Standard Logics Are Valuation-Nonmonotonic

Standard Logics Are V aluation-Nonmonoton ic Mladen P a viˇ ci ´ c Physics Chair, F aculty of Civil Engine ering University of Zagr eb, Zagr eb, Cr o atia pavi cic@ grad .hr ; http ://m 3k.g rad.hr/pavicic Norma n D. Megill Boston Information Gr oup, 19 L o cke Ln. L exington, M A 02420, USA nm@a lum. mit. edu ; http:/ /www .metamath .org June 3, 2018 Abstract It has r ecen tly b een disco v ered th at b oth qu an tum and classical prop ositional log ics can be mod elled b y classes o f non-orthomo dular and th us non-distributive lattice s that prop erly con tain standard or- thomo dular and Boolean classes, resp ectiv ely . In this pap er we pr ov e that these log ics are complete eve n for those classes of the former lat- tices from w hic h the stand ard orthomo d ular lattices and Bo olean al- gebras a re excl uded. W e also sho w that neither quan tum nor classical computers can b e founded on the latter mo dels. It follo ws that logics are valuation-nonmonot onic in the s en se th at their p ossible mo d els (corresp onding to their p ossible hardwa re imp lemen tations) and the v aluations for th em dr astically c h ange w hen we add new conditions to their defining conditions. These v aluations can ev en b e completely separated b y pu tting them int o disjoint lattice classes by a tec hnique present ed in the pap er . Keywor ds: nonmonotonic logic, classical logic , quant um logic, non- distributive non-orthomo d ular lattice, weakly orthomo du lar lattice, Bo olean algebra, wea kly distribu tiv e lattice, artificial in tellige nce 1 1 In t r o du ction A go o d deal of artificial in telligence researc h is fo cused on artificial neu- ral netw orks, on the one hand, and on default/ nonmonotonic logic, on the other. Neural net w orks a r e characterize d b y hea vy reliance on logic gates. On the other hand, nonmonotonic inference rules formalize generalizations of standard log ic that admit c hanges in the sense tha t v alues of prop ositions ma y change when new information (axioms) is added to or old informatio n is deleted from the system. In this pap er, w e show that already standard logics (classical as w ell as quan tum)—whose mo no tonicit y is usually tak en for g ran ted—are nonmonotonic at b oth the lev el of lo gic gates that imple- men t them and t he leve l of its v aluations, i.e., mappings f r o m the logic to its mo dels. W e consider tw o standard log ics (in contrast to, e.g., mo dal log ics) in this pap er: prop ositional classical logic and prop ositional quan tum logic. In pra c- tice, classical logic relies almost exclusiv ely on the { 0,1 } v aluation, i.e., the t w o - v alued truth table v aluation, for its prop ositional part. This v aluat io n extends to the sen t ences of all theories that ma ke use of classical logic, suc h as set theory , mo del theory , and the foundations of mathematics. How ev er, there are also non- standard v aluations generated by non-distributive lattices, whic h correctly mo del classical prop ositional log ic, and by non- o rthomo dular lattices, whic h correctly mo del quantum logic. An immediate consequence of this v aluatio n dic hotom y is that classical lo gic mo delled b y suc h non- distributiv e lattices do es not underlie presen t-day classical computers, since non-standard v aluations cannot b e used to run them. Only classical log ic mo delled by a Bo olean algebra a nd having a { 0,1 } v aluation can serv e us fo r suc h a purp ose. Hence, whenev er w e w an t to utilize a logic for a pa rticular application w e hav e to sp ecify the mo del w e would use as w ell. Before we go in to details in the next sections, w e should b e more sp e- cific ab out our distinction of standard vs. non-standar d v a luations. Let us illustrate it with a gr aphical represen tation of the O6 lattice given in F ig- ure 1, whic h can serv e as a mo del for classical logic in the same wa y that { 0,1 } Bo olean algebra can. Lines in the figure mean ordering. Thus w e hav e 0 ≤ x ≤ y ≤ 1 a nd 0 ≤ y ′ ≤ x ′ ≤ 1, where 0 and 1 are the least and the greatest elemen ts of the lattice, resp ectiv ely . Can this mo del b e given a linearly ordered o r n umerical in terpretation, for instance the in terpretation pro vided by the probabilistic seman tics for classical logic [1]? The answ er is no, b ecause when x 6 = y 6 = 0 , 1, an o rdering b et w een x and either x ′ or y ′ and 2 b et w een y and either x ′ or y ′ is not defined, and it is assumed that it cannot b e defined. Hence, sym b ols 1 a nd 0 in the figur e cannot b e inte rpreted as the numb ers 1 and 0. If they w ere num b ers, 0 < x < y < 1 and 0 < y ′ < x ′ < 1 w ould imply that x, y and x ′ , y ′ w ere also num b ers a nd w e would, for example ha v e x = 0 . 3 and x ′ = 0 . 7. This means w e w ould hav e x < x ′ and it yields x ∩ x < x ′ ∩ x = 0, i.e., x = 0, whic h is a con tradiction, since x 6 = 0. Therefore when w e sp eak of s tandar d valuation of prop o sitions of classical logic, w e mean any v aluation for whic h w e can establish a corresp ondence with r eal num b ers and their ordering, i.e., whose corresp onding mo del can b e totally ordered. F or instance, with tw o-v alued ( { TRUE,F ALSE } ) Bo olean algebra w e can ascrib e the numb er 1 to TRUE and the numb er 0 to F ALSE , and in the probabilistic inte rpretation of classical logic [1] all v a lues from the in terv al [0,1] are real num b ers whic h are totally ordered. When we deal with v alues from our O6 example ab o v e, there is no w a y t o establish a cor- resp ondence of O6 elemen ts with real n um b ers, and w e shall call suc h a v aluation non-s tan d a r d . The p oint here is that the latter v aluation cannot b e implemen t ed in presen t-day binary computers—whose hardw a re usually deals with numerical v alues suc h as v olta ge—and consequen tly also not in the corresp onding artificial in telligence, at the lev el of the underlying logic gates building their hardw are. This means that a statemen t from a lo g ic can b e “t r ue” or “false” in o ne mo del in one wa y and in some o t her mo del in another w a y . When it “holds” (i.e., is “true”) in a standard mo del, sa y the tw o-v alued Bo olean algebra, w e can ascrib e a n um b er to it , say “1 ” . When it “holds” in a non-standar d mo del, meaning, e.g., that it is equal to 1 in Figure 1, we cannot do so and w e cannot ev aluate the mo del for the statemen t directly with binary log ic gates. It is usually tak en for grante d that logic is ab out prop ositions a nd their v alues. F or example, w e are tempted to a ssume that prop o sition p meaning “Material p oint q is at p osition r at time t ” is either true or false . Ho w- ev er, with non- standard v aluations x and y from Figure 1, w e can ascrib e neither a truth v alue nor ev en a probabilit y to p , although “ p or non- p ” is certainly alwa ys v alid meaning p ∪ p ′ = 1. The { 0,1 } Bo olean algebra and the pro babilistic mo del, on the other hand, are the only known classical log- ical mo dels that allo w ascribing { 0,1 } standard (i.e., n umerical) v alues to prop ositions and hence “found[ing] the mathematical theories of logic a nd probabilities” [2]. Classical logic defined by nothing but its axiomatic syn tax is a more general theory , in terms of the p ossible v aluations it may ha v e, than 3 its no n- isomorphic seman tics (e.g., a predicate logical calculus with standard v aluation 1 whic h is nothing but a “predicate Bo olean a lgebra”). The standard-non-standard dic hoto m y can b e ev en b etter understo o d with the example of quantum logic whic h—when taken together with its or- thomo dular lattice mo del—underlies Hilb ert space and therefore could b e implemen ted in to w o uld- b e quan t um computers and eve n tually in to quan- tum artificial in telligence. According to the Ko c hen-Sp ec ke r theorem, a { 0 , 1 } v aluation for quan t um logic do es not exist, 2 but there is an analogy b et w een a Bo olean algebra (distributiv e orthola ttice) and an orthomo dular (ortho)lattice that underlies the Hilb ert space of quan tum mech anics. Ev- ery orthomo dular lattice is a mo del of quantum logic just as eve ry Bo olean algebra (distributiv e ortholattice) is a mo del of classical log ic. Ho w ev er, as with classical logic, there are also non-orthomo dular lattices which are mo d- els of quantum logic but on whic h no Hilb ert space can b e built. Therefore quan tum logic in general (not mo delled b y an y mo del, i.e., without an y se- man tics), or more precisely its syn tax, would b e o f limited use if we w an ted to implemen t it in to quan tum computers. Only one of it s mo dels—an ortho- mo dular latt ice—can serve us for this goa l, and therefore w e call v a lua tions defined on the elemen ts o f the latt er mo del— s tand ar d valuations , as opp osed to non-stand ar d valuations on the former non-orthomo dular mo dels. In this pap er, w e pro v e the no nmonotonicit y of b oth classical and quan- tum logic with resp ect to particular in trinsically differen t, disjoin t classes of mo dels. The result separates t w o kinds of mo dels t ha t hav e so fa r b een assumed to b elong to o v erlapping classes. In particular, g eneral families o f non-distributiv e and non-ort ho mo dular latt ices called w eakly o r thomo dular and weakly distributiv e orthola ttices ( WOML and WDOL) that are mo dels of quantum and classical prop ositional logics, respective ly , fo r whic h w e pre- 1 “A quantificational s chema is v alid if it co mes out true under all interpretations in all nonempt y universes. . . [T]he truth v alue of a co mp o und statement depends on no fea - tures o f the co mpo nent se nt ences and ter ms ex cept their truth v alues a nd their exten- sions. . . [Qua ntificational] schema [co nt aining sent ence letters] will b e v alid, clearly , just in case it resolves to ‘ ⊤ ’ o r to a v alid schema under each s ubs titution o f ‘ ⊤ ’ and ‘ ⊥ ’ fo r its sentence letter s. So [its] tes t is truth-v alue analysis .”