On 3/8/2015 2:07 PM, LizR wrote:
On 9 March 2015 at 05:36, Bruno Marchal <[email protected] <mailto:[email protected]>> wrote:On 07 Mar 2015, at 09:36, LizR wrote:I thought <>P meant P was possible?In the alethic interpretation of modal logic, <> means possible, and [] means necessary. Before I get lost in logic, just going by the verbal descriptions... Am I right in thinking that "p" means "p is true" ? So p -> <>p would mean "p is true implies that it's possible p is true" In the temporal interpretation of modal logic, <> means sometime, and [] means always. p is true implies that p is sometimes true In the locus interpretation of modal logic, <> means somewhere, and [] means everywhere. ...somewhere... In the deontic interpretation of modal logic, <> means permitted, and [] means obligatory. p is permitted (by whom?!) etc. Note that all "<>" interpretation are form of possibility (alethic, temporal, locative, ...). In our interview of the Löbian machine, <> is translated in arithmetic with Gödel beweisbar predicate: In particular: <>t is consistency. <>t = ~[] f = ~ beweisbar ("0 = 1"), with "0=1" being a number coding the sentence "0 = s(0)".OK, so this is saying that p -> <>p would mean "if p is true then that implies that p is consistent" - which, roughly speaking, is what Godel showed to be wrong.Beweisbar(x) = Ey proof(y, x), that is: it exist a proof (y) of x. Proof must be mechanically checkable, and so, like sentences, they can be coded into numbers, and the predicate proof just decode the proof named by y and looks if it proves the sentence coded by x. <>t = ~[] f means intuitively, as said by PA: "PA does not prove the false", or "PA is consistent".
So to summarize you don't want p->[]p as a modal axiom because it particularizes to t->[]t which says all true propositions are provable, contrary to Godel's theorem. Godel proved PA incompleteness by diagonalization on classes of numbers. But this applies to PA, not to every axiom set. So why not conclude there is something wrong with PA? To me it seems more intuitively compelling to say p-><>p than to say every number has a successor and deny p-><>p.
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