I was reading something the other day that said at least once, probably more than once, that modus ponens is invalid in LP, and I find this a bit annoying because ⊃ just isn't really a conditional at all in LP, and (I think) you could add a conditional → with conditional proof as intro rule and >> - ThreadSky

mikebenchcapon.bsky.social • 8 days ago

I was reading something the other day that said at least once, probably more than once, that modus ponens is invalid in LP, and I find this a bit annoying because ⊃ just isn't really a conditional at all in LP, and (I think) you could add a conditional → with conditional proof as intro rule and >>

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gdivinetz.bsky.social•8 days ago

lol

mikebenchcapon.bsky.social•8 days ago

>> modus ponens as elimination rule to LP and get a non-trivial logic where modus ponens was valid; I imagine it'd be a sublogic of R but don't really know

mattweiner19.bsky.social•8 days ago

I just skimmed through a Priest paper on LP and it seems like part of his motivation was wanting to have a truth predicate, and with a truth predicate modus ponens collapses into triviality (because you can do "if this sentence is true, then P")

mikebenchcapon.bsky.social•8 days ago

As I understand it Priest does want a conditional that obeys modus ponens but doesn't want it to validate (P→(P→Q))→(P→Q); in classical logic the material conditional does validate that but I think that LP can probably be (conservatively? at least non-trivially) extended with one that doesn't

mikebenchcapon.bsky.social•8 days ago

The reason this rule is the key one for Curry's paradox is that if you've got a sentence C that says C→F then we get C↔(C→F) pretty automatically, and C→(C→F) from that, and then if you can get C→F from that you can then get C and then get F, but rejecting (C→(C→F))→(C→F) kind of works out

mattweiner19.bsky.social•8 days ago

hmyeah. guess he needs that.

as far as I can remember/tell (A->(A->B))->(A->B) is OK in R and even E, so those systems aren't going to help I think?

I'm in this weird position where I like paraconsistency but not dialetheism so I don't vibe with LP that much

mikebenchcapon.bsky.social•8 days ago

Yeah R would be no good for that reason but there are weaker relevant logics that should; looking at the chart in his intro to non-classical logic I think DWX or TWX would probably fit the bill

mikebenchcapon.bsky.social•8 days ago

I get that the first part alone isn't grounds for annoyance — part of why I think it's not a conditional is it doesn't obey modus ponens — but I think the second part makes it OK for me to be annoyed; tbh I don't think ⊃ has any business being in the language of LP anyway as it only confuses people

aidanmcglynn.bsky.social•8 days ago

Is this what people mean when they get mad about 'horseshoe theory'?

mikebenchcapon.bsky.social•8 days ago

Writing it as → instead of ⊃ seems obviously wrongheaded and would at the very least need a lot of justification

mikebenchcapon.bsky.social•8 days ago

I've thought some more and it wouldn't be a sublogic of R unless conditional proof was restricted the way R restricts it, but I think the general point ⊃ isn't really a conditional in LP and LP can probably be extended with a proper conditional still stands

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