```Date: Dec 11, 2012 9:24 PM
Author: fom
Subject: Re: fom - 01 - introduction

In 1999, Pavicic and Megill published"Non-orthomodular Models for BothStandard Quantum Logic and StandardClassical Logic: Repercussions forQuantum Computers".  In that document,they examined the sentential logicaxiomatized withDef:  (A -> B) = (-(A \/ B))axiom schemataAx1: |- (A \/ A) -> AAx2: |- A -> (A \/ B)Ax3: |- (A \/ B) -> (B \/ A)Ax4: |- (A -> B) -> ((C \/ A) -> (C \/ B))and inference ruleDet: ((|- A) /\ (|- (A -> B)) => (|- B))They concluded that Boolean algebras werenot the canonical model for this logic.What they discovered was that the latticemapping(a /\ (b /\ c)) = ((a \/ b) /\ (a \/ c)) |--> 1did not force a lattice model to be distributive.According to their analysis, the source ofconfusion on the models comes from interpretingA=B <=> |- ((-(A \/ B)) /\ (-(B \/ A)))where A=B would mean either( (A |--> 1) /\ (B |--> 1))or( (A |--> 0) /\ (B |--> 0))in standard completeness proofs.  This, ofcourse, reflects the fact that the logic beginswith a restriction that its terms be propositionshaving no other valuations.They claim that the canonical model is aweakly distributive lattice obtained fromformula algebras of the formL = <Phi(L), -, /\, \/>satisfying the ortholattice axioms,OL1: ((A \/ B) = (B \/ A))  |--> 1OL2: (((A \/ B) \/ C) = (A \/ (B \/ C)))  |--> 1OL3: ((-(-(A)) = A)  |--> 1OL4: ((A \/ (B \/ (-(B))))) = (B \/ (-(B))))  |--> 1OL5: ((A \/ (A /\ B)) = A)  |--> 1OL6: ((A /\ B) =(-((-(A)) \/ (-(B)))))  |--> 1and the distributivity mapping that they hadbeen investigatingDL1: ((A /\ (B /\ C)) = ((A \/ B) /\ (A \/ C)))  |--> 1With the understanding that these latticemodels for the logic need not be Booleanlattices, they concluded thatA=B <=>def{Antecedents} |- ((-(A \/ B)) /\ (-(B \/ A)))and(For every map o from the formulas of the languageinto the lattice O^6)(For every formula X in{Antecedents})...... ((o(X)  |--> 1) => ((o(A) = o(B)))The lattice O^6 has the form                 1               /   \             /       \           /           \         /               \       /                   \     /                       \   -B                        -A    |                         |    |                         |    |                         |    |                         |    |                         |    |                         |    |                         |    |                         |    |                         |    |                         |    |                         |    A                         B     \                       /       \                   /         \               /           \           /             \       /               \   /                 0The constructions that have been presentedwith the "fom" tag use other means toestablish anticipated properties oftruth functions.Specifying the namespace of logical constantswith a geometry established the relationsof negation, contraposition, and conjugationwithout truth tables.Specifying the connectivity algebra ofintensional functions established functionalitywithout truth tables.Formulating a syntactic form to label a geometryso that any particular selections following analgorithmic procedure could be interpreteduniformly as logical equivalence established"pretheoretic" truth table semantics.Formulating an algorithm to enumerate thenamespace of the connectivity algebra so thatan order would be given for the first twocomponents of a truth table relative toa form interpretable as a complete connectiveestablished "pretheoretic" conditions formulageneration.Topologizing the connectivity algebraestablished possibility of interpreting thetruth table forms in accordance with Fregeannotions of "the True" and "the False."With these conditions established, thetruth conditions for expressions taken tobe propositions was given relative tothe extension of NOR functionality toarbitrary symbol strings.  This partof the strategy applies principlesof presupposition,A presupposes B <=>def Neither A nor -A is true if B is falseIn this case, B=(NOR is a truth-functional connective)and free Demorgan lattices on one generatorprovided the algebraic form for mapping thetruth values of the expression and a truthfunctional connective simultaneously andcoherentlyTRU --> TRUNOR(NOR(A,NOR(A,A)),NOR(A,NOR(A,A))) --> TRUNTRU --> NOTNOR(NOR(A,A),NOR(NOR(A,A),NOR(A,A))) --> NOTThe last item, "lexical blocking" explainswhy the lattice O^6 had been essential to thework of Pavicic and Megill.  For, in thedefinition of propositions, it was also requiredto map the proposition and its (NOR form)negation to an orthocomplemented pair differentfrom 1 or 0 (TRU or NOT).  In order fora typical deductive system to be sound, thepropositions it generates as consequences ofits assumptions must be compatible withthose assumptions.  Mapping the free DeMorganalgebras into O^6 as sets of propositionscreates tree structures that work likemodels of modal logic.I need to think more about that.  Anyway,while I doubt anyone has been lookingat those constructions, this is how theexplanation begins.
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