Classical logic is not distributive
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From: herbzet on 13 May 2010 01:36 In http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.2698v1.pdf "Is Quantum Logic a Logic?" by sometimes sci.logic contributor Norm Megill and co-authored by Mladen Pavicic, it is shown that classical propositional logic has a sound and complete model that is not a complemented distributive lattice, i.e., is not a Boolean algebra. In particular, the theorem of classical logic [a ^ (b v c)] <-> [(a^b) v (a^c)] is true in this model for all elements a,b,c, but we also have that [a ^ (b v c)] =/= [(a^b) v (a^c)] for some elements a,b,c of the model -- unlike in a Boolean algebra. A brief statement of the result is also at the Metamath site http://us.metamath.org/qlegif/mmql.html#prop at the paragraph "Classical propositional calculus has non-Boolean lattice models". -- hz
From: herbzet on 13 May 2010 01:59 herbzet wrote: > ... it is shown that classical propositional logic has a sound > and complete model that is not ... I guess that should read "... classical propositional logic is sound and complete for a model that is not blah blah blah". Whatever. -- hz
From: herbzet on 14 May 2010 01:45 herbzet wrote: > ...(which is not a Boolean algebra, not having 2^n elements > for n < 0)... Should read, "for n > 0". -- hz
From: herbzet on 14 May 2010 05:07
herbzet wrote: > The axioms CL 1-4 are ...complete for classical logic. Emil Post [1921] (doctoral dissertation at Columbia University 1920). A fifth axiom (p v (q v r)) -> (q v (p v r)) was found to be redundant by Paul Bernays [1926]. -- hz |