[3, p. 131] 2 In 2004 we g av e exhaus tive algorithms for genera tion o f K o chen-Speck er v ector sys- tems with a rbitrary n umber of v ectors in Hilb ert spa ces o f arbitrary dimension. [4 , 5, 6 ] The alg orithms use MMP (McK ay-Megill-Pa viˇ ci´ c) diagr ams for whic h in 3-dim Hilb ert space a dir ect corr esp ondence to Greechie and Has se diagrams can be esta blis hed. Thus, we also hav e a constructive pro of of the no n-existence of a { 0 , 1 } v a lua tion within the lattice itself. 4 viously pro v ed soundness and completeness [7, 8], do include their standard mo dels, ortho mo dular lattices (OML) and Bo olean a lgebras (BA) [distribu- tiv e ortholattices (DOL)]. Here w e prov e that these lattices can b e separated in t he sense that the logics can also b e mo delled by WOML and WDOL from whic h the standar d orthomo dular and Bo olean algebras are ex c lude d . 3 Soundness and completeness of these prop ositional logics are pro v ed. Sp ecifically , w e consider t he pr op er sub classes of t hese lattice families that exclude those la ttices that are orthomo dular (f or the WOML case) and dis- tributiv e (for the WDOL case), i.e., W OML \ OML and WDOL \ BA (where “ \ ” denotes set-theoretical difference). Using them as the basis for a mo d- ification of the standard Lindenb aum algebra t ec hnique, w e presen t a new result show ing that quantum and classical prop ositional logics are resp ec- tiv ely complete for these prop er sub classes, in and of themselv es, as mo dels. In other words, ev en after remo ving ev ery lattice f rom W O ML (WDOL) in whic h the orthomo dular (distributiv e) la w holds, quan tum (classical) prop o- sitional lo g ic is still complete for the remaining lattices. In b ot h classical and quan tum logics, when w e add new conditions to the defining conditions of the la t t ices that mo del the logics, we get new lattices that also mo del these logics but with changed v alua tions f or the prop osi- tions from the logics. This prop erty of standard logics and v aluations of their prop ositions is what w e call valuation-nonmo n otonicity . The more con- ditions w e add, the few er c hoices we hav e for v aluations. This is why w e consider sub classes that exclude la ttices obta ined b y adding new conditions. F or instance, W OML \ OML will pro vide us only with v a lua tions on w eakly orthomo dular lattices that are not o rthomo dular, and by adding the ortho- mo dularity condition to W OML we get O ML, whic h con tains only v a lua tions on ort ho mo dular lattices. Apart from the ort ho mo dularit y condition, there are many mor e (if no t infinitely many) conditions in b etw een W O ML and OML that all provide different v aluations and new pr o p er sub classes, as w e sho w and discuss in Sections 8 and 9 b elo w. W e will study the quan tum log ic case first, since t he results w e obta in for WOMLs will automatically ho ld for WDOLs and simplify our subsequen t presen tatio n of the latter. In Section 2, w e define o rthomo dular and w eakly 3 The names weakly o rthomo dular and weakly distr ibutive or tholattices stem from the fact that in g eneral these lattice families contain o rthomo dular a nd distributive ones, although in the lig ht of the prese nt “disjointness results” the names s eem to be so mewhat inappropria te. Recall also that at the b eg inning orthomo dular lattices were called we akly mo dular lattices. [9] 5 orthomo dular (ortho)lattices, and in Section 3 distributiv e and w eakly dis- tributiv e ones. In Section 4, w e define the classes of prop er w eakly ortho- mo dular and pro p er w eakly distributive ortholattices. In Section 5, w e define quan tum and classical lo gics and prov e their soundness fo r the mo dels de- fined in Section 4. In Sections 6 and 7, we pro v e the completeness of quantum logic for W OML \ OML and WDOL \ BA mo dels resp ectiv ely . In Section 8, w e define v aluat ion-nonmonotonicity , and in Sections 8 and 9, w e discuss the differences b et w een the completeness pro ofs f or WOM L \ OML, WDOL \ BA, W OMLi \ OML, WDOLi \ BA, W O ML \ W OMLi, and WDOL \ WD OLi w e ob- tain in Sections 6-9 and the completeness pro ofs f or W OML and WDOL w e obtained in [7, 8]. And finally , w e discuss and summarize the results w e obtained in this pap er in Section 10. 2 Orthomo dular and W eakly Or t homo dular Lattices Definition 2.1 An ortholattice , O L , is an algebr a hO L 0 , ′ , ∪ , ∩i such that the fol lo w ing c onditions a r e satisfie d for any a, b, c ∈ O L 0 [10] : a ∪ b = b ∪ a (1) ( a ∪ b ) ∪ c = a ∪ ( b ∪ c ) (2) a ′′ = a (3) a ∪ ( b ∪ b ′ ) = b ∪ b ′ (4) a ∪ ( a ∩ b ) = a (5) a ∩ b = ( a ′ ∪ b ′ ) ′ (6) In addition, sinc e a ∪ a ′ = b ∪ b ′ for any a, b ∈ O L 0 , we define the greatest elemen t of the la ttice (1) and the least elemen t of the lattice (0): 1 def = a ∪ a ′ , 0 def = a ∩ a ′ (7) and the ordering r elat io n ( ≤ ) on the lattice: a ≤ b def ⇐ ⇒ a ∩ b = a ⇐ ⇒ a ∪ b = b (8) 6 Connectiv es → 1 ( Sasaki ho ok ), → 2 ( Dishkant implic ation ), → 5 ( r elevanc e implic ation ), → 0 ( classic al implic ation ), ≡ ( quantum e quivale nc e ), a nd ≡ 0 ( classic al e q uivalenc e ) are defined as fo llo ws: Definition 2.2 a → 1 b def = a ′ ∪ ( a ∩ b ) , a → 2 b def = b ′ → 1 a ′ , a → 5 b def = ( a ∩ b ) ∪ ( a ′ ∩ b ) ∪ ( a ′ ∩ b ′ ) , a → 0 b def = a ′ ∪ b . Definition 2.3 4 a ≡ b def = ( a ∩ b ) ∪ ( a ′ ∩ b ′ ) . Definition 2.4 a ≡ 0 b def = ( a → 0 b ) ∩ ( b → 0 a ) . Connectiv es bind f rom w eak est to strongest in the order → 1 ( → 0 ), ≡ ( ≡ 0 ), ∪ , ∩ , and ′ . Definition 2.5 If, i n an ortholattic e, a = ( a ∩ b ) ∪ ( a ∩ b ′ ) , we say that a c ommutes with b , which we write as aC b . Definition 2.6 If, in an o rtholattic e, a ≡ (( a ∩ b ) ∪ ( a ∩ b ′ )) = 1 , we say that a we akly c ommutes with b , and we write this as aC w b . Definition 2.7 The c ommutator of a and b , C ( a, b ) , is defin e d as ( a ∩ b ) ∪ ( a ∩ b ′ ) ∪ ( a ′ ∩ b ) ∪ ( a ′ ∩ b ′ ) . Definition 2.8 (P a vi ˇ ci ´ c and Megill [7]) A n ortholattic e in which the fol low- ing c ond i tion holds: ( a ′ ∩ ( a ∪ b )) ∪ b ′ ∪ ( a ∩ b ) = 1 (9) is c al le d a weak ly ortho mo dular orthola ttice (WOM L) . Using Definition 2.2, w e can also express Eq. ( 9 ) as either o f t he t w o follo wing equations, whic h are equiv alen t in an o r tholattice: ( a → 2 b ) ′ ∪ ( a → 1 b ) = 1 (10) ( a → 1 b ) ′ ∪ ( a → 2 b ) = 1 . (11) 4 In every orthomo dular la ttice a ≡ b = ( a → 1 b ) ∩ ( b → 1 a ), but not in every orthola t- tice. 7 Definition 2.9 An ortholattic e in which either of the fo l lowing c onditions hold: [11] a ≡ b = 1 ⇒ a = b (12) a ∪ ( a ′ ∩ ( a ∪ b )) = a ∪ b (13) is c al le d an o rthomo dular lattice (OML) . The equations of D efinition 2 .1 determine a (prop er) class of lattices, called a n e quational variety , [1 2, p. 3 5 2] tha t we designate OL. Thus the term OL will ha v e tw o meanings, dep ending on con text. When w e sa y a la t t ice is an OL, w e mean that the equations o f Definition 2.1 hold in that la t tice. When w e sa y a lattice is in OL , we mean t ha t it b elongs to the equational v ariet y O L determined b y those equations. While these t w o statemen t s are of course equiv alen t, the distinction will matter when we say suc h things as “the class OL prop erly includes the class OML.” Similar remarks a pply to OML, WOML, and t he other v a rieties in this pap er. W e recall that whereas ev ery OML is a W OML, there are W OMLs that are not OMLs. [7] In