flexibeast.space - gemlog - 2020-09-29

“[T]he irreducibility of rejection to negation”

i recently read the Stanford Encyclopedia of Philosophy entry for ‘dialetheism’[a], by Graham Priest. Though i felt i understood most of the arguments made, i found myself tarpitted by the following:

We now come back to the issue flagged in Section 4.2, on the irreducibility of rejection to negation. That rejecting A is tantamount to accepting its negation is a common view, famously endorsed and defended (more precisely in terms of the corresponding speech acts of assertion and denial) by Frege and Peter Geach. But dialetheists have argued that this fusion is a confusion (see Berto 2008 on this issue). The point can be made independently of the issue of dialetheism: a paracompletist may well want to deny A, but it would be unfair to take such a denial as equivalent to the assertion of ¬A, since if A is truth-valueless, ¬A is normally considered truth-valueless, too, not a truth, and so not to be asserted. A dual position can hold for dialetheism: given that accepting ¬A is different from rejecting A, a dialetheist can do the former and not the latter -- exactly when she thinks that A is a dialetheia.

i'm not sure i grok what's going on here. Is the argument that, although denying A is usually equivalent to asserting ¬A, it's not always the case, since A might possibly be truth-valueless?

This reminds me of learning to differentiate between ¬¬A and A in logics without LEM[b]. Proving that something is not not true isn't necessarily the same as directly proving that something is true: saying that an object must theoretically exist is not the same as actually providing that object. Even if one doesn't object to LEM on philosophical grounds, there are still practical reasons to prefer direct proof, as described by set theorist Joel David Hamkins:

“Reductio ad absurdum or the contrapositive?”

i'm obviously going to have to do more reading to understand the dialetheist perspective on rejection vs. negation.

🏷 maths


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[a] Stanford Encyclopedia of Philosophy: ‘dialetheism’

[b] Wikipedia: ‘law of excluded middle’