I just had a weird thought
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From: Marshall on 13 May 2010 09:48 On May 12, 10:33 pm, herbzet <herb...(a)gmail.com> wrote: > etc., etc.! > > Plus ça change, eh? > (herbzet's axiom in French) "The more things change, the less they stay the same." Marshall
From: herbzet on 14 May 2010 01:21 Marshall wrote: > herbzet wrote: > > > etc., etc.! > > > > Plus �a change, eh? > > (herbzet's axiom in French) > > "The more things change, the less they stay the same." Bonsoir, Marshall -- nice to hear from you. Thanks for your reply! -- hz
From: Marshall on 15 May 2010 09:10 On May 13, 10:21 pm, herbzet <herb...(a)gmail.com> wrote: > Marshall wrote: > > herbzet wrote: > > > > etc., etc.! > > > > Plus ça change, eh? > > > (herbzet's axiom in French) > > > "The more things change, the less they stay the same." > > Bonsoir, Marshall -- nice to hear from you. Thanks for your reply! C'est mon plaisir. Glad to see you back, and I hope (for entirely selfish reasons) you will again be a frequent contributor. Marshall
From: herbzet on 16 May 2010 01:03 Marshall wrote: > herbzet wrote: > > Marshall wrote: > > > herbzet wrote: > > > > > > etc., etc.! > > > > > > Plus �a change, eh? > > > > (herbzet's axiom in French) > > > > > "The more things change, the less they stay the same." > > > > Bonsoir, Marshall -- nice to hear from you. Thanks for your reply! > > C'est mon plaisir. > > Glad to see you back, and I hope (for entirely selfish reasons) you > will again be a frequent contributor. My goodness, what an agreeable remark! I'm touched! (Perhaps it's best not to inquire into your "entirely selfish reasons"(!)) Basically, I don't think I'm quite up to be a frequent contributor for awhile longer -- though I intend to come back more often in the not-to-distant future. I just wanted to direct people's attention to the startling fact of non-Boolean-algebra structures for which classical propositional logic is a complete theory. I was hoping for some discussion by the smart guys of this, but it may take some time for people to get used to the idea. On the subject of models of classical (propositional) logic: both Schroder and C.I.Lewis (inventor, with C.H.Langford, of the S1-5 modal logics) tried to get rid of the models of Boole's algebra that have more than two elements (and so make things nicer for true/false logic) by including the formula A = (A = 1) in their axiomizations for propositional logic -- an axiomization that is otherwise identical to that for their algebra of classes. Later workers have passed over this meaningless axiom in charitable silence. Indeed, I myself have occasionally wondered how to get rid of all the "extra" models ever since I started to learn about logic. Letting '[=]' be ascii-speak for the triple bar symbol of logical equivalence, my latest effort was to add the formula (A[=]B) v (B[=]C) v (C[=]A) as an axiom. "Hah!" I thought. "Not elegant, but that oughta do it!" thinks I -- but alas: that formula turns out to denote the universal class (as does any tautology interpreted as a class) when A,B,C are any classes, 'v' is interpreted as class union, and '[=]' is inter- preted (as is usual) as the complement of the symmetric difference. Then the penny finally dropped: classical propositional logic is Post complete -- adding *any* formula as an axiom that is not already a theorem makes the logic inconsistent: every formula then becomes provable. The "extra" models cannot be gotten rid of by adding new axioms, except at the cost of inconsistency (and thus having no models at all). What an exciting life I lead! -- hz
From: herbzet on 16 May 2010 18:15
herbzet wrote: > Then the penny finally dropped: classical propositional logic is Post > complete -- adding *any* formula as an axiom that is not already a > theorem makes the logic inconsistent: every formula then becomes > provable. The "extra" models cannot be gotten rid of by adding > new axioms, except at the cost of inconsistency (and thus having > no models at all). David Ullrich slapped me on this a few years ago. Of course we *can* add further assumptions to the classical propositional logic -- call them "postulates" -- we just can't treat them like tautologies. With a tautology, we can validly substitute any formula phi for any propositional variable w in the tautology, provided that we uniformly substitute phi for every occurrence of w in the tautology. If we add a postulate to the calculus that is not already a logical theorem (i.e., not a tautology) then either it is already a contradiction or it is a contingent formula. If we add a contingent formula we cannot uniformly substitute any formula phi for any propositional variable w, or else we can get a contradiction -- if we add, for example, the postulate (p v q) -> r then by uniform substitution we can get ((p->p) v (p->p)) -> ~(p->p), a contradiction. Another way of saying this is that we can't add a non-tautologous propositional formula /scheme/ as an *axiom* scheme, because some of its instances will be contradictory. Of course, I've neglected the case of adding first-order formulae as axioms/postulates. -- hz |