Over the last couple of days on Twitter, I was involved in a thread, kicked off by Dan Piponi, discussing closed set logic—the natural dual of intuitionistic logic in which the law of the excluded middle holds but the law of non-contradiction fails, and which has models in the closed sets of any topological space, as opposed to the open sets, which model intuitionistic logic.
\(\def\ydash{\succ}\)This logic also has a nice sequent calculus in which sequents have one premise (or zero) and multiple conclusions. In the thread I made the claim that this is a natural and beautiful sequent calculus (it is!) but that the structure of the sequents means that the logic doesn’t have a natural conditional. The dual to the conditional (subtraction) can be defined, for which \(A\ydash B\lor C\) if and only if \(A-B\ydash C\). But the traditional conditional rules don’t work so well.
I realised, when I thought about it a bit more, that this fact is something I’ve just believed for the last 20 years or so, but I’ve never seen written down, so now is as good as a time, and here is as good as a place as any to explain what I mean.
While I was busy writing my most recent paper, “Proof Terms for Classical Derivations”, I heard that Raymond Smullyan had died at the age of 97. I posted a tweet with a photo of a page from the draft of the paper I was writing at the time, expressing loss at hearing of his death and gratitude for his life.
There are many reasons to love Professor Smullyan. I learned combinatory logic from his delightful puzzle book To Mock a Mockingbird, and he was famous for many more puzzle books like that. He was not only bright and sharp, he was also warmly humane. However, the focus of my gratitude was something else. In my tweet, I hinted at one reason why I’m especially grateful for Smullyan’s genius—his deep understanding of proof theory. I am convinced that his analysis of inference rules in the tableaux system for classical logic rewards repeated reflection. (See his First-Order Logic, Chapter 2, Section 1 for details.) I’ll try to explain why it’s important and insightful here.
I’m Greg Restall, and this is my personal website. I teach philosophy and logic as Professor of Philosophy at the University of Melbourne. ¶ Start at the home page of this site—a compendium of recent additions around here—and go from there to learn more about who I am and what I do. ¶ This is my personal site on the web. Nothing here is in any way endorsed by the University of Melbourne.
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