tag:blogger.com,1999:blog-3891434218564545511.post8412119794048176229..comments2024-03-28T19:56:42.305-05:00Comments on Alexander Pruss's Blog: A quick route from mathematics to metaphysical necessityAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-3891434218564545511.post-61012363969264197512015-06-15T08:26:49.612-05:002015-06-15T08:26:49.612-05:00For something to be logically necessary, must it b...For something to be logically necessary, must it be provable in a computable way, using a finite number of steps? Or can we expand the notion of logical proof to include things that could be proved by hypothetical machines that could perform an infinite number of steps and draw conclusions about the results (a form of <a href="http://en.wikipedia.org/wiki/Hypercomputation" rel="nofollow">hypercomputation</a>), for example verifying the Goldbach conjecture by showing explicitly that for each integer N, there exist two primes that add to N?JesseMhttps://www.blogger.com/profile/09993568347649474812noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-20175894186939320132015-05-21T10:06:11.002-05:002015-05-21T10:06:11.002-05:00Let L be a logically possible, but metaphysically ...Let L be a logically possible, but metaphysically impossible proposition. Suppose we understand L is logically possible iff. we cannot derive a contradiction from L in S. But then L is either a theorem or contingent. Either way, L is consistent with the rules of S. There has to be some model on which L comes out true in some world. But it is false that L is consistent with the rules of S, since L is metaphysically impossible. So, it can't be right that L is logically possible iff. we cannot derive a contradiction from L in S.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-82789733140682850342015-05-21T08:41:42.410-05:002015-05-21T08:41:42.410-05:00If their non-contingency is based on no contradict...If their non-contingency is based on no contradiction following from them in any world, as you say, then that just IS logical necessity of consistency, no? To prove X is usually just to show a contradictory entailment in denying X, right?Michael Gonzalezhttps://www.blogger.com/profile/05279261871735286117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-49116754108327459382015-05-19T07:22:22.907-05:002015-05-19T07:22:22.907-05:00logical possibility is possibility with respect to...<i>logical possibility is possibility with respect to a subset of the real rules</i>.<br /><br />That makes it sound like p is logically possible only if it is consistent with the real rules. But that's false. Lot's of 'logically possible' propositions are not consistent with any set of rules.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-34928741287109211752015-05-18T21:44:16.207-05:002015-05-18T21:44:16.207-05:00I am inclined to agree that logical possibility is...I am inclined to agree that logical possibility isn't a species of possibility. It's more like "conjunction-elimination possibility", where a proposition is conjunction-elimination possible iff one cannot derive a contradiction from it by applying conjunction-elimination. I.e., logical possibility is possibility with respect to a subset of the real rules.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-66643140488202814452015-05-17T08:46:34.370-05:002015-05-17T08:46:34.370-05:00van Inwagen offers some useful advice here. Logica...van Inwagen offers some useful advice here. Logical possibility is not a species of possibility (similarly for necessity, of course). It is not as though there are propositions that are logically possible but not metaphysically so. If the proposition is not metaphysically possible, it's not possible simpliciter. There are no worlds in which it is true, so it surely isn't logically possible either: there's no larger set of worlds in some of which it is true. It's not any-wise possible. Healthful advice, I think, http://www.andrewmbailey.com/pvi/Modal_Epistemology.pdf.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.com