Ancestors

Toot

Written by Zack Weinberg on 2025-01-18 at 18:22

be me

see pointless internet argument about the principle of explosion

shrug, close tab

background thread in my brain going "well actually the problem with the principle of explosion is that the definition of → in classical logic fails to align with the natural language meaning of 'if A then B', in this essay I will..." for the next several hours

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Descendants

Written by Zack Weinberg on 2025-01-18 at 18:30

(essay would not be saying anything particularly new, see e.g. https://en.wikipedia.org/wiki/Relevance_logic and https://en.wikipedia.org/wiki/Connexive_logic )

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Written by Zack Weinberg on 2025-01-18 at 19:59

I am still stuck thinking about this and the problem isn't the definition of material implication. It can't be, explosion can be described without using implications at all. Rather, the problem is that the standard presentation of the principle of explosion makes it look silly. Or trivially fallacious, if you prefer.

The standard presentation goes like this:

1. Assume both P and not-P.

2. P, therefore also (P or A).

3. not-P, therefore we can strike out P from (P or A), leaving A. QED.

• Obviously*, either step 1 or step 3 of this chain of deductions is illegitimate. If P and not-P cannot simultaneously be true (classical logic) then we've already gone wrong in step 1 by assuming them both. If they can (e.g. intuitionistic logic) then striking out P in step 3 is not justified by not-P.

So, to the casual observer, it seems ridiculous for logicians to have been arguing about this since the days of Aristotle.

The reason it's not ridiculous is, what logicians are actually worried about is different. Nobody starts off with "assume both P and not-P." But we start off with "assume both P and Q" all the time, and what if it's possible to prove not-P from Q (alone)? Then we might execute step 3 of the above without realizing we have done this!

And Gödel proved that whenever our initial set of assumptions is sufficient to do arithmetic, we cannot be sure none of them allows us to prove the negation of another, so we can't rule out the mistake in step 1.

• That's* why some logicians want to find a different set of deduction rules, that still lets us prove things, but might catch it when we're about to execute an illegitimate step 3.

[#]logic #mathematics #proof

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