particular, the lattice O6 (Fig. 1) is a W OML but is not an OML. ❅ ❅     ❅ ❅ 0 x y ′ y x ′ 1 r r r r r r Figure 1 : Orthola ttice O6, also called b en z e ne ring a nd hexagon . On the one hand, the equations that hold in OML prop erly include tho se that hold in WOM L, since WOML is a strictly more general class of lattices. But there is also a sense in whic h the equations of W OML can b e considered to prop erly include those of OML, via a mapping that Theorem 2.11 b elow describes. Fir st, we need a tec hnical lemma. Lemma 2.10 The fol lo w ing c on d itions hold in al l WOML s: a ≡ a = 1 (14) 8 a ≡ b = 1 ⇒ b ≡ a = 1 (15) a ≡ b = 1 ⇒ a ′ ≡ b ′ = 1 (16) a ≡ b = 1 ⇒ ( a ∪ c ) ≡ ( b ∪ c ) = 1 (17) a ≡ b = 1 ⇒ ( a ∩ c ) ≡ ( b ∩ c ) = 1 (18) a ≡ b = 1 & b ≡ c = 1 ⇒ a ≡ c = 1 (19) ( a ∪ b ) ≡ ( b ∪ a ) = 1 (20) (( a ∪ b ) ∪ c ) ≡ ( a ∪ ( b ∪ c )) = 1 (21) a ′′ ≡ a = 1 (22) ( a ∪ ( b ∪ b ′ )) ≡ ( b ∪ b ′ ) = 1 (23) ( a ∪ ( a ∩ b )) ≡ a = 1 (24) ( a ∩ b ) ≡ ( a ′ ∪ b ′ ) ′ = 1 (25) ( a ∪ ( a ′ ∩ ( a ∪ b ))) ≡ ( a ∪ b ) = 1 (26) a ≡ (( a ∩ b ) ∪ ( a ∩ b ′ )) = a ≡ 0 (( a ∩ b ) ∪ ( a ∩ b ′ )) (27) a = 1 ⇔ a ≡ 1 = 1 (28) a = 1 ⇔ a ≡ 0 1 = 1 (29) In addition, Eqs. (14)–(16) a nd (20)–(29) ho l d in al l ortholattic es. Pr o of. Most of these conditions are prov ed in [7 ], and the o thers are straigh t- forw ard.  Theorem 2.11 The e quational the ory of O ML s c a n b e simulate d by a pr op er subset of the e quational the ory of W OML s. Pr o of. The equational theory of OML consists of equalit y axioms ( a = a , a = b ⇒ b = a , a = b ⇒ a ′ = b ′ , a = b ⇒ a ∪ c = b ∪ c , a = b ⇒ a ∩ c = b ∩ c , and a = b & b = c ⇒ a = c ); the OL axioms, Eqs. (1)–(6); and the O ML la w, Eq. (12). An y theorem of the equational v ariet y of OMLs can b e pr ov ed with a sequenc e of applications of these axioms. W e construct a mapping from these a xioms in to equations that hold in WOMLs as follow s. W e map eac h axiom, whic h is an equation in the fo rm t = s or an inference of the form t 1 = t 2 . . . ⇒ t = s (where t , s , and t 1 , t 2 , . . . are terms), to the equation t ≡ s = 1 or the inference t 1 ≡ t 2 = 1 . . . ⇒ t ≡ s = 1. These mappings hold in any WOML b y Eqs. (14)–(26), resp ectiv ely , o f Lemma 2.1 0. W e then sim ulat e t he OML pro of b y replacing eac h a xiom reference in the pro of with 9 its corresp onding W OML mapping. The result will b e a pro o f t ha t ho lds in the equational v ariety of WOMLs. Suc h a mapp ed pro of will use o nly a prop er subset of the equations that hold in W OML: any equation whose r ig h t-hand side do es not equal 1, such as a = a , will neve r b e used.  Theorem 2.12 L et t 1 , ..., t n , t b e any terms ( n ≥ 0 ). If the infer enc e t 1 = 1 & . . . & t n = 1 ⇒ t = 1 holds in al l OML s, then it holds in any WOML . Pr o of. In any ortholattice, t = 1 iff t ≡ 1 = 1 b y Eq. (28). Therefore, the inference of the theorem can b e restated as follo ws: t 1 ≡ 1 = 1 & . . . & t n ≡ 1 = 1 ⇒ t ≡ 1 = 1. But this is exactly what w e prov e when w e sim ulate the original OML pro of of the inference in W OML, using the metho d in the pro of of Theorem 2.11. Th us by Theorem 2.11 , the inference holds in W OML.  Corollary 2.13 No set of e quations of the form t = 1 that hold in OML , when adde d to the e quations of an ortholattic e, determines the e quational the ory of OML s. Pr o of. Theorem 2.12 show s that all equations of this form hold in a W OML.  Lemma 2.14 In any W OML , aC w b iff C ( a, b ) = 1 . Pr o of. In any OML, aC b implies a ′ C b . Therefore, by Theorem 2.11, aC w b implies a ′ C w b in any W OML. Using Eqs. ( 1 8) and (20) to com bine these tw o conditions, w e obtain ( a ∪ a ′ ) ≡ ((( a ∩ b ) ∪ ( a ∩ b ′ )) ∪ (( a ′ ∩ b ) ∪ ( a ′ ∩ b ′ ))) = 1 i.e., C ( a, b ) ≡ 1 = 1, from whic h we o btain C ( a, b ) = 1 b y Eq. (28 ) . Con vers ely , if C ( a, b ) = 1, then in any OL, 1 = ( a ∩ b ) ∪ ( a ∩ b ′ ) ∪ ( a ′ ∩ b ) ∪ ( a ′ ∩ b ′ ) ≤ ( a ∩ b ) ∪ ( a ∩ b ′ ) ∪ a ′ = ( a ∩ (( a ∩ b ) ∪ ( a ∩ b ′ )) ∪ ( a ′ ∩ (( a ∩ b ) ∪ ( a ∩ b ′ )) ′ ) = a ≡ (( a ∩ b ) ∪ ( a ∩ b ′ )), so aC w b .  Theorem 2.15 (F oulis-Holland theorem, F-H) In any OML, if at le ast two of the thr e e c onditions aC b , aC c , and bC c hold, then the distributive law a ∩ ( b ∪ c ) = ( a ∩ b ) ∪ ( a ∩ c ) holds. Pr o of. See [12, p. 25 ].  10 Theorem 2.16 (W eak F oulis-Holland theorem, wF-H) I n any WOML , if at le ast two of the thr e e c onditions C ( a, b ) = 1 , C ( a, c ) = 1 , and C ( b, c ) = 1 hold, then the w eak distributiv e la w ( a ∩ ( b ∪ c )) ≡ (( a ∩ b ) ∪ ( a ∩ c ) ) = 1 holds. Pr o of. By Lemma 2.14, w e can replace the conditions with aC w b , aC w c , and bC w c . Then the conclusion follows from F-H and Theorem 2.1 1.  As Theorem 2.11 sho ws, if t and s are terms, then the equation t ≡ s = 1 holds in all W OMLs iff the equation t = s holds in a ll O MLs. One migh t naiv ely exp ect, then, that if t = s is the OML law, then t ≡ s = 1 will b e t he W OML law . This is not alw a ys the case: the O ML la w giv en b y Eq. ( 13), when con verted to ( a ∪ ( a ′ ∩ ( a ∪ b )) ≡ ( a ∪ b ) = 1, is not the W OML law ; in fact, it holds in an y OL. How ev er, t here is a v ersion of the OML la w with this prop ert y , a s the follow ing theorem show s. Theorem 2.17 An ortholattic e is an OML iff it satisfies the fol lowing e qua- tion: a ∪ ( b ∩ ( a ′ ∪ b ′ )) = a ∪ b (30) A n ortholattic e is a W OML iff it satisfies the fol lowing e quation: ( a ∪ ( b ∩ ( a ′ ∪ b ′ ))) ≡ ( a ∪ b ) = 1 (31) Pr o of. F o r Eq. (30): It is easy to v erify that Eq. (30) holds in an OML, for example by applying F-H: a ∪ ( b ∩ ( a ′ ∪ b ′ )) = ( a ∪ b ) ∩ ( a ∪ a ′ ∪ b ) = ( a ∪ b ) ∩ 1 = a ∪ b . On the other hand, this equation fails in lattice O6 (Fig. 1), meaning it implies t he orthomo dular la w by Theorem 2 of [12, p. 22]. It is also instructiv e to pr ov e Eq. (13) directly: a ∪ ( a ′ ∩ ( a ∪ b )) = a ∪ (( a ∪ b ) ∩ a ′ ) = a ∪ (( a ∪ b ) ∩ ( a ′ ∪ ( a ′ ∩ b ′ )) = a ∪ (( a ∪ b ) ∩ ( a ′ ∪ ( a ∪ b ) ′ )) = a ∪ ( a ∪ b ) = a ∪ b , where the p en ultimate step follows f rom Eq. (30) with a ∪ b substituted for b , and all ot her steps hold in OL. F or Eq. (3 1): Since a ∪ ( b ∩ ( a ′ ∪ b ′ )) = a ∪ b holds in an y OML by Eq. (30), ( a ′ ∪ ( b ′ ∩ ( a ∪ b ))) ≡ ( a ∪ b ) = 1 holds in W OML b y Theorem 2.11. On the o t her hand, substituting b ′ and a ′ for a and b in Eq. (31), w e hav e 1 = ( b ′ ∪ ( a ′ ∩ ( b ′′ ∪ a ′′ ))) ≡ ( b ′ ∪ a ′ ) = (( b ′ ∪ ( a ′ ∩ ( b ∪ a ))) ∩ ( b ′ ∪ a ′ )) ∪ (( b ∩ ( a ∪ ( b ′ ∩ a ′ ))) ∩ ( b ∩ a )) = ( b ′ ∪ ( a ′ ∩ ( b ∪ a )) ∪ ( b ∩ a )) = ( a ′ ∩ ( a ∪ b )) ∪ b ′ ∪ ( a ∩ b ), whic h is the W OML law Eq. (9) .  Another ve rsion of t he W OML law will b e useful later. 11 Theorem 2.18 An ortholattic e is a WOML iff it satisfies the fol lowing c on- dition: a → 1 b = 1 ⇒ a → 2 b = 1 (32) Pr o of. See Theorem 3.9 of [7].  3 Distribu tiv e and W eakly Distributiv e Ortholattic e s Definition 3.1 (P a vi ˇ ci ´ c and Megill [7]) A n ortholattic e in which the fol low- ing e quation holds: ( a ≡ b ) ∪ ( a ≡ b ′ ) = ( a ∩ b ) ∪ ( a ∩ b ′ ) ∪ ( a ′ ∩ b ) ∪ ( a ′ ∩ b ′ ) = 1 (33) is c al le d a weak ly distributiv e ortholattice , WDOL . A WDOL is thu s an o r t holattice in whic h the condition C ( a, b ) = 1 holds. This condition is kno wn as c ommensur ability . [9, D ef. (2.13) , p. 3 2]. Definition 3.2 An ortholattic e to which the fol lowin g c ondition is a dde d: a ∩ ( b ∪ c ) = ( a ∩ b ) ∪ ( a ∩ c ) (34 ) is c al le d a di stributive ortholattic e (DOL) o r (much mor e o ften) a Bo olean algebra (BA) . Eq. ( 3 4) is called the distributive l a w . W e recall that whereas eve ry BA is a WDOL, there are WDOLs that are not BAs. [7] In particular, the la t t ice O6 (Fig. 1) is a WDO L but is not a BA. The first part of the following theorem will tur n out to b e v ery useful, b ecause it will let us reuse a ll o f the results w e hav e already obtained fo r W OMLs. Theorem 3.3 Every WD O L is a WOML , but n o t every WOML is a WDOL . 12 ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ❩ ❩ ❩ ❩ ✚ ✚ ✚ ✚ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✚ ✚ ✚ ✚ ❩ ❩ ❩ ❩ 0 x y ′ y x ′ 1 r r r r r r ❅ ❅ ❅ ❅         ❅ ❅ ❅ ❅     ❅ ❅ ❅ ❅     ❅ ❅ ❅ ❅ 0 x w z ′ y y ′ z w ′ x ′ 1 r r r r r r r r r r Figure 2: (a) OML M02; (b) Non-WDOL from [13 ], Fig. 3. Pr o of. Since a ′ ∩ b ≤ a ′ ∩ ( a ∪ b ) and ( a ∩ b ′ ) ∪ ( a ′ ∩ b ′ ) ≤ b ′ in any OL, the WDOL la w, Eq. (33 ), giv es us 1 = ( a ∩ b ) ∪ ( a ∩ b ′ ) ∪ ( a ′ ∩ b ) ∪ ( a ′ ∩ b ′ ) ≤ ( a ′ ∩ ( a ∪ b )) ∪ b ′ ∪ ( a ∩ b ), whic h is the WOML law, Eq. (9). On the other ha nd, the mo dular (and therefore W OML) lattice MO2 (Fig. 2a) violates Eq. (33). If w e put x for a and y for b , the equation ev aluates to 0 = 1.  