tag:blogger.com,1999:blog-3891434218564545511.post5902292811002310533..comments2024-03-28T13:23:50.623-05:00Comments on Alexander Pruss's Blog: Logical closure accounts of necessityAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-3891434218564545511.post-57046313786270037512018-03-20T10:01:11.360-05:002018-03-20T10:01:11.360-05:00William:
That's a good point, but it depends ...William:<br /><br />That's a good point, but it depends on what and how "(1)" designates. If it rigidly designates the sentence type or the proposition, we don't have the contingency problem.<br /><br />Anyway, once (1) is rewritten syntactically, as in the diagonal lemma, the contingency problem disappears completely.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-39249541978508220342018-03-16T12:43:29.002-05:002018-03-16T12:43:29.002-05:00"Thus (1) is true but there is a possible wor..."Thus (1) is true but there is a possible world w where (1) is false."<br /><br />(1) is contingent on the post having been made. So the opposite of "(1) is true" could also be <br /><br />"(1) is true (when asserted) but there is a possible world where (1) does not exist."<br /><br /><br /><br />Williamhttps://www.blogger.com/profile/09292602256213936359noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-83153272130745286252018-03-16T05:32:12.021-05:002018-03-16T05:32:12.021-05:00Without that argument, I would be tempted to concl...Without that argument, I would be tempted to conclude that Grim's (4) was nonsense from the assumption that God was omniscient. That would deal with my "technically" above.<br /><br />There are other reasons to think of (4) as nonsense, of course (such as would defeat my argument above, via L being nonsense), all of which indicates that your (1) is also nonsense. I imagine that a fan of logical closure accounts would feel that way.<br /><br />Perhaps you would agree that in the case of omniscience it would be reasonable? There is not the same almighty authority behind the closure accounts, but I wonder if you can use a kind of argument there that you would find it reasonable to reject in such a blunt way elsewhere.Martin Cookehttps://www.blogger.com/profile/11425491938517935179noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-49556014284365000722018-03-16T01:58:27.564-05:002018-03-16T01:58:27.564-05:00I like your argument, it's got more to it than...I like your argument, it's got more to it than Patrick Grim's 2000 argument against omniscience, for example; but, if there was a being that knew everything, all about His creation, all about your future, and mine, and so forth ... except that it did not, because it could not, know Grim's (4):<br />God doesn't believe that Grim's (4) is true<br />then surely that being would, really, be omniscient. I mean, technically He wouldn't be, but really, if He knew everything else, except Grim's (4) and maybe infinitely many related things? And I feel the same way about your argument.<br /><br />Logical closure accounts of necessity cannot be false because of your (1).<br />Your argument is really all about the Liar paradox.<br /><br />The assumption that propositions, assertions, must be either true or else not true is plausible, of course, because to want the truth is to want things to be made clear. It is to want vagueness to be eliminated. But, the Liar paradox arises with sentences that are not normal sentences, and so the question is whether there are exceptional circumstances in which it would be highly implausible were ‘is true’ not behaving like a vague predicate; and indeed there are.<br /><br />Suppose that @ is originally an apple but has its molecules replaced, one by one, with molecules of beetroot. Suppose that the question ‘What is @?’ is asked, after each replacement, and that the answer ‘It is an apple’ is always given. Originally it is a correct answer, because originally it is true that @ is an apple, but eventually it is not. If the proposition that @ is an apple must be true or else not, then an apple can be turned into something else (presumably a mixture of apple and beetroot) by replacing just one of its original molecules with a molecule of beetroot. And that is highly implausible. It is surely possible, since much more plausible, that @ would be, at such a stage, no less an apple than apple/beetroot mix, that it would be as much an apple as not, so that the proposition that @ is an apple would be as true as not.<br /><br />Given that possibility, the following is a proof by reductio ad absurdum that ‘is true’ is indeed a vague predicate. I call the following assertion ‘L’:<br />The assertion that you are currently considering, this assertion, is not true.<br />L is clearly an assertion; it is the assertion that L is not true. So if L is true, then L is not true, but L cannot be true and not true, so L cannot be true. But, if L is not true, then L is true, and L must be either true or else not true. Contradiction; but, were ‘is true’ vague, L would be true insofar as L was not true. It would follow only that L was as true as not. No contradiction.Martin Cookehttps://www.blogger.com/profile/11425491938517935179noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-86811487165618216662018-03-15T13:39:48.554-05:002018-03-15T13:39:48.554-05:00Oh yes, I see.
Still, Liar-style sentences prove (...Oh yes, I see.<br />Still, Liar-style sentences prove (in ordinary logic) that "is true" can be a vague predicate in just such abnormal circumstances as (1), so that there is really no reason why (1) should be either true or else false.<br />(I don't know about Godel, but I'd guess that that was more formal?Martin Cookehttps://www.blogger.com/profile/11425491938517935179noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-79831328691781315532018-03-15T12:43:07.409-05:002018-03-15T12:43:07.409-05:00The strategy iterates. Let F' be the union of ...The strategy iterates. Let F' be the union of F with (1), (2), etc. Then form (1') which is just like (1) but with F' in place of F. No matter how much you add to F, the problem iterates.<br /><br />This point is analogous to the observation that you can't get rid of incompleteness just by adding the Goedel sentences to the axiom set.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-65400321841473096752018-03-15T11:18:08.814-05:002018-03-15T11:18:08.814-05:00If you are right about this being a problem, then ...If you are right about this being a problem, then there is an obvious solution.<br />The threat is to necessary truths being all and only those things provable from F,<br />so why not have the necessary truths being anything provable from F, or (1), and so on, and only such things? The difference is minimal (my "and so on" is for "(2) Statement 2 cannot be proved from F and (1)" and so on), but necessary, if you are right.<br />(Given that this is essentially Liar territory, I do not think that you are.)Martin Cookehttps://www.blogger.com/profile/11425491938517935179noreply@blogger.com