Abstract
John Bell’s arrival at the University of Western Ontario was a turning point in my life. At the time I was a PhD student, a mere few overdue course credits away from beginning a dissertation on Carnap’s Logical Syntax of Language. What better way to make up a couple of those credits than by taking the set theory course to be taught by the hotshot new professor the philosophy department had recruited from England, thanks in part to Margaret Thatcher’s campaign to drive intellectual talent out of the UK? This turned out to be a doubly lucky choice for me. First, being new to the department, John was on the look-out for PhD students who seemed able to do their sums, so the class was a chance to catch his eye. More significantly, the class was a revelation to me.
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A similar argument can be made, I think, for the other star example from the book, the simultaneous correctness of classical and relevant consequence relations. “Disjunctive syllogism is valid when GTT’s cases x are taken to be possible worlds, and it is invalid when those cases x are taken to be situations. That is that. To whether Disjunctive Syllogism is valid, the classical and relevant accounts give different but not rival answers” (Beall and Restall, 2006, p. 56). While there are differences of detail between what I would say about the examples, these differences do not warrant covering essentially the same ground twice here.
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And that various flavours of constructivism disagree with each other. In this connection, it is interesting that Beall and Restall choose the Axiom of Choice as their example. For that axiom, and indeed other choice principles, have a disputed and complicated status within constructive logic, one that John Bell has done as much to illuminate as anyone. See (Bell, 1993, 2006, 2009; Maietti and Valentini, 1999; DeVidi, 2004, 2006). By advocates of some constructive systems AC is regarded as a principle of logic that follows from the constructivist account of the existential quantifier; but in other systems of constructive reasoning it implies the law of excluded middle, and so all of classical logic.
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This is a version of the intuition behind the “formality” condition identified as part of the “core” of the concept of logical concept by (2006, p. 21).
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But not all. See (Read, 2006) for an emphatic example.
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This paper has been knocking around for a long time in one form or another. I first presented it in 2003 to an audience that included John Bell at the University of Western Ontario, and in the intervening years at the Western Canadian Philosophical Association and to the philosophy departments at Concordia University and the University of Waterloo. I thank members of those audiences, and in particular Bill Demopoulos, Bill Harper, Tim Kenyon, Greg Lavers, Jeff Pelletier and Jim Van Evra, for useful comments and questions. The financial support of the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. The best thing about the paper is surely the title, so I am sorry to have to confess that I owe it to my hilarious colleague Kenyon.
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DeVidi, D. (2011). The Municipal By-Laws of Thought. In: DeVidi, D., Hallett, M., Clarke, P. (eds) Logic, Mathematics, Philosophy, Vintage Enthusiasms. The Western Ontario Series in Philosophy of Science, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0214-1_5
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