W e are now in a p osition to prov e t w o imp orta n t equiv alents to the WDOL la w. W e call them we ak distributive law s , since they pro vide analogs to the distributiv e la w of Bo olean a lg ebras. Theorem 3.4 An ortholattic e is a WDOL iff it satisfies either of the fol - lowing e q uations: ( a ∩ ( b ∪ c )) ≡ 0 (( a ∩ b ) ∪ ( a ∩ c )) = 1 (35) ( a ∩ ( b ∪ c )) ≡ (( a ∩ b ) ∪ ( a ∩ c )) = 1 (36) Pr o of. First, w e pro v e these law s can b e deriv ed from eac h other in any OL. Assuming Eq. (35) and using the fact that ( a ∩ b ) ∪ ( a ∩ c ) ≤ ( a ∩ ( b ∪ c ), in any OL we hav e 1 = (( a ∩ ( b ∪ c )) → 0 (( a ∩ b ) ∪ ( a ∩ c ))) ∩ ((( a ∩ b ) ∪ ( a ∩ c )) → 0 ( a ∩ ( b ∪ c )) ) = (( a ∩ ( b ∪ c )) → 0 (( a ∩ b ) ∪ ( a ∩ c )). Putting b ′ for c , 1 = (( a ∩ ( b ∪ b ′ )) → 0 (( a ∩ b ) ∪ ( a ∩ b ′ )) = ( a → 0 (( a ∩ b ) ∪ ( a ∩ b ′ )) = ( a ′ ∪ (( a ∩ b ) ∪ ( a ∩ b ′ )) ≤ ( b ′ ∩ ( b ∪ a )) ∪ a ′ ∪ ( b ∩ a ), whic h is the WOML la w. This lets us use our previous W OML results. Starting from the last equality in the first sen tence of the previous paragra ph, in any OL we also hav e 1 = (( a ∩ ( b ∪ c )) → 0 (( a ∩ b ) ∪ ( a ∩ c )) = ( a ∩ ( b ∪ c )) → 1 13 (( a ∩ b ) ∪ ( a ∩ c )) = ( ( a ∩ ( b ∪ c )) → 1 (( a ∩ b ) ∪ ( a ∩ c ))) ∩ ((( a ∩ b ) ∪ ( a ∩ c )) → 1 ( a ∩ ( b ∪ c ))). Therefore, using the fo o tnote to Definition 2.3 and Theorem 2.12, it follo ws that in any W OML, a nd therefore (b y the previous paragra ph) in any OL, Eq. (35) implies 1 = ( a ∩ ( b ∪ c )) ≡ (( a ∩ b ) ∪ ( a ∩ c )). Con v ersely , Eq. (35) f o llo ws immediately from Eq. (36) in an y OL. Th us these t wo equations are equiv alen t la ws when added to the equations for OL. Next, we pro v e that Eq. (36) is equiv alen t to the WDOL law in the presence of the equations for OL. Since C ( a, b ) = 1 for an y a, b in a WDOL, Eq. (36) follow s immediately from wF-H (Theorem 2.16) , whic h holds in ev ery W OML and thus , by Theorem 3.3, in ev ery WDO L . Con v ersely , in OML, w e can prov e C ( a, b ) = 1 if w e use instances of the distributiv e la w as h yp otheses. Using Theorem 2.11, suc h a pro of can b e con v erted to a W OML pro of, replacing the instances of the distributiv e law with instances of Eq. (3 6). This will yield a pro of of C ( a, b ) ≡ 1 = 1, whic h in any OL implies C ( a, b ) = 1 by Eq. (28) . This prov es that Eq. (36) implies the WDOL la w, Eq. (33).  Theorem 3.5 An ortholattic e is a WDOL iff it satisfies either of the fol - lowing e q uations: a ≡ (( a ∩ b ) ∪ ( a ∩ b ′ )) = 1 (37) a ≡ 0 (( a ∩ b ) ∪ ( a ∩ b ′ )) = 1 . (38) Pr o of. In any OL, a ≡ (( a ∩ b ) ∪ ( a ∩ b ′ )) = ( a ∩ (( a ∩ b ) ∪ ( a ∩ b ′ ))) ∪ ( a ′ ∩ (( a ′ ∪ b ′ ) ∩ ( a ′ ∪ b ))) = (( a ∩ b ) ∪ ( a ∩ b ′ )) ∪ a ′ = ( a → 0 b ) ′ ∪ ( a → 1 b ). Th us Eq. (37) implies 1 = ( a → 0 b ) ′ ∪ ( a → 1 b ) ≤ ( a → 2 b ) ′ ∪ ( a → 1 b ), whic h is the WOML law in the f orm of Eq. (10). By Lemma 2.14, in any W OML Eq. (37) implies C ( a, b ) = 1, whic h is the WD OL law. F or the conv erse, Eq. (37) holds in an WDO L b y Lemma 2.14. Eq. (38) is equiv alen t to Eq. (37 ) in an y OL by Eq. (27).  W e men t ion that Eq. (37) is the definition of aC w b . Theorem 3.6 An ortholattic e is a WDOL iff it satisfies the fol lowing c on- dition: a ≡ 0 b = 1 ⇒ ( a ∪ c ) ≡ 0 ( b ∪ c ) = 1 (39) 14 Pr o of. First, w e sho w that Eq. (39) implies t he WOML law. Putting d for a and d ∩ e for b , the h ypot hesis b ecomes, in an OL, 1 = d ≡ 0 ( d ∩ e ) = ( d ′ ∪ ( d ∩ e )) ∩ (( d ′ ∪ e ′ ) ∪ d ) = ( d ′ ∪ ( d ∩ e )) ∩ 1 = d → 1 e . Also putting e for c , the conclusion b ecomes, in an O L , 1 = ( d ∪ e ) ≡ 0 (( d ∩ e ) ∪ e ) = ( d ∪ e ) ≡ 0 e = (( d ′ ∩ e ′ ) ∪ e ) ∩ ( e ′ ∪ ( d ∪ e )) = (( d ′ ∩ e ′ ) ∪ e ) ∩ 1 = d → 2 e . The condition d → 1 e = 1 ⇒ d → 2 e = 1 is the W OML law b y Eq. (32). Ha ving our previous WOML results no w a v ailable to us, w e next sho w that Eq. (39) implies the WDOL law. W e put d ′ ∩ ( d ∪ e ′ ) for a , e ′ ∩ ( e ∪ d ′ ) for b , a nd d ′ for c . T o satisfy the h yp othesis, we m ust sho w that in an y WOML, ( d ′ ∩ ( d ∪ e ′ )) ≡ 0 ( e ′ ∩ ( e ∪ d ′ )) = 1 , i.e., tha t (( d ∪ ( d ′ ∩ e )) ∪ ( e ′ ∩ ( e ∪ d ′ ))) ∩ (( e ∪ ( e ′ ∩ d )) ∪ ( d ′ ∩ ( d ∪ e ′ ))) = 1. F or the first conjunct, w e apply wF-H to ( d ∪ ( d ′ ∩ e )) ∪ ( e ′ ∩ ( e ∪ d ′ )) = (( d ∪ ( d ′ ∩ e )) ∪ ( e ′ ∩ ( e ∪ d ′ ))) ≡ 1 = 1 to obtain )( d ∪ ( d ′ ∩ e ) ∪ e ′ ) ∩ ( d ∪ ( d ′ ∩ e ) ∪ e ∪ d ′ )) ≡ 1 = 1, whic h reduces to (1 ∩ 1) ≡ 1 = 1. The other conjunct is satisfied similarly , b y symmetry . The conclusion b ecomes (( d ′ ∩ ( d ∪ e ′ )) ∪ d ′ ) ≡ 0 (( e ′ ∩ ( e ∪ d ′ )) ∪ d ′ ) = d ′ ≡ 0 (( e ′ ∩ ( e ∪ d ′ )) ∪ d ′ ) = 1. Expanding the definition of ≡ 0 and discarding the left - hand conjunct, we ha v e (( e ∪ ( e ′ ∩ d )) ∩ d ) ∪ d ′ = 1. Using wF- H, this b ecomes 1 = (( e ∩ d ) ∪ (( e ′ ∩ d ) ∩ d )) ∪ d ′ = (( e ∩ d ) ∪ ( e ′ ∩ d )) ∪ d ′ = ((( e ∩ d ) ∪ ( e ′ ∩ d )) ∪ d ′ ) ≡ 1. Conjoining b oth sides of the ≡ with d using Eq. (18 ), w e hav e (((( e ∩ d ) ∪ ( e ′ ∩ d )) ∪ d ′ ) ∩ d ) ≡ (1 ∩ d ) = 1. Applying wF-H t wice, we obta in 1 = (((( e ∩ d ) ∩ d ) ∪ (( e ′ ∩ d ) ∩ d )) ∪ ( d ′ ∩ d )) ≡ (1 ∩ d ) = ((( e ∩ d ) ∪ ( e ′ ∩ d )) ∪ 0) ≡ d = (( e ∩ d ) ∪ ( e ′ ∩ d )) ≡ d , whic h is the WDOL la w in the f o rm of Eq. (3 7). Con v ersely , to sho w that Eq. (39) holds in any WD O L, we apply Eq. (40) b elo w (whic h do es not dep end on the presen t theorem) to the h yp othesis a nd conclusion, conv erting it to Eq. (17).  An essen tial c haracteristic of the WDOL la w and its equiv alents is t ha t they m ust fail in the mo dular (and therefore OML and W OML) lattice MO2. Ho w eve r, suc h a fa ilure is not sufficien t to ensure that we ha v e a WDOL law equiv alen t. Theorem 3.7 The fol lo w ing c on d ition h olds i n al l WDOL s: a ≡ 0 b = 1 ⇔ a ≡ b = 1 (40) It also fails in mo dular lattic e MO2 . However, when adde d to the e q uations for OL , it do es not determine the e quations of WDOL . Pr o of. T o v erify tha t this condition holds in any WD OL, w e first conv ert the h ypo thesis to the OL-equiv alen t hypothesis ( a ≡ 0 b ) ≡ 1 = 1 using Eq. (29). 15 By using the WDO L la w C ( a, b ) = 1 to satisfy the h yp otheses of an y uses of wF- H, it is then easy to prov e that this condition ho lds in any WDOL. In particular, the rev erse implicatio n holds in an y OL. The failure of Eq. (40) in MO2 is verifie d by putting x for a and y for b ; t hen the left- hand side holds but the right-hand side b ecomes 0 = 1. On the other hand, it do es not imply the WDOL law nor ev en the W OML la w: it passes in the non-WOML lattice of Figure 2b.  On the one hand, the equations that hold in BA prop erly include those that hold in WDOL, since WDOL is a strictly more general class of lattices. But there is also a sense in whic h the equations of WDOL can b e considered to prop erly include those o f BA, via a mapping that Theorem 3.8 b elow describes. Theorem 3.8 The e quational the o ry of BA s c an b e simulate d by a pr op er subset of the e quational the ory of WDOL s. Pr o of. The equational theory of BA consists of equalit y axioms (see the pro of of Theorem 2.11); the OL axioms, Eqs. (1 )–(6); and the distributiv e la w, Eq. (34). Any theorem of t he equational v ariety of BAs can b e prov ed with a sequenc e of applications of these axioms. W e construct a mapping from these axioms in to equations that hold in WDOL s as follo ws. W e map eac h axiom, whic h is an equation in the fo rm t = s or an inference of the form t 1 = t 2 . . . ⇒ t = s (where t , s , and t 1 , t 2 , . . . are terms), to the equation t ≡ 0 s = 1 or the inference t 1 ≡ 0 t 2 = 1 . . . ⇒ t ≡ 0 s = 1. These mappings hold in an y WDOL by Eqs. (14)– ( 25) a nd (35), resp ectiv ely , after con v erting ≡ t o ≡ 0 with Eq. (40). W e then sim ulate the BA pro of by replacing eac h axiom reference in the pro of with its corresp o nding WDOL mapping. The result will b e a pro of that holds in the equational v a riet y of WDOLs. Suc h a mapp ed pro of will use o nly a prop er subset of the equations that hold in WDOL: any equation whose right-hand side do es not equal 1 , suc h as a = a , will neve r b e used.  Theorem 3.9 L et t 1 , ..., t n , t b e any terms ( n ≥ 0 ). If the infer enc e t 1 = 1 & . . . & t n = 1 ⇒ t = 1 holds in al l BA s, then it holds in an y WDOL . Pr o of. In a ny ortholattice, t = 1 iff t ≡ 0 1 = 1 by Eq. ( 2 9). Therefore, the inference of the theorem can b e restated as fo llo ws: t 1 ≡ 0 1 = 1 & . . . & t n ≡ 0 1 = 1 ⇒ t ≡ 0 1 = 1. But this is exactly what we prov e when we sim ulat e 16 the original BA pro of of t he inference in WDOL, using the metho d in the pro of of Theorem 3.8. Th us by Theorem 3.8, the inference holds in WDO L .  