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b(from)f(KA2,)i(KA9,)25 3611 y(KA12,)k(KA13,)f(and) i(KMP0.)191 3736 y(Less)38 b(ob)n(vious)f(is)i(a)f(main)h(result)g(of)f (this)h(section,)i(whic)n(h)e(is)g(that)g(KA11)e(is)i(redundan)n(t.)70 b(This)39 b(is)f(somewhat)25 3861 y(surprising,)f(since)e(the)i(resem)n (blance)d(b)r(et)n(w)n(een)i(KA11)f(and)h(a)g(\\prop)r(er")e(orthomo)r (dular)g(axiom)h(at)h(\014rst)g(suggests)e(it)25 3985 y(migh)n(t)27 b(b)r(e)h(essen)n(tial.)25 4226 y Fu(Theorem)i(2.6.)40 b Fv(Axiom)30 b Ft(KA11)f Fv(is)h(derivable)i(fr)l(om)e Ft(KA1)p Fv({)p Ft(KA10)p Fv(,)f Ft(KA12)p Fv({)p Ft(KA13)p Fv(,)f(and)i Ft(KMP0)p Fv(.)25 4417 y(Pr)l(o)l(of.)43 b Fh(`)336 4429 y Ff(K0)450 4417 y Fg(A)30 b Ft(or)e(just)i Fh(`)c Fg(A)k Ft(will)g(mean)f(\\)p Fg(A)g Ft(is)h(a)f(theorem)g(of)g (K0.")41 b(In)30 b(our)f(pro)r(ofs)f(w)n(e)h(will)h(sometimes)f(not)h (men)n(tion)25 4541 y(p)r(ossible)g(use)g(of)g(KA2,)g(KA12,)g(KA13,)g (or)f(KMP0.)43 b(These,)31 b(together)e(with)i(KA3,)f(KA4,)h(KA9,)f (and)g(KA10,)g(allo)n(w)f(us)25 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(y)f(ortholattice)h(analogy)-7 b(.")33 b(Of)24 b(course)e(ortholattice) h(analogs)f(can)h(include)h(only)f(theorems)25 5289 y(of)k(K0)g(of)h (the)g(form)f Fh(`)c Fg(A)g Fh(\021)g Fg(B)t Ft(.)p eop %%Page: 5 5 5 4 bop 25 -257 a Ft(M.)28 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c)24 b(and)j(N.)h(Megill,)55 b Fv(Binary)31 b(Ortholo)l(gic)g(with)g(Mo)l (dus)f(Ponens)1480 b Ft(5)191 33 y(F)-7 b(rom)24 b(KA8)g(w)n(e)g(ha)n (v)n(e)f Fh(`)g(:)p Fg(A)12 b Fh(^)g Fg(A)25 b Fh(\021)e Ft(\()p Fh(:)p Fg(A)12 b Fh(^)g Fg(A)p Ft(\))g Fh(^)g Ft(\()p Fh(:)p Fg(B)20 b Fh(^)12 b Fg(B)t Ft(\))26 b(and)e Fh(`)f(:)p Fg(B)17 b Fh(^)12 b Fg(B)28 b Fh(\021)22 b Ft(\()p Fh(:)p Fg(B)17 b Fh(^)12 b Fg(B)t Ft(\))g Fh(^)g Ft(\()p Fh(:)p Fg(A)g Fh(^)g Fg(A)p Ft(\).)41 b(Using)24 b(KA5)25 158 y(to)k(connect)g(the)h(righ)n(t-hand)e(sides)i(w)n(e)f (conclude)g Fh(`)c(:)p Fg(A)19 b Fh(^)h Fg(A)k Fh(\021)g(:)p Fg(B)g Fh(^)19 b Fg(B)t Ft(.)40 b(This)28 b(theorem)g(together)g(with)g (the)h(equalit)n(y)25 282 y(metatheorem)e(ab)r(o)n(v)n(e)f(allo)n(w)h (us)g(to)h(de\014ne)f(constan)n(ts)g(0)1862 235 y Ff(def)1876 282 y Ft(=)40 b Fh(:)p Fg(A)19 b Fh(^)g Fg(A)28 b Ft(and)g(1)2512 235 y Ff(def)2525 282 y Ft(=)41 b Fh(:)p Ft(0.)191 407 y(W)-7 b(e)28 b(de\014ne)g Fh(`)23 b Fg(A)g Fh(\024)g Fg(B)915 360 y Ff(def)929 407 y Ft(=)63 b Fh(`)23 b Fg(A)c Fh(_)g Fg(B)27 b Fh(\021)c Fg(B)t Ft(.)191 531 y(\(i\))29 b(W)-7 b(e)29 b(ha)n(v)n(e)e(the)h(la)n(w)g(of)g(excluded)g(middle)h Fh(`)24 b Fg(A)19 b Fh(_)h(:)p Fg(A)29 b Ft(whic)n(h)f(follo)n(ws)f (from)h(KA1,)g(KA13,)f(and)h(KA5.)39 b(This)28 b(can)25 656 y(b)r(e)g(restated)f(as)g Fh(`)c Ft(1.)191 780 y(\(ii\))29 b(By)f(ortholattice)f(analogy)g(w)n(e)h(ha)n(v)n(e)f(that)h Fh(`)c Fg(A)h