarchive/OSSA2/papersandcommentaries/40COMMENTS ON GIRLE'S "BELIEF SETS AND COMMITMENT STORES"John WoodsDepartment of PhilosophyUniversity of Lethbridge©1998, John WoodsI am awkwardly positioned to offer a critique of Rod Girle's paper. The awkwardness is that I agree with itsassertions and I like the case he makes for them. If we academics were disposed to do our work efficiently, Iwould in present circumstances end my remarks here, so that we could move on to the enlightenment of truecontention elsewhere, perhaps in the bar.I am not confident enough to make a break with a tradition as hallowed as this one, so I have tried to castaround for something useful to say about this system DL3, apart from what we already know about it. I myselfam interested in the nature and very existence of strategies for conflict resolution in the abstract sciences. Amongthe matters that I am currently looking into are such old chestnuts as ex falso quodlibet and the set theoretic andsemantic paradoxes. These issues are instructive in their way, but differently. From the point of view of conflictresolution, the principal fact about ex falso is the sheer awfulness of the arguments made by its detractors and itsproponents. And the chief moral to be learned from this fact is that logicians have developed scant expertise forhandling such contentions in their own backyard. Ex falso says that from a contradiction everything follows.There is a proof of this dating from the first year of the thirteenth century but, in an act of crass American culturalappropriation, it is known these days as the Lewis-Langford proof.1 Here it is:1. F ÙØF assumption2. F 1, simplification3. FÚY 2, addition4. ØF 1, simplification5. Y 3,4, disjunctive syllogism Conditionalizing, we have it that éFÙØFù entails arbitrary Y.We get a whiff of the awfulness of arguments which rage over ex falso by noting that, in the submission ofAnderson and Belnap, the Lewis-Langford proof is "self-evidently preposterous", that the move to line 5 fromlines 3 and 4 is "fallacy", indeed that disjunctive syllogism commits a "fallacy of relevance".2 For his part, StephenRead finds that there is no single sense of "or" that will concurrently deliver the goods for the rules of addition anddisjunctive syllogism. Speaking for the side opposite, Quine complains that giving up on ex falso amounts togiving up on classical negation and that it would be difficult to abandon something so basic. This is not argument.This is whimsy.However, it is hopeful to note that DL3 gives the appearance of possessing the technical wherewithal forsignificant improvement upon this bad record, especially in its challenge and resolution rules. Let us see.Revisiting the proof, S is the classical supporter of ex falso and the Lewis-Langford proof; H is the other side.Though DL3 doesn't offer specific guidance about how to make and share assumptions, let us take it that S andH agree to assume 1, and that in so doing éFÙØFù enters the commitment store (CS) of both parties, with itsprovisional status suitably flagged. Subsequent steps come by way of DL3's standard rules. Thus line 2 entersthe CS of both S and H by way of the rule of simplification together with DL3's rule, LogChall. Similarly forlines 3 and 4. It is the same way with line 5, or so it would appear. Girle's Imcon and LogChall "prevent thewithdrawal or challenge of logical principles". The prevention embeds an ambiguity. Does it, in the case beforeus, disenjoin the challenge or abandonment of DS because it is a principle of logic? Or does it forbid this if DS isa principle of logic? Or yet again, does it forbid it when S and H agree that DS is a principle of logic? Bearing inmind that S is our classicist and H our relevantist and that they disagree about whether DS is a principle of logic,we see that Imcom and LogChall fail to be usable rules precisely where a usable rule is needed. It is importantto emphasize that the Lewis-Langford problem stops cold any prospect of DL3-resolution. We might think thatthe dispute now moves to the question of whether DS is a principle of logic. That is not an askable challenge untilit is established that DS is not a rule of logic. Girle's rules oblige us not to challenge DS unless it is invalid. But itsinvalidity is precisely what S and H are deadlocked over. The present result easily generalizes. DL3 is unable toresolve any disagreement about any "logical principle". This is a disappointment, of course. We should not,however, rush to judgement. DL3 is no good for disputes about logical principles. But it is not good for nothing,as I shall now show.Let me now mimic the (epistolary) conversation between Russell and Frege in 1902. Russell is S and Frege is H.1. S: If R (the set of all non-self-membered sets) exists then RÎR and RÏR.2. H: Yes.3. S: By axioms we both accept, R exists.4. H: Agreed.5. S: So we're in trouble.6. H: You can say that again!At this juncture, I break away from the actual, historical conversation between Russell and Frege. It is well-known, and requires no elaboration here, that in the view of the historical Russell and Frege, the paradox of sets,at a minimum, destroyed the intuitive concept of set, the very idea of set, so to speak. (This is another matterabout which, as I suggested at the beginning, instruction is required). Elsewhere I have called this interpretation ofthe import of the Russell paradox "Frege's Sorrow".Frege's Sorrow would not have been the conclusion had this conversation been regulated by Girle-rules. To seehow this comes to pass we re-start the conversation at step 7.7. S: Since our resolution rules tell us to drop a conjunct if a statement in our CS is an immediatecontradiction, let's drop "RÎR".8. H: But there is also a rule about honouring immediate consequences of what's left, i.e., "RÏR".The trouble is that "RÏR" immediately restores "RÎR"; and we're right back where we started.9. S: Worse still, the rules drive us into an endless cycle of resolution and paradox-rebirth.10. H: Of course, there is no prospect under the rules of wriggling out of Excluded Middle, isthere?11. S: No; it's a principle of logic.12. H: But look, S. You've shown that if R exists then RÎR and RÏR.13. S: Unfortunately.14. H: Now the consequent of that conditional is a logical falsehood, n`est ce pas?15. S: Yes, and of course its negation is a logical truth.16. H: Right, And we can't give up that logical truth and we can't give up your fateful conditional.17. S: Nor can we give up modus tollens, another principle of logic.18. H: Which, together with the conditional and our logical truth produces as an immediateconsequence the negation of its antecedent.19. S: You mean, that R doesn't exist, after all?20. H: Yip.21. S: So arithmetic isn't toppling ?22. H: Yip.What is so striking about the Girle apparatus is that it imposes, perhaps inadvertently, the Barber solution on theparadox of sets. Whether it is right to do so is not here at issue. The point rather is that it takes a radicallydeflationary stand against paradoxical hysteria. This is so not only for the Russell paradox but also for theparadox of the Liar, putting "If L is a statement, then L is true and L is not true" in place of "If R is a set, thenRÎR and RÏR" and repeating the Girle conversation.To sum up: DL3 is of no consequence for disputes about what are and aren't principles of logic. But it is tailor-made for paradoxers, and should be welcomed by them.Notes1. C.I. Lewis and C.H. Langford, Symbolic Logic, New York: Dover 1932, 252.2. Alan Ross Anderson and Nuel D. Belnap, Jr., Entailment, vol. 1, Princeton: University of Princeton Press,1975.  View Paper by R. GirleView Index of Papers and CommentariesReturn to Main Menu

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