Corollary 3.10 No set of e quations of the form t = 1 that hold in BA , when adde d to the e quations of an o rtholattic e, determin e s the e quational the ory of BA s. Pr o of. Theorem 3.9 sho ws that all equations of this form hold in a WDOL.  4 The Classe s of Prop er W eakly Orthomo dular and Pro p er W eakly Distribu tiv e Ortho lattices One of the main aims of our pap er is to prov e t ha t b oth quantum and clas- sical logics are sound and complete with resp ect to at least a class of all w eakly orthomo dular lattices (WOMLs ) in whic h o rthomo dularity fails for ev ery lattice and a class of all w eakly distributive lattices (WDOLs) in whic h distributivit y fails fo r ev ery la ttice, resp ectiv ely . T o prov e the soundness and completeness of quantum log ic we shall con- sider a new class of lattices that b elong to the class W OML but not to the class OML. W e will denote the resulting class WOML-OML. In ot her w ords, WOML-OML denotes the set-theoretical difference WOML \ OML. A member of the class WOML-OML is a lattice, sp ecifically a mem b er of the class WOML, and w e will call suc h a la ttice a p r op er WOML. Thus a prop er W OML is one that satisfies the W OML equations but violates the OML equations. La t t ice O6 is an example o f a prop er W OML. Lat t ice MO2 is an example of a W OML tha t is not a pro p er W OML, i.e., that do es not b elong to the class WOM L-OML, since it also b elongs to the class OML. Notice t ha t W OML-OML is not an equational v ariety like W OML, b e- cause w e cannot turn W OML into W OML-OML b y adding new equational conditions to those defining W OML. If w e try to add the orthomo dularit y condition (1 2) [14, 11] to WOM L-OML, w e will get the empt y set. In Section 6 we shall sho w that quantum logics is complete for W OML- OML: ev ery wff whose v aluation equals 1 for all members of WOM L-OML 17 is a pro v able statemen t in quan t um logic. This is not necessarily ob vious a priori : quan tum lo g ic ( QL ) is not necessarily complete for an arbitrary collection of W OMLs. F or example, it is not complete f or the subset o f W OML-OML consisting of the singleton set { O6 } , since O6 is a mo del for classical log ic. The significance of this result can b e explained as follows. Since QL is already complete for OML mo dels, it might b e argued that completeness for the more general W OML mo dels ([7]) has its origin in the OML mem b ers of the equational v a r iety WOM L, rather than b eing an intrinsic prop ert y of the non- OML members. W e show that this is not the case by completely remo ving a ll OMLs from t he picture. In order fo r the completeness pro of to go through, w e will hav e to con- struct a sp ecial Linden ba um algebra that b elongs to W OML-OML. This requires a mo dification to the standard Linden baum algebra (whic h, in the standard pro o f, “wan ts” to b e an OML). The tech nique that w e use, in v olv- ing cutting dow n the equiv alence classe s for the Linden baum algebra to f orce it to b elong to WOML-OML, migh t b e useful for other completeness pro ofs that are not amenable to the standard Linden baum-algebra appro ac h. F ollowing an analogous blueprin t, in Section 7 w e will also show tha t classical logic is complete for the class of mo dels WDOL-BA, defined as the set-theoretical difference WDOL \ BA (where WD OL and BA here de- note equational v arieties), whic h again b y definition has nothing to do with Bo olean a lgebras. In fact, a simpler result is p ossible: Sc hech ter [15, p. 27 2] has pro v ed that classical logic ( C L ) is complete for the single WDOL lattice O6. Sc hec hter’s result can b e strengthened to show that classical logic is complete for an y subset o f WDO L . This is an immediate consequence o f the fact that classical lo gic is maximal, i.e., no extension o f it can b e consisten t. So if classical logic is sound for a mo del, it is automatically complete for that mo del. 5 Logics and Th e ir Sound ness for Our Mo dels Logic ( L ) is a language consisting of prop ositions and a set of conditions and rules imp osed on them called axioms and rules of inference. The prop ositions we use are w ell-formed form ulae (wffs), defined as fol- 18 lo ws. W e denote elemen tary , or primitiv e, prop o sitions b y p 0 , p 1 , p 2 , ... , and ha v e t he following primitiv e connectiv es: ¬ (negation) and ∨ (disjunction). The set of wffs is defined recursiv ely as follows: p j is a wff for j = 0 , 1 , 2 , ... ¬ A is a wff if A is a wff. A ∨ B is a wff if A and B are wffs. W e in tro duce conjunction with the follow ing definition: Definition 5.1 A ∧ B def = ¬ ( ¬ A ∨ ¬ B ) . The op erations of implication are the following ones (classical, Sasaki, and Kalmbac h) [16]: Definition 5.2 A → 0 B def = ¬ A ∨ B . Definition 5.3 A → 1 B def = ¬ A ∨ ( A ∧ B ) . Definition 5.4 A → 3 B def = ( ¬ A ∧ B ) ∨ ( ¬ A ∧ ¬ B ) ∨ ( A ∧ ( ¬ A ∨ B )) . W e also define the e quivalenc e op erations as follows : Definition 5.5 A ≡ B def = ( A ∧ B ) ∨ ( ¬ A ∧ ¬ B ) . Definition 5.6 A ≡ 0 B def = ( A → 0 B ) ∧ ( B → 0 A ) . Connectiv es bind from w eak est t o strongest in the order → , ≡ , ∨ , ∧ , ¬ . Let F ◦ b e the set of all prop ositions, i.e., of all wffs. Of the ab ov e connectiv es, ∨ and ¬ are primitive ones. Wffs con taining ∨ and ¬ within logic L are used to build an algebra F = hF ◦ , ¬ , ∨i . In L , a set of axioms and rules of inference are imp osed on F . F ro m a set of axioms b y means of rules o f inf erence, w e get other expressions which we call theorems. Axioms themselv es are also theorems. A sp ecial sym b ol ⊢ is used to denote the set of theorems. Hence A ∈ ⊢ iff A is a theorem. The statemen t A ∈ ⊢ is usually written as ⊢ A . W e read t his: “ A is prov able” since if A is a theorem, then there is a pro of for it . W e presen t the axiom systems of our prop ositional logics in sc hemata form (so that we disp ense with the rule of substitution). 19 5.1 Qu ant um Logic and Its Soundn ess for W OM L -OML Mo dels W e presen t Kalmbac h’s quan tum log ic b ecause it is the sys tem that has b een in v estigated in the greatest detail in her b o ok [1 2 ] and elsewh ere [17, 13]. Quan tum logic ( QL ) is defined as a languag e consisting of pro p ositions and connectiv es (op erations) as in tro duced ab o v e, and the following axioms and a rule of inference. W e will use ⊢ QL to denote prov a bilit y from the axioms and rule of QL and omit the subscript when it is clear from conte xt (suc h as in the list of axioms that f ollo w). Axioms A1 ⊢ A ≡ A (41) A2 ⊢ A ≡ B → 0 ( B ≡ C → 0 A ≡ C ) (42) A3 ⊢ A ≡ B → 0 ¬ A ≡ ¬ B (43) A4 ⊢ A ≡ B → 0 A ∧ C ≡ B ∧ C (44) A5 ⊢ A ∧ B ≡ B ∧ A (45) A6 ⊢ A ∧ ( B ∧ C ) ≡ ( A ∧ B ) ∧ C (46) A7 ⊢ A ∧ ( A ∨ B ) ≡ A (47) A8 ⊢ ¬ A ∧ A ≡ ( ¬ A ∧ A ) ∧ B (48 ) A9 ⊢ A ≡ ¬¬ A (49) A10 ⊢ ¬ ( A ∨ B ) ≡ ¬ A ∧ ¬ B (50) A11 ⊢ A ∨ ( ¬ A ∧ ( A ∨ B )) ≡ A ∨ B (51) A12 ⊢ ( A ≡ B ) ≡ ( B ≡ A ) (52) A13 ⊢ A ≡ B → 0 ( A → 0 B ) (53) A14 ⊢ ( A → 0 B ) → 3 ( A → 3 ( A → 3 B )) (54) A15 ⊢ ( A → 3 B ) → 0 ( A → 0 B ) (5 5 ) Rule of Inference ( Mo dus Ponens ) R1 ⊢ A & ⊢ A → 3 B ⇒ ⊢ B (56) In Kalmbac h’s presen tation, the connectiv es ∨ , ∧ , and ¬ are primitive . In the base set o f a n y mo del (suc h as a n OML or W OML mo del) that b elongs to OL, ∩ can b e defined in terms of ∪ and ′ , as justified by DeMorgan’s la w, and th us the corr esp o nding ∧ can b e defined in terms of ∨ and ¬ [using 20 Eq. (6)]. W e shall do this for simplicit y . Regardless of whether w e consider ∧ primitiv e or defined, w e can drop axioms A1, A11, and A15 b ecause it has b een prov ed that they are redundan t, i.e., can b e deriv ed from the other axioms. [13] Definition 5.7 F or Γ ⊆ F ◦ we say A is deri v able fr om Γ and write Γ ⊢ QL A or just Γ ⊢ A if ther e is a finite se quenc e of formulae, the last of which is A , an d e ach of w hich is either one of the ax i o ms of QL or is a memb er of Γ or is obtaine d fr om its pr e cursors wi th the help of a rule of in f e r enc e of the lo gic. T o prov e soundness means to pro v e that all axioms as w ell as the rules of inference (and therefore all theorems) of QL hold in its mo dels. Definition 5.8 We c al l M = h L, h i a m o del if L is an algebr a and h : F ◦ − → L , c al le d a valuation, is a morphism o f formulae F ◦ into L , pr eserv- ing the op er ations ¬ , ∨ while turning them into ′ , ∪ . Whenev er the base set L of a mo del b elongs to W OML-OML, w e say (informally) that the mo del b elongs to W OML-OML. In particular, if w e sa y “for all mo dels in W OML-OML” o r “for all prop er WOML mo dels,” w e mean f or a ll ba se sets in W OML-OML and for all v a luations on each base set. The term “mo del” ma y refer either to a sp ecific pair h L, h i or to all p ossible suc h pa ir s with the base set L , dep ending on con text. Definition 5.9 We c al l a formula A ∈ F ◦ valid in the mo del