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

In 1999, Pavicic and Megill published

"Non-orthomodular Models for Both

Standard Quantum Logic and Standard

Classical Logic: Repercussions for

Quantum Computers". In that document,

they examined the sentential logic

axiomatized with

Def: (A -> B) = (-(A \/ B))

axiom schemata

Ax1: |- (A \/ A) -> A

Ax2: |- A -> (A \/ B)

Ax3: |- (A \/ B) -> (B \/ A)

Ax4: |- (A -> B) -> ((C \/ A) -> (C \/ B))

and inference rule

Det: ((|- A) /\ (|- (A -> B)) => (|- B))

They concluded that Boolean algebras were

not the canonical model for this logic.

What they discovered was that the lattice

mapping

(a /\ (b /\ c)) = ((a \/ b) /\ (a \/ c)) |--> 1

did not force a lattice model to be distributive.

According to their analysis, the source of

confusion on the models comes from interpreting

A=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, of

course, reflects the fact that the logic begins

with a restriction that its terms be propositions

having no other valuations.

They claim that the canonical model is a

weakly distributive lattice obtained from

formula algebras of the form

L = <Phi(L), -, /\, \/>

satisfying the ortholattice axioms,

OL1: ((A \/ B) = (B \/ A)) |--> 1

OL2: (((A \/ B) \/ C) = (A \/ (B \/ C))) |--> 1

OL3: ((-(-(A)) = A) |--> 1

OL4: ((A \/ (B \/ (-(B))))) = (B \/ (-(B)))) |--> 1

OL5: ((A \/ (A /\ B)) = A) |--> 1

OL6: ((A /\ B) =(-((-(A)) \/ (-(B))))) |--> 1

and the distributivity mapping that they had

been investigating

DL1: ((A /\ (B /\ C)) = ((A \/ B) /\ (A \/ C))) |--> 1

With the understanding that these lattice

models for the logic need not be Boolean

lattices, they concluded that

A=B <=>def

{Antecedents} |- ((-(A \/ B)) /\ (-(B \/ A)))

and

(For every map o from the formulas of the language

into 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

\ /

\ /

\ /

\ /

\ /

\ /

0

The constructions that have been presented

with the "fom" tag use other means to

establish anticipated properties of

truth functions.

Specifying the namespace of logical constants

with a geometry established the relations

of negation, contraposition, and conjugation

without truth tables.

Specifying the connectivity algebra of

intensional functions established functionality

without truth tables.

Formulating a syntactic form to label a geometry

so that any particular selections following an

algorithmic procedure could be interpreted

uniformly as logical equivalence established

"pretheoretic" truth table semantics.

Formulating an algorithm to enumerate the

namespace of the connectivity algebra so that

an order would be given for the first two

components of a truth table relative to

a form interpretable as a complete connective

established "pretheoretic" conditions formula

generation.

Topologizing the connectivity algebra

established possibility of interpreting the

truth table forms in accordance with Fregean

notions of "the True" and "the False."

With these conditions established, the

truth conditions for expressions taken to

be propositions was given relative to

the extension of NOR functionality to

arbitrary symbol strings. This part

of the strategy applies principles

of presupposition,

A presupposes B <=>def Neither A nor -A is true if B is false

In this case, B=(NOR is a truth-functional connective)

and free Demorgan lattices on one generator

provided the algebraic form for mapping the

truth values of the expression and a truth

functional connective simultaneously and

coherently

TRU --> TRU

NOR(NOR(A,NOR(A,A)),NOR(A,NOR(A,A))) --> TRU

NTRU --> NOT

NOR(NOR(A,A),NOR(NOR(A,A),NOR(A,A))) --> NOT

The last item, "lexical blocking" explains

why the lattice O^6 had been essential to the

work of Pavicic and Megill. For, in the

definition of propositions, it was also required

to map the proposition and its (NOR form)

negation to an orthocomplemented pair different

from 1 or 0 (TRU or NOT). In order for

a typical deductive system to be sound, the

propositions it generates as consequences of

its assumptions must be compatible with

those assumptions. Mapping the free DeMorgan

algebras into O^6 as sets of propositions

creates tree structures that work like

models of modal logic.

I need to think more about that. Anyway,

while I doubt anyone has been looking

at those constructions, this is how the

explanation begins.