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b(and)g(to)f(whic)n(h)h(the)g (follo)n(wing)e(\(\\distributivit)n(y"\))i(axiom)f(is)g(added:)1899 6016 y(2)p eop %%Page: 3 3 3 2 bop -25 540 a Fp(A8.)165 b Fj(`)23 b Fk(A)c Fj(_)f Fq(\()p Fk(B)23 b Fj(^)c Fk(C)6 b Fq(\))51 b Fj($)938 552 y Fh(cl)1044 540 y Fq(\()p Fk(A)19 b Fj(_)g Fk(B)t Fq(\))g Fj(^)g Fq(\()p Fk(A)g Fj(_)f Fk(C)6 b Fq(\))-25 723 y(The)27 b(afore)g(de\014ned)h(quan)n(tum)f(and)h(classical)e (logics)g(are)h(equiv)-5 b(alen)n(t)27 b(to)h(an)n(y)e(textb)r(o)r(ok)i (de\014nition.)3200 693 y Fo(4,6)-25 858 y Fq(Let)34 b(us)f(no)n(w)h(lo)r(ok)f(at)g(p)r(ossible)h(mo)r(dels)g(for)f(the)h (ab)r(o)n(v)n(e)f(logics.)54 b(Closed)34 b(subspaces)e(of)i(Hilb)r(ert) h(space)e(form)g(an)h(algebra)-150 958 y(called)21 b(Hilb)r(ert)h (lattice.)35 b(A)22 b(Hilb)r(ert)g(lattice)g(is)g(a)f(kind)g(of)h (orthomo)r(dular)e(lattice)i(whic)n(h)f(w)n(e,)i(in)e(this)h(section,)h (in)n(tro)r(duce)e(starting)-150 1057 y(with)33 b(an)g(ortholattice.)52 b(In)33 b(an)n(y)g(Hilb)r(ert)g(lattice)g(the)h(op)r(eration)e Fg(me)l(et)8 b Fq(,)33 b Fk(a)22 b Fj(\\)h Fk(b)p Fq(,)34 b(corresp)r(onds)d(to)i(set)g(in)n(tersection,)g Fj(H)3726 1069 y Fh(a)3780 995 y Ff(T)3863 1057 y Fj(H)3933 1069 y Fh(b)3967 1057 y Fq(,)-150 1157 y(of)d(subspaces)f Fj(H)399 1169 y Fh(a)439 1157 y Fk(;)14 b Fj(H)546 1169 y Fh(b)609 1157 y Fq(of)30 b(Hilb)r(ert)h(space)e Fj(H)q Fq(,)h(the)h(ordering)d(relation)h Fk(a)e Fj(\024)f Fk(b)k Fq(corresp)r(onds)e(to)i Fj(H)2986 1169 y Fh(a)3053 1157 y Fj(\022)c(H)3214 1169 y Fh(b)3248 1157 y Fq(,)k(the)h(op)r(eration)e Fg(join)6 b Fq(,)-150 1257 y Fk(a)18 b Fj([)h Fk(b)p Fq(,)27 b(corresp)r(onds)f(to)i(the)g(smallest)f(closed)g(subspace)g (of)g Fj(H)i Fq(con)n(taining)d Fj(H)2350 1269 y Fh(a)2404 1194 y Ff(S)2487 1257 y Fj(H)2557 1269 y Fh(b)2591 1257 y Fq(,)i(and)f Fk(a)2847 1226 y Fi(?)2931 1257 y Fq(corresp)r(onds)e (to)j Fj(H)3559 1226 y Fi(?)3558 1277 y Fh(a)3615 1257 y Fq(.)-25 1391 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Fj(\024)f Fk(b)189 b Fj(\))g Fk(a)23 b Fj(!)g Fk(b)g Fq(=)g(1)1540 b(\(1\))1899 6016 y(3)p eop %%Page: 4 4 4 3 bop -150 540 a Fq(and)24 b(therefore)f(all)h(the)g(axioms)f(of)h (form)g Fj(`)f Fk(A)g Fj(!)g Fk(B)28 b Fq(can)c(b)r(e)h(represen)n(ted) d(b)n(y)i Fk(a)f Fj(!)g Fk(b)g Fq(=)g(1.)35 b(Also,)24 b(it)h(is)f(straigh)n(tforw)n(ard)d(to)j(pro)n(v)n(e)-150 639 y(that)i(the)f(set)h(of)f(form)n(ulas)f(from)i(the)f(logic)g(is)g (closed)g(under)g(the)h(rules)f(of)g(inference.)36 b(Therefore,)25 b(the)h(soundness)e(for)h(quan)n(tum)-150 739 y(logic)i(can)g(b)r(e)h (pro)n(v)n(ed)e(b)n(y)h(b)r(oth,)h(W)n(OML)g(and)f(OML,)g(and)h(for)f (classical)f(logic)h(b)n(y)g(b)r(oth,)h(WDL)g(and)g(DL.)16 874 y(So,)e(the)h(clue)f(for)g(the)g(existence)g(of)g(t)n(w)n(o)g(mo)r (dels)g(for)g(eac)n(h)f(logic)g(w)n(as)h(ob)n(viously)e(hidden)j(in)f (the)h(completeness)f(pro)r(of)f(\(for)-150 974 y(150)30 b(y)n(ears)g(for)g(classical)g(logic)h(and)g(for)g(65)f(y)n(ears)g(for) h(quan)n(tum)g(logic\).)48 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b(More)24 