I want to post on more recent reading soon, but meanwhile I have one last footnote to the discussion of Barwise and Etchemendy. I didn't include this in the main line of discussion, both because it might be of more general interest and because it can be intelligibly separated from the rest of their argument.
Remember that the oddest upshot of their modelling of propositions as hyper-sets is that (a) classical bivalent logic is correct, such that everything is false if it is not true and true if it is not false, (b) the Liar Sentence expresses a truth-evaluable proposition, and (c) the Liar Sentence somehow gets to be false without also being true. Various purely formal moves nominally validated this status for both the Russellian and Austinian ways of thinking about propositions, but I think the authors end the book all-too-aware of how capricious and counter-intuitive all this sounds. Thus, they end by gesturing in the direction of the distinction between negation and denial. They claim that if they had factored in denial as well as assertion and negation, it would have been clear that the logic of their notion of propositions was entirely classical, but that it would “involve us in untangling one of the most basic conflations in the logical literature, the conflation of negation and denial” and that this “would take us far from the topic of the book.” It's hard, on a snarky level, not to think of Fermat and proofs to long to include in the margins, but never mind that right now. There's still an interesting issue about what role this distinction could contribute here.
The problem is that Barwise and Etchemendy themselves tell us very little about the distinction, or what role they think it should play. They tell us that speech-act theorists are always telling logicians to take the distinction more seriously. Fair enough. But how exactly would it be helpful here?
In one of the best turns of phrase in the book, they say that just as ignoring relativistic effects doesn't cause any problems on a trip to the corner supermarket, but knowledge of those effects becomes vitally important when approaching the speed of light, “when approaching sentences like the Liar, we risk paradox if we ignore the difference between negation and denial.”
Excellent. Sadly, we never find out what exactly the import of that distinction is here.
If anyone has any suggestions to make in the comments, I'd be glad to hear them. Tentatively, though, here's my position:
Is there a distinction? Yes. Is that distinction relevant or useful for the purposes Barwise and Etchemendy are gesturing towards? Absolutely not.
Of course, the distinction between negation and denial could be very relevant to a discussion of the semantic paradoxes if, for example, we were working within a deviant logic that denied the Law of the Excluded Middle and posited extra possibilities 'between' P and ~P. (This is precisely what Barwise and Etchemendy repeatedly deny that they are suggesting.) If, however, we are assuming classical bivalent logic, then denial and negation are still distinct, but the category of propositions we are logically or epistemically warranted in denying will, it seems to be, clearly be necessarily co-extensive with the category of propositions we are logically or epistemically warranted in asserting the negations of. As such, for someone trying to fit the square peg of granting that the Liar is truth-evaluable and that every sentence that is not true is false and vice versa into the round hole of denying dialetheism, it doesn't look like this distinction can be of any use.
Meanwhile, my friend Ryan pointed me towards another webcomic that references dialetheism , although as far as I'm concerned there's nothing in the comic itself that's quite about dialetheism.
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Here's a quick idea: even if we accept classical logic, what counts with respect to assertion and denial is (at least, in part) what we are justified in either accepting or rejecting in the way of believing.
If I have no evidence to settle whether P or ~P, and I adhere to classical logic, then it seems like I am in the following situation: one of P or ~P is true, but I can neither accept/assert nor reject/deny either one. Maybe the Liar is one of these cases where I just can't justifiably believe it or its negation (although that would be a funny thing for Barwise and Etchemendy to argue since they have already claimed that the Liar is, in fact, false).
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