M , an d w rite  M A , if h ( A ) = 1 for a l l valuations h on the mo del, i.e., for al l h ass o ciate d with the b ase s e t L of the mo del. We c al l a formula A ∈ F ◦ a c onse quenc e of Γ ⊆ F ◦ in the mo del M an d write Γ  M A if h ( X ) = 1 for al l X in Γ implies h ( A ) = 1 , for al l valuations h . F or brevit y , whenev er w e do not mak e it explicit, the not a tions  M A and Γ  M A will alw a ys b e implicitly quantified o v er all mo dels of the appropria t e t yp e, in this section for all prop er W O ML mo dels M . Similarly , when w e sa y “v alid” without qualification, w e will mean v a lid in all mo dels of that t yp e. The follow ing theorem sho ws that if A is a theorem of QL , then A is v alid in any prop er WOML mo del. 21 In [7, 8] we prov ed the soundness for W OML and OML. W e no w pro v e the soundness of quantum log ic b y means of W OML-OML, i.e., that if A is a theorem in QL , then A is v alid in an y prop er W OML mo del, i.e., in an y W OML-OML mo del. Theorem 5.10 [Soundness] Γ ⊢ A ⇒ Γ  M A Pr o of. By Theorem 29 of [18], an y WOML is a mo del for QL . Therefore, an y prop er W OML is also a mo del.  5.2 Classical Logic and Its Soundness for WDOL-BA Mo dels W e mak e use of the PM classical logical system C L (Whitehead and Russell’s Principia Mathematic a axiomatization in Hilb ert and Ac kermann’s presen- tation [19] but in sc hemata form so that w e disp ense with their rule of sub- stitution). In this system, the connectiv es ∨ and ¬ are primitiv e, and the → 0 connectiv e sho wn in the axioms is implicitly understo o d to b e expanded according to its definition. W e will use ⊢ C L to denote prov abilit y from the axioms and rule of C L , omitting the subscript when it is clear from con text. Axioms A1 ⊢ A ∨ A → 0 A (57) A2 ⊢ A → 0 A ∨ B (58) A3 ⊢ A ∨ B → 0 B ∨ A (59) A4 ⊢ ( A → 0 B ) → 0 ( C ∨ A → 0 C ∨ B ) (60) Rule of Inference ( Mo dus Ponens ) R1 ⊢ A & A → 0 B ⇒ ⊢ B (61) W e assume that the only legitimate w a y of inf erring theorems in C L is by means of these a xioms and the Mo dus Ponens rule. W e mak e no assumption ab out v aluations of the primitiv e pro p ositions from whic h wffs are built, but instead a r e inte rested in wffs that are v alid in the underlying mo dels. Soundness and completenes s will sho w that tho se theorems t ha t can b e inferred from the axioms and the rule of inference are exactly tho se t ha t are v alid. 22 W e define deriv ability in C L , Γ ⊢ C L A or just Γ ⊢ A , in the same wa y as w e do for system QL . The mo dels and v alidity of form ulae in a mo del are also defined as fo r QL ab ov e. The follo wing theorem sho ws that if A is a theorem of C L , then A is v alid in any prop er WD O L mo del. In [7, 8] w e prov ed the soundness f o r WDOL and BA. W e no w pro v e the soundness of classical logic b y means o f WDOL - BA, i.e., that if A is a theorem in C L , then A is v alid in any prop er WDOL mo del, i.e., in any WDOL-BA mo del. Theorem 5.11 [Soundness] Γ ⊢ A ⇒ Γ  M A Pr o of. By Theorem 30 of [1 8 ], any WDOL is a mo del fo r C L . Therefore, an y prop er WDOL is also a mo del.  6 The Comple t eness of Quan tum Lo gic for W OML-OML Mo de l s Our main task in proving the soundness of QL in the previous section w a s to sho w that all axioms as w ell as the rules of inference (a nd therefore all theo- rems) from QL hold in W OML-OML. The task of proving the completeness of QL is the opp osite one: w e hav e t o imp ose the structure of W OML-OML on the set F ◦ of formulae of QL . W e start with a relation of congruence, i.e., a relation of equiv alence compatible with the op erations in QL . W e mak e use of an equiv alence re- lation to establish a corresp ondence b et wee n form ulae of QL and fo rm ulae of WOML-OML. The resulting equiv alence classes stand for elemen ts o f a prop er W OML (i.e., a mem b er of WOML-OML) and enable the complete- ness pro of of QL b y means of W OML-OML. Our definition of congruence inv olv es a sp ecial set of v a luations on la ttice O6 (show n in Figure 1) called O 6 and defined a s follows. Definition 6.1 L etting O6 r epr esent the lattic e fr om Figur e 1, we d e fine O 6 as the set of al l mappin gs o : F ◦ − → O6 such that for A, B ∈ F ◦ , o ( ¬ A ) = o ( A ) ′ , and o ( A ∨ B ) = o ( A ) ∪ o ( B ) . The purp ose of O 6 is to let us refine the equiv alence classes used fo r the completeness pro of, so that the Linden baum a lg ebra will b e a prop er WOML, 23 i.e., one that is not orthomo dular. This is accomplished by conjoining the term ( ∀ o ∈ O 6)[( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( A ) = o ( B )] to the equiv alence re- lation definition, meaning that for equiv alence we require also that (whenev er the v aluations o of the wffs in Γ are all 1) the v a luations of wffs A and B map to the same p oint in the lattice O6. Th us wffs A ∨ B and A ∨ ( ¬ A ∧ ( A ∨ B )) b ecome mem b ers of t w o separate equiv alence classes, what b y Theorem 6.7 b elo w, amounts to non-ortho mo dularity of WOM L. Without t he conjoined term, these t w o wffs would b elong to the same equiv alence class. The p oint of doing this is to provide a completenes s pro of that is no t dep enden t in a n y w a y o n the orthomo dular law and to sho w that completeness do es not r equire that any of t he underlying mo dels b e OMLs. Theorem 6.2 The r el a tion of equiv alence ≈ Γ , QL or just ≈ , define d as A ≈ B (62) def = Γ ⊢ A ≡ B & ( ∀ o ∈ O 6)[( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( A ) = o ( B )] , is a r elation of c ongruenc e in the algebr a F , wher e Γ ⊆ F ◦ Pr o of. Let us first pro v e that ≈ is an equiv alence relation. A ≈ A follo ws from A1 [Eq. (41)] of system QL and the identit y law of equalit y . If Γ ⊢ A ≡ B , we can detac h the left-hand side o f A12 to conclude Γ ⊢ B ≡ A , through the use of A13 and rep eated uses of A14 and R1. F rom this and comm utativit y of equalit y , we conclude A ≈ B ⇒ B ≈ A . (F or brevity w e will not usually men tion further uses of A12, A13, A14, and R1 in what follo ws.) The pro of of tra nsitivity runs as follo ws. A ≈ B & B ≈ C (63) ⇒ Γ ⊢ A ≡ B & Γ ⊢ B ≡ C & ( ∀ o ∈ O 6 ) [( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( A ) = o ( B )] & ( ∀ o ∈ O 6 ) [( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( B ) = o ( C )] ⇒ Γ ⊢ A ≡ C & ( ∀ o ∈ O 6 ) [( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( A ) = o ( B ) & o ( B ) = o ( C )] . In the last line ab o v e, Γ ⊢ A ≡ C follo ws from A2, a nd the la st metacon- junction reduces to o ( A ) = o ( C ) b y transitivity of equalit y . Hence the conclusion A ≈ C b y definition. 24 In order to b e a relation o f congruence, the relation of equiv alence mus t b e compatible with the op erations ¬ and ∨ . These pro ofs run a s follows. A ≈ B (64) ⇒ Γ ⊢ A ≡ B & ( ∀ o ∈ O 6 )[( ∀ X ∈ Γ)( o ( X ) = 1 ) ⇒ o ( A ) = o ( B )] ⇒ Γ ⊢ ¬ A ≡ ¬ B & ( ∀ o ∈ O 6 )[( ∀ X ∈ Γ)( o ( X ) = 1 ) ⇒ o ( A ) ′ = o ( B ) ′ ] ⇒ Γ ⊢ ¬ A ≡ ¬ B & ( ∀ o ∈ O 6 )[( ∀ X ∈ Γ)( o ( X ) = 1 ) ⇒ o ( ¬ A ) = o ( ¬ B )] ⇒ ¬ A ≈ ¬ B A ≈ B (65) ⇒ Γ ⊢ A ≡ B & ( ∀ o ∈ O 6 )[( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( A ) = o ( B )] ⇒ Γ ⊢ ( A ∨ C ) ≡ ( B ∨ C ) & ( ∀ o ∈ O 6 )[( ∀ X ∈ Γ)( o ( X ) = 1) ⇒ o ( A ) ∪ o ( C ) = o ( B ) ∪ o ( C )] ⇒ ( A ∨ C ) ≈ ( B ∨ C ) In the second step of Eq. (64), w e used A3. In the second step of Eq. (65 ), w e used A4 and A10. F or the quan tified part of these expressions, w e applied the definition of O 6.  Definition 6.3 The e q uivalenc e c la ss fo r wff A under the r elation of e quiv- alenc e ≈ is defi n e d as | A | = { B ∈ F ◦ : A ≈ B } , and we d enote F ◦ / ≈ = {| A | : A ∈ F ◦ } . The e quivalenc e class e s define the natur al mo rp hism f : F ◦ − → F ◦ / ≈ , w h ich giv e s f ( A ) = def | A | . We write a = f ( A ) , b = f ( B ) , etc. Lemma 6.4 The r el a tion a = b on F ◦ / ≈ i s given by: | A | = | B | ⇔ A ≈ B (66) Lemma 6.5 The Lindenb aum alge b r a A = hF ◦ / ≈ , ¬ / ≈ , ∨ / ≈i is a W OML , i.e., Eqs. (1)–(6) and Eq. (9) hold for ¬ / ≈ and ∨ / ≈ as ′ and ∪ r esp e ctively [where—for simplicit y—w e use the same sym b ols ( ′ and ∪ ) as fo r O6, since there are no am biguous expressions in whic h the origin of the op eratio ns w ould not b e clear f rom the con text] . 25 Pr o of. F or the Γ ⊢ A ≡ B part of the A ≈ B definition, the pro ofs of the ortholattice conditio ns, Eqs. (1)–(6), f ollo w from A5, A6, A9, the dual of A8, the dua l of A7, and D eMorga n’s law s resp ectiv ely . (The duals follow from DeMorgan’s law s, deriv ed fro m A10, A9, and A3.) A11 giv es us an analog of the OML la w for the Γ ⊢ A ≡ B par t , and the W OML la w Eq. ( 9) follow s from the OML la w in an ortholattice. F or the quan tified part of the A ≈ B definition, lattice O6 is a (prop er) W OML.  