b(o)n(v)n(er,)g Fp(A7)h Fq(turns)f(out)h(to)g(b)r(e)-150 2422 y(redundan)n(t:)35 b(it)25 b(can)f(b)r(e)h(inferred)f(from)g Fp(A1-6)p Fq(.)1339 2392 y Fo(4)1401 2422 y Fq(It)g(is)h Fk(a)e Fq(=)f Fk(b)48 b Fj(,)24 b(`)f Fk(A)g Fj($)2155 2434 y Fh(q)r(l)2236 2422 y Fk(B)29 b Fq(what)c(turns)f Fp(A7)h Fq(in)n(to)f Fk(a)12 b Fj([)g Fq(\()p Fk(b)g Fj(\\)g Fq(\()p Fk(a)3403 2392 y Fi(?)3473 2422 y Fj([)g Fk(b)3576 2392 y Fi(?)3632 2422 y Fq(\)\))24 b(=)f Fk(a)12 b Fj([)g Fk(b)p Fq(.)16 2558 y(Can)33 b(w)n(e)g(de\014ne)g(classes)f(of)h(equiv)-5 b(alence)33 b(in)h(another)e(w)n(a)n(y?)53 b(Y)-7 b(es,)35 b(w)n(e)d(can.)54 b(The)33 b(clue)g(is)h(to)f(prev)n(en)n(t)f(turning)h Fp(A7)g Fq(in)n(to)-150 2657 y Fk(a)19 b Fj([)h Fq(\()p Fk(b)f Fj(\\)h Fq(\()p Fk(a)226 2627 y Fi(?)302 2657 y Fj([)g Fk(b)413 2627 y Fi(?)468 2657 y Fq(\)\))26 b(=)f Fk(a)20 b Fj([)g Fk(b)p Fq(.)41 b(And)29 b(this)h(is)e(what)h(the)h (lattice)f(O6)f(sho)n(wn)h(b)r(ello)n(w)f(do)r(es.)41 b(An)n(y)29 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b(me)l(et)e(and)i(join)f(of)h(any)f(subset)f(of)i Fq(HL)f Fg(always)h(exist.)-52 1577 y(2.)43 b Fq(A)n(tomic:)37 b Fg(Every)29 b(non-zer)l(o)f(element)g(in)h Fq(HL)f Fg(is)h(gr)l(e)l(ater)f(than)g(or)h(e)l(qual)f(to)h(an)f(atom.)38 b(\(A)n(n)28 b(atom)g Fk(a)g Fg(is)h(a)g(non-zer)l(o)f(lattic)l(e)58 1676 y(element)h(with)i Fq(0)22 b Fk(<)h(b)f Fj(\024)h Fk(a)30 b Fg(only)g(if)h Fk(b)22 b Fq(=)h Fk(a)p Fg(.)38 b(A)n(n)29 b(atom)h(c)l(orr)l(esp)l(onds)h(to)f(a)g(pur)l(e)g(state.\)) -52 1842 y(3.)43 b Fq(Sup)r(erp)r(osition)27 b(Principle:)37 b Fg(\(The)31 b(atom)f Fk(c)f Fg(is)h(a)g(sup)l(erp)l(osition)g(of)h (the)e(atoms)h Fk(a)g Fg(and)g Fk(b)f Fg(if)h Fk(c)23 b Fj(6)p Fq(=)g Fk(a)p Fg(,)30 b Fk(c)23 b Fj(6)p Fq(=)f Fk(b)p Fg(,)30 b(and)g Fk(c)23 b Fj(\024)g Fk(a)18 b Fj([)g Fk(b)p Fg(.\))88 2008 y(\(a\))42 b(Given)d(two)g(di\013er)l(ent) g(atoms)g Fk(a)f Fg(and)h Fk(b)p Fg(,)i(ther)l(e)e(is)g(at)f(le)l(ast)h (one)g(other)g(atom)g Fk(c)p Fg(,)i Fk(c)e Fj(6)p Fq(=)f Fk(a)h Fg(and)g 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b(reason)e(for)h(that)g(is)h(that)g(it)g(is)f(extremely)g (di\016cult)i(to)e(deal)h(with)g(suc)n(h)f(equations.)1899 6016 y(5)p eop %%Page: 6 6 6 5 bop 16 540 a Fq(Since)33 b(already)e(equations)g(with)i(4)f(v)-5 b(ariables)31 b(con)n(tain)h(at)g(least)h(ab)r(out)f(30)f(terms)i(whic) n(h)f(one)g(cannot)g(further)g(simplify)-7 b(,)-150 639 y(a)32 b(prop)r(er)g(to)r(ol)g(for)g(\014nding)g(and)h(handling)f(the)h (equations)e(is)i(indisp)r(ensable.)51 b(As)33 b(a)f(great)f(help)i (came)f(Greec)n(hie)g(lattices)g(in)-150 739 y(whic)n(h)26 b(suc)n(h)g(equations)g(m)n(ust)g(either)g(fail)h(or)e(hold)h(\(as)g (in)h(O6)e(ab)r(o)n(v)n(e\).)36 b(E.g.,)25 b(to)i(\014nd)f(that)h(t)n (w)n(o)e(equations)h(cannot)g(b)r(e)g(inferred)-150 839 y(from)h(eac)n(h)g(other)g(it)h(su\016ces)f(to)h(\014nd)g(t)n(w)n(o)f (Greec)n(hie)g(lattices)g(whic)n(h)g(the)h(equations)f(in)n(terc)n (hangeably)f(pass)h(and)g(fail.)16 974 y(The)34 b(\014rst)g(attempt)h 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