```Date: Dec 11, 2012 9:31 PM
Author: fom
Subject: Re: fom - 09 - CORRECTED - a canonical complete connective

In "Cylindrical Algebras" byTarski and Monk, there is a chapteron duality.  Because the topicof the material involves thecoincidence of certain geometricnotions with first-order logic,Tarski and Monk associate thenegation symbol with the set-theoreticnotion of complementation.The free cylindrical algebra  offormulas has a signature:<Phi, \/, /\, -, F, T, E_i, v_i=v_j>, i,j<omegathe signature for the dual algebrais<Phi, /\, \/, -, T, F, A_i, -(v_i=v_j)>, i,j<omegawith respect to which Tarski andMonk make the observation:"...we easily see that theduals of the dual notionsare the original notions,e.g.,dual(dual(E_k))=E_kA notion is called self-dualif it coincides with its dual:thus, complementation isself-dual."This assertion is correct whenwhen working in a mathematicalcontext such as a real Euclideanspace.  But, in so far as itmight be applied to the negationsymbol in general -- that is, asthe transformation of the signaturesmight imply for the cylindricalalgebras of formulas-- the apparentself-duality masks the fact thatthe complete connectives NOR andNAND can be used to eliminatenegation.  Just as the OR and ANDconnectives in the signatureabove are dual to one another,fixing a canonical form for theelimination of negation exchangesdual connectives.Indeed, it exchanges canonicalrepresentations for all thewell-formed formulas.  That is,there is aPhi_NORof formulas that exchanges withthe dual formulasPhi_NANDIn his work on algebraic logic,Paul Halmos observed that logiciansfocus on provability whereasmathematicians seem to focus onfalsifiability.  In a manner thatcan be made precise, this reflectsthe duality of NAND and NOR.The construction here fixes apreference for the use of a completeconnective in the recursive generationof well-formed formulas.The namespace of logical constantshas been given as a 21-pointprojective geometry.  The namingscheme for the geometry uses the samenames for lines as for points.  Thisreflects the relationship of dualstructures in projective geometry.The names isolated to form the noralgebra demarcate an affine plane withinthe projective plane.  That affineplane has 20-lines.We use the line names to label anortholattice given by the orthogonalitydiagram:               OTHER              *            / |           /  |          /   |         /    |        /     | NOR           NIF  SOME *------*-------------*              |\           /|              |  \       /  |              |    \   /    |              |      X      |              |    /   \    |              |  /       \  |              |/           \|              *-------------*          NIMP                ANDThe Greechie diagram for this latticeshows that it is an amalgam of two Booleanblocks, one on four atoms and one onthree atoms:O-----O-----O-----O                    \                     \                      O                       \                        \                         OThe top of this lattice is TRUThe bottom of this lattice is NOT From the orthogonality diagram, itcan be seen that the atoms of thefour-atom Boolean block are justAND, NOR, NIF, NIMPthe four-coatoms areNAND, OR, IF, IMPand the connectivity in the latticeis precisely what one is accustomedto seeing for the free Boolean algebraon two generators.  The namescorresponding to the generatorsareFIX, LETand their orthocomplements areFLIP, DENYFor the three-atom Boolean block,the three atoms are justNOR, SOME, OTHERwhile the three coatoms areOR, NO, ALLThe amalgam is an atomic amalgamso that the shared vertices inthe lattice areTRU, OR, NOR, NOTWith regard to the earlier remarks,had the namespace been constructedfor emphasizing falsifiability,the loci labelled withTRU, NTRUwould have their names exchanged,the top of the ortholattice wouldhave been THIS, and the bottomof the ortholattice would havebeen NTRU.Moreover, the orthogonality diagramwould have had the form               NO              *            / |           /  |          /   |         /    |        /     | AND           NIMP   ALL *------*-------------*              |\           /|              |  \       /  |              |    \   /    |              |      X      |              |    /   \    |              |  /       \  |              |/           \|              *-------------*           NIF                NORwhere the names exchange according tocontraposition, placing AND and NANDamong the vertices common to bothBoolean blocks.  That is, the commonvertices becomeTRU, AND, NAND, NOTFor this alternative construction, thecomplete connective used for therecursive generation of well-formed formulaswould be NAND and the symbol of negationwould have been an abbreviation accordingto-A <->  NAND(A,A)For the construction chosen, the recursivegeneration of well-formed formulas usesthe NOR connective and the symbol ofnegation is an abbreviation accordingto-A <->  NOR(A,A)
```