Lemma 6.6 In the Lindenb aum alge b r a A , i f f ( X ) = 1 for a l l X in Γ implies f ( A ) = 1 , then Γ ⊢ A . Pr o of. Le t us assume that f ( X ) = 1 fo r all X in Γ implies f ( A ) = 1 i.e., | A | = 1 = | A | ∪ | A | ′ = | A ∨ ¬ A | , where the first equalit y is from Definition 6 .3, the second equalit y follows from Eq. ( 7 ) (the definition of 1 in an ortholattice), and the third from the fa ct that ≈ is a congruence. Thus A ≈ ( A ∨ ¬ A ), whic h b y definition means Γ ⊢ A ≡ ( A ∨ ¬ A ) & ( ∀ o ∈ O 6)[( ∀ X ∈ Γ)( o ( X ) = 1 ) ⇒ o ( A ) = o (( A ∨ ¬ A ))]. This implies, in par t icular (by dro pping the second conjunct), Γ ⊢ A ≡ ( A ∨ ¬ A ). No w in any ortho lattice, a ≡ ( a ∪ a ′ ) = a holds. By mapping the steps in the pro of of this ortholattice iden tit y to steps in a pro of in the logic, w e can pro v e ⊢ ( A ≡ ( A ∨ ¬ A )) ≡ A from QL axioms A1–A15. (W e call this a “pro of b y ana logy ,” whic h is closely related to the metho d of Theorem 2.11. A direct pro of of ⊢ ( A ≡ ( A ∨ ¬ A )) ≡ A is also not difficult.) Detac hing the left-ha nd side (using A12 , A13, A14, and R1), w e conclude Γ ⊢ A .  Theorem 6.7 The orthomo dular law d o es not h o l d in A . Pr o of. This is Theorem 3.27 fro m [7], and the pro of pro vided t here runs as follo ws. W e assume F ◦ con tains at least t w o eleme n tary (primitiv e) prop o- sitions p 0 , p 1 , . . . . W e pick a v aluation o that maps t w o o f them, A and B , to distinct no des o ( A ) and o ( B ) o f O6 that are neither 0 nor 1 suc h that o ( A ) ≤ o ( B ) [i.e., o ( A ) and o ( B ) are on the same side of hexagon O6 in Figure 1]. F rom the structure of O6, w e obtain o ( A ) ∪ o ( B ) = o ( B ) and o ( A ) ∪ ( o ( A ) ′ ∩ ( o ( A ) ∪ o ( B ))) = o ( A ) ∪ ( o ( A ) ′ ∩ o ( B )) = o ( A ) ∪ 0 = o ( A ). Therefore o ( A ) ∪ o ( B ) 6 = o ( A ) ∪ ( o ( A ) ′ ∩ ( o ( A ) ∪ o ( B )), i.e., o ( A ∨ B ) 6 = o ( A ∨ ( ¬ A ∧ ( A ∨ B ))). This falsifies ( A ∨ B ) ≈ ( A ∨ ( ¬ A ∧ ( A ∨ B ) ) . There- fore a ∪ b 6 = a ∪ ( a ′ ∩ ( a ∪ b )), pro viding a coun terexample to the orthomo dular la w for F ◦ / ≈ .  26 Lemma 6.8 M A = hA , f i is a pr op er W OML mo de l . Pr o of. F ollows from Lemma 6.5 and Theorem 6.7.  No w w e are able to prov e the completeness of QL , i.e., tha t if a formula A is a conseque nce of a set of wffs Γ in a ll W OML-OML mo dels, then Γ ⊢ A . In particular, when Γ = ∅ , all v alid formulae are prov able in QL . (Recall from the note b elo w D efinition 5.9 that the left-hand side of the metaimplication b elo w is implicitly quan tified o v er all prop er W OML mo dels M .) Theorem 6.9 [Completeness] Γ  M A ⇒ Γ ⊢ A . Pr o of. Γ  M A means that in all prop er WOML mo dels M , if f ( X ) = 1 for all X in Γ, then f ( A ) = 1 holds. In particular, it holds fo r M A = hA , f i , which is a prop er W OML mo del b y Lemma 6.8. Therefore, in the Linden ba um algebra A , if f ( X ) = 1 for a ll X in Γ , then f ( A ) = 1 holds. By Lemma 6.6, it follo ws that Γ ⊢ A .  7 The Complet e ness of Class i cal Logic for WDOL-BA Mo dels W e hav e to imp ose the structure of WDO L-BA on the set F ◦ of form ulae of C L . W e star t with a relation of congruence, i.e., a relation of equiv alence compatible with t he o p erations in C L . W e mak e use of an equiv alence rela- tion to establish a corresp ondence b et w een formulae of QL and fo rm ulae of WDOL-BA. The resulting equiv alence classes stand for elemen ts of a prop er WDOL (i.e., a mem b er of WDOL- BA) and enable the completeness pro of of QL b y means of WDOL-BA. W e will closely f ollo w the pro cedure outlined in Section 6 a nd will often implicitly assume that definitions a nd theorems giv en in that section for QL hav e a completely a nalogous form for C L . Theorem 7.1 The r el a tion of equiv alence ≈ Γ , C L or just ≈ , define d as A ≈ B (67) def = Γ ⊢ A ≡ 0 B & ( ∀ o ∈ O 6)[( ∀ X ∈ Γ) ( o ( X ) = 1) ⇒ o ( A ) = o ( B )] , is a r elation of c ongruenc e in the algebr a F . Pr o of. As give n in [18].  27 Lemma 7.2 The Lindenb aum algebr a A = hF ◦ / ≈ , ¬ / ≈ , ∨ / ≈i is a WDOL , i.e., Eq s . (57)–(60) and Eq. (61) h old for ¬ / ≈ a n d ∨ / ≈ as ′ and ∪ r esp e c- tively. Pr o of. In analogy to Lemma 6.5 a nd following [18].  Lemma 7.3 In the Lindenb aum alge b r a A , i f f ( X ) = 1 for a l l X in Γ implies f ( A ) = 1 , then Γ ⊢ A . Pr o of. As given in [18].  Theorem 7.4 Distributivity do es not hold in A . Pr o of. ( a ∩ ( b ∪ c )) = (( a ∩ b ) ∪ ( a ∩ c )) fails in O6.  Lemma 7.5 M A = hA , f i is a pr op er WDOL mo del. Pr o of. F o llo ws from L emma 7.2 and Theorem 7.4.  Theorem 7.6 [Completeness] Γ  M A ⇒ Γ ⊢ A Pr o of. Analogous to the pro of of Theorem 6.9.  8 V aluation-Nonmono t onicit y In Sections 5, 6, a nd 7 w e prov e the soundness and completeness o f b oth quan- tum ( QL ) and classical ( C L ) standard lo g ic for pro p er weakly orthomo dular (W OML-OML) and w eakly distributiv e (WDOL-BA) ortholattices, resp ec- tiv ely . As w e stressed in the In tro duction and in Section 4, WOML-OML is the class of all those ortho la ttices (see D efinition 2.1) that satisfy Definition 2.8 (WOML) but do not satisfy Definition 2.9. Analogously , WDOL- BA in- cludes all those ortholatt ices tha t satisfy Definition 3.5 but do not satisfy Definition 3.2. The set-theoretical differences WOM L \ OML (W OML-OMLs) and WDOL \ BA (WDOL-BAs) determine v aluations that quantum and classical log ic can re- sp ectiv ely make use of. The set o f v aluations that can b e assigned to logical prop ositions are simply elemen t s of any of particular lattices, e.g., O6 giv en in Figure 1. Of course, any standard Bo olean v aluation set suc h as, e.g., { 0,1 } , i.e., { TRUE,F ALSE } , is then precluded b y definition. On the other 28 hand, if w e decide to use, e.g., { 0,1 } - v aluation, i.e., t w o-v alued BA as our mo del, then w e cannot use WDOL-BAs v aluations a n y more. Both WOML-OMLs and WDOL-BAs, on the one ha nd, and OMLs and BAs, on the other, are mo dels for whic h w e can prov e soundness and com- pleteness o f quantum and classical lo g ic, resp ectiv ely . Whic h ones we will use, i.e., whic h v a luations we will c ho ose, dep ends on t he hardw are, i.e., the kind of implemen ta tion w e adopt. F o r an implemen tatio n of the { 0,1 } v al- uation, we use to da y’s binary chips ; for the O6 or any other non-Bo olean v aluation, we migh t design appro priate chips and circuits in the future. Ac- tually there are certainly man y more non-Bo olean v aluations t ha n the O6 one, if not infinitely many . F or example, in [11, Th. 3.2 ] we prov ed that equation ( a ≡ b ) ∩ (( b ≡ c ) ∪ ( a ≡ c )) = (( a ≡ b ) ∩ ( b ≡ c )) ∪ (( a ≡ b ) ∩ ( a ≡ c )) , (68) whic h ho lds in any OML, do es not hold in all W OMLs, since it fa ils in the Rose-Wilkinson ortholattice in Figure 3 which satisfies the W OML condition Eq. (9). ❍ ❍ ❍ ❍ ❍      ❍ ❍ ❍ ❍ ❍    ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ✟ P P P P ❍ ❍ ❍ ❍ ❍      ❍ ❍ ❍ ❍ ❍    ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ✟ P P P P r r r r r r r r r r r r r r r r r r r r w z y x t s r 0 v ′ u ′ u v 1 r ′ s ′ t ′ x ′ y ′ z ′ w ′ Figure 3: Rose-Wilkinson lattice If w e add Eq. (68) to the WOML conditions, w e get a family of lattices— let us call it WOMLi—whic h is strictly smaller than WOM L and strictly larger tha n OML. One of its v aluations is obv iously on the O6 lattice but not on the Ro se-Wilkinson lattice. In analogy to the wa y we in tro duced prop er W OMLs in Section 4, w e can define WOMLi-OML as the class W OMLi \ OML, eac h mem b er of whic h is a pr op er W OMLi. Now the class W OML con ta ins 29 b oth the Rose-Wilkinson and O6 lattices. The class WOM Li-OML will con- tain O6 but not the Rose-Wilkinson lattice. The class OML will con tain neither Rose-Wilkinson nor O6. A sligh t mo dification of the pro of of Section 6 (b y replacing W OML with W OMLi) sho ws that quan tum logic is complete for W OMLi-OML, and it is also complete for W OMLi itself as follo ws from the completeness pro of s of quantum logic for WOML giv en in [7, 8]. Alternativ ely , w e can obtain a hierarch y of classes of mo dels for quantum logic by a dding conditions to the equations determining the class W O ML. Rather than r estricting W OML by subtracting OML from it (to obta in W OML-OML), w e restrict W OML b y adding new conditions (stronger than the W OML la w but we ak er t ha n the orthomo dular la w) to its defining equa- tions to o btain smaller equational v arieties, in b etw een OML and W OML. W e obtain the a nalogous hierarch y for classical logic by substituting “WDOL” for “W OML,” “BA” for “ O ML,” and “distributiv e” for “or thomo dular.” F or in- stance, if w e start with WOM L, w e can c ho ose any mo del from it w e wish: