Value

Conditionals in Closed Set Logic

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. ¶ ¶ Consider the conditional rules in the classical sequent calculus. They look something like this (give or take variations in the presentation, all equivalent given the classical structural rules): Classical sequent rules for the conditional. ¶ If we restrict these rules to multiple conclusion single... -

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Permissibility of the natural

The usual way to argue that an action is permissible is to argue that the arguments against the action’s permissibility fail. But it would be really nice to be able to give a more positive argument for an action’s permissibility. Sometimes one can do so by showing that the action is obligatory, but (a) that doesn’t help with the permissibility of non-obligatory actions, and (b) often an argument for the obligatoriness of a positive action presupposes the action’s permissibility (e.g., the obligation to kill a dog that is attacking one’s child when no other means of defense is available presupposes the general permissibility of killing dogs with good reason). ¶ Here is a place where Natural Law NL) can provide something quite useful, namely... -

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Sean Carroll: Extracting the Universe from the Wave Function

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george romero (1940 - 2017)

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Conditionals in Closed Set Logic 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... Consequently.org -


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