O6, Rose-Wilkinson, Beran 7 b [20, Fig. 7b], o r an y other W OML lattice. When w e add the condition (68) we can no longer use, e.g., the Rose-Wilkinson la t - tice/v aluation. When w e add the o r thomo dular law , w e can no longer use O6 or Rose-Wilkinson or Beran 7b v aluations. Th us b y adding conditions to the definitions of WOML and WDOL, w e change v alues (v aluatio ns) of logical prop ositions and we call this valuation non-monotonicity . More formally: Theorem 8.1 Quantum (classic al) lo gic is so und and c omplete with r esp e ct to either the W O ML (WDOL) o r the OML (BA) mo del families or any mo del family which is in b etwe en WOML (WDOL ) and OML (BA) (such as W OMLi ab ove). Particular W OML, WDOL, OML, BA, W OMLi lat- tic es r epr esent valuation sets for lo gic a l p r op osition s . By a dding c ond i tion s to Definitions 2.1 a n d 2.8 (WOM L) , 2.1 . and (68) (W OMLi) , 2.1 and 3.1 (WDOL) , etc. we ch ange the sets of val uations that c an b e as crib e d to pr op o- sitions. Th is pr op erty of lo gic al p r op ositions getting ne w sets of values, when we add new c onditions to the origi n al definition of la ttic es to mo de l o ur lo gic with, we c al l v aluation- nonmonotonicit y Pr o of. The soundness and completeness pro ofs for W OML and WDOL are giv en b y Theorems 29 & 39 and 3 0 & 47 of [8] (or b y Theorems 3.1 & 3.29 and 4.3 & 4.11 of [7]), resp ectiv ely . The soundness and completenes s pro ofs for OML and BA are we ll kno wn. See, e.g., [17] and [19]. Soundness and com- pleteness pro ofs fo r an y lattice in b et wee n W OML and OML and in b etw een 30 WDOL a nd BA fo llo w fr om the resp ective pro of s for W OML and OML. F or the soundness part of the pro of, this is b ecause an y suc h W OMLj or WDOLj (j = 1 , 2 , . . . ) is a WOML or WDOL, resp ectiv ely . W e can obtain a pro of that quan tum (classical) logic is complete for W OMLj (WDOLj) by rewriting the completeness pro of of Section 6 (7) so that the set of mappings to O6 that refines the equiv alence relations is replaced b y a set of mappings to a lat- tice that satisfies W OMLj (WDOLj) but violates W OMLj+1 (WDOLj+1), e.g., t he Rose-Wilkinson lattice for W OMLj = W OML and W OMLj+1 = W OMLi. The part of the pro of that refers to adding conditions is obv ious from the v ery definitions of WOML, W OMLj, OML, WDOL, WDO Lj, and BA.  W e stress here t ha t w e cannot mix up the tw o alternative w ays of c ho os- ing v aluations (restricting classes and fo rming set differences vs. v aluation- nonmonotonicit y), b ecause if w e added, e.g., the conditions defining OML (BA) to W OML-OML (WDOL-BA), w e w ould simply get empt y sets. 9 Complete n ess for Smaller Mo de l S ub classes The reader familiar with the authors’ earlier completeness pro o fs in [7] will notice that the new pro ofs here, in Sections 6 a nd 7, are iden tical except for the replacemen t of W OML (WDOL) with WOML-OML (WDOL-BA) in certain pla ces. This yields a strong er result for eac h log ic ( QL and C L ), i.e., eac h is complete for a smaller class of mo dels. If a logic is complete for a class of mo dels, it obv iously contin ues to b e complete if more mo dels f o r the logic are added t o that class. Th us the earlier completeness results follow immediately from the new ones, since WOML is o btained from W OML-OML b y adding bac k the OML mo dels for QL ( a nd ana lo gously WDOL for C L ). The k ey idea that allo w ed us to exclude OML from W OML in t he QL completeness pro of w as refinemen t of the equiv alence relatio n in Theorem 6.2 with the set of mappings O 6 . This resulted in smaller equiv alence classes, al- lo wing us to construct a Linden baum algebra that violated the orthomo dular la w and is t hus a prop er WOML. In fact, the O 6 “tr ick” is not limited to the use of lat t ice O6. W e can rewrite the completeness pro of for e.g. QL using any lattice that is a prop er W OML (a WOML but not an OML) in place of O6 . This will r esult in a completeness pr o of for a differen t class of mo dels that can b e an ev en smaller sub class o f W OML. 31 F or example, the Rose-Wilkinson lattice of Figure 3 is a prop er WOM L. If we use it in place of O6, an ana logous completenes s pro of show s that QL is complete f or the class WOML \ W OMLi, whic h is strictly smaller than W OML-OML. Since W OML \ W O MLi do esn’t include O6, this show s t hat QL is complete for a class of mo dels that is not only unrelated to OMLs but is ev en unrelated to the “natural” OML coun terexample O6, whic h up to now has serv ed as our prototypical W OML example. As men t ioned earlier, for classical logic C L , w e ha v e an ev en stronger completeness result that it is complete for single WDOL lattices, not just classes of them. F or example, it turns out that the Rose-Wilkinson lattice is also a prop er WDOL (as w ell as a prop er WOML). Th us the Rose-Wilkinson lattice, b y it self, provide s a mo del for whic h classical logic is sound and complete, sho wing that the hexagon O6 is not the only “ exotic” non-Bo olean lattice mo del for C L . 10 Concl u sion The main result we obtained in the previous sections is that logics can b e mo delled by disjoin t classes of differen t ortholattices. Classical logic can b e mo delled b y non-distributiv e lattices and quantum logic b y non- orthomo dular lattices. These lattices represen t differen t disjoint v aluation sets, where the v alua tion is a mapping fr o m prop ositions to a lattice. Thus b y adding conditions (axioms) to the original definition of an ortholat tice w e determine classes of lat t ices that in turn determine v aluatio ns that one can ascrib e to logical prop ositions. W e call the latter pro p ert y of logical prop o- sitions valuation-no n monotonicity (see Theorem 8.1). But b y considering disjoin t classes of lattices w e can f ur t her restrict v aluations we wan t to use. This can b e done as follow s. W e considered v arieties of classical non- distributiv e w eakly distributive lattice (WDOL, see Definition 3 .4) mo dels of classical prop ositiona l logic and non-orthomo dular w eakly orthomo dular lattice (WOML, D efinition 2.9) mo dels of quan tum quan tum prop ositional logics and pro v ed their soundness and completeness for those mo dels (see Theorems 5 .10, 5.11, 6.9, a nd 7.6) In particular, w e considered sub classes of WDOL and W OML that do not con tain Bo olean alg ebras (BAs, Definition 3.2) and ort homo dular lat- tices (OMLs, Definition 2.9), resp ectiv ely , while in Sections 8 and 9 w e also considered a p ossibly infinite sequence of sub classes of WDOL a nd W OML 32 that do not contain lattices WDOLi and W O MLi, resp ectiv ely , whic h in turn prop erly con t a in BA and OML, a nd for all of whic h w e ha v e prov ed the soundness and completeness . W e denoted these classes (v a rieties of WDOL and WMOL) a s WDOL-BA, W OML-OML, WDOL- WDOLi, and WMOL- W OMLi. The v aluations of W OML-OML and OML, of WDOL-BA and BA, of W ODL-WODLi and W ODLi, of WOML-W OMLi and W OMLi [Eq. (6 8)], and of W OMLi-OML and OML do not o v erlap. F or instance, v aluations from WDOL-BA cannot b e numeric ( { 0,1 } or { TRUE,F ALSE } ) at all since it do es no t con tain the tw o-v alued Bo olean algebra. A t t he lev el of logical ga tes, classical or quan tum, with to da y’s tec hnol- ogy for computers and artificial in telligence, we can use only bits and qubits, resp ectiv ely , i.e., only v aluat io ns corr esp o nding to { 0 ,1 } BA and OML, re- sp ectiv ely . And when we talk ab o ut log ics to day , we tak e fo r gran ted that they hav e the la t t er v aluation— { T RUE,F ALSE } in the case of classical logic and Hasse (Greec hie) diagrams in the case o f quan tum logic [21]. This is b ecause a v alua t io n is all w e use to implemen t a logic. In its final applica- tion, w e do not use a logic as giv en b y its axioms and rules of inferences but instead a s giv en by its mo dels. Actually , logics g iven only b y their axioms and rules of inferences (in Sections 5.1 and 5.2), i.e., without an y mo dels and an y v aluatio ns, cannot b e implemen ted in any hardw a re at all. It w ould b e in teresting to in v estigate how other v aluations, i.e., v arious ortholattices, might b e implemen t ed in complex circuits. That would prov ide us with the p o ssibility of con trolling essen tially differen t alg ebraic structures (logical mo dels) implemen ted in to radically different hardw ar e (logic circuits consisting of logic g a tes) b y the same logic as defined b y its axioms a nd rules of inference. Ac kno wledgemen t Supp orted by the Ministry of Science, Education, and Sp ort o f Croatia through the pro j ect No. 082-09 82562- 3 160. References [1] H. 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