Fh(\024)e Fg(B)t Ft(,)29 b(i.e.)39 b Fh(`)24 b Fg(A)19 b Fh(_)h Fg(B)28 b Fh(\021)c Fg(B)t Ft(,)k(implies)h Fh(`)24 b Fg(A)19 b Fh(^)g Fg(B)29 b Fh(\021)23 b Fg(A)29 b Ft(\(conjoin)f Fg(A)25 905 y Ft(to)f(b)r(oth)h(sides)g(and)f(apply)g (KA7\).)191 1029 y(\(iii\))i Fh(`)23 b Fg(A)c Fh(_)g Ft(\()p Fh(:)p Fg(A)h Fh(^)f Ft(\()p Fg(A)g Fh(_)h Fg(B)t Ft(\)\))k Fh(\024)f Fg(A)c Fh(_)g Fg(B)33 b Ft(follo)n(ws)27 b(b)n(y)g(ortholattice)h(analogy:)35 b(in)29 b Fh(`)23 b(:)p Fg(A)d Fh(^)f Ft(\()p Fg(A)g Fh(_)g Fg(B)t Ft(\))24 b Fh(\024)g Fg(A)19 b Fh(_)g Fg(B)t Ft(,)28 b(place)25 1154 y Fg(A)g Ft(in)g(a)f(disjunction)h(on)f(b)r(oth)h(sides.)191 1279 y(\(iv\))g Fh(`)23 b(:)p Ft(\()p Fg(A)c Fh(_)g Fg(B)t Ft(\))g Fh(_)g Ft(\()p Fg(A)g Fh(_)f Ft(\()p Fh(:)p Fg(A)i Fh(^)f Ft(\()p Fg(A)g Fh(_)f Fg(B)t Ft(\)\)\))24 b Fh(\021)f Ft(1)k(is)h(pro)n(v)n(ed)e(as)h(follo)n(ws.)36 b(By)27 b(DeMorgan's)f(la)n(ws)h(w)n(e)g(ha)n(v)n(e)357 1428 y Fh(`)c(:)p Fg(A)c Fh(^)g Ft(\()p Fg(A)g Fh(_)f Fg(B)t Ft(\))24 b Fh(\021)f(:)p Ft(\()p Fg(A)c Fh(_)g(:)p Ft(\()p Fg(A)g Fh(_)g Fg(B)t Ft(\)\).)25 1577 y(By)27 b(comm)n(utativit)n(y)g (of)h(disjunction)g(w)n(e)f(ha)n(v)n(e)357 1727 y Fh(`)c(:)p Ft(\()p Fg(A)c Fh(_)g Fg(B)t Ft(\))g Fh(_)g Fg(A)k Fh(\021)g Fg(A)18 b Fh(_)h(:)p Ft(\()p Fg(A)g Fh(_)g Fg(B)t Ft(\).)25 1876 y(The)28 b(disjunction)g(of)f(these)h(yields)357 2026 y Fh(`)23 b Ft(\()p Fh(:)p Ft(\()p Fg(A)c Fh(_)g Fg(B)t Ft(\))g Fh(_)g Fg(A)p Ft(\))g Fh(_)g Ft(\()p Fh(:)p Fg(A)g Fh(^)g Ft(\()p Fg(A)g Fh(_)g Fg(B)t Ft(\)\))k Fh(\021)g Ft(1)25 2175 y(and)k(asso)r(ciativit)n(y)f(of)i(disjunction)g (giv)n(es)e(us)i(the)g(result.)191 2300 y(\(v\))22 b(F)-7 b(rom)22 b Fh(`)h Fg(A)g Fh(\024)g Fg(B)j Ft(and)c Fh(`)h(:)p Fg(B)11 b Fh(_)c Fg(A)25 b Fh(\021)d Ft(1)g(w)n(e)f(can)h(infer)g Fh(`)h Fg(A)g Fh(\021)g Fg(B)t Ft(,)g(pro)n(v)n(ed)e(as)g(follo)n(ws.) 34 b(F)-7 b(rom)22 b(the)g(\014rst)g(h)n(yp)r(othesis:)25 2424 y(b)n(y)32 b(de\014nition)h(of)g Fh(\024)f Ft(and)h(DeMorgan's)e (la)n(ws,)i(w)n(e)f(ha)n(v)n(e)g Fh(`)f(:)p Fg(A)22 b Fh(^)h(:)p Fg(B)36 b Fh(\021)31 b(:)p Fg(B)t Ft(;)k(b)n(y)e(\(ii\))g(w) n(e)f(ha)n(v)n(e)g Fh(`)f Fg(A)22 b Fh(^)g Fg(B)36 b Fh(\021)31 b Fg(A)p Ft(.)53 b(The)25 2549 y(disjunction)36 b(of)g(b)r(oth)h(sides)e(yields)h Fh(`)h Ft(\()p Fg(A)g Fh(\021)g Fg(B)t Ft(\))h Fh(\021)e Fg(A)24 b Fh(_)h(:)p Fg(B)t Ft(.)62 b(F)-7 b(rom)36 b(this)g(and)g(the)g(second)g(h)n(yp)r (othesis)f(w)n(e)h(obtain)25 2673 y Fh(`)23 b Ft(\()p Fg(A)g Fh(\021)g Fg(B)t Ft(\))g Fh(\021)g Ft(1.)37 b(Using)27 b(KA12,)g(KA13,)f(and)i(KMP0,)e(w)n(e)h(detac)n(h)g(excluded)h(middle)g (\(i\))g(to)g(obtain)f(the)h(result.)191 2798 y(\(vi\))h Fh(`)c Fg(A)20 b Fh(_)g Ft(\()p Fh(:)p Fg(A)g Fh(^)g Ft(\()p Fg(A)f Fh(_)h Fg(B)t Ft(\)\))26 b Fh(\021)f Fg(A)19 b Fh(_)h Fg(B)33 b Ft(follo)n(ws)28 b(immediately)h(from)g(\(iii\),)h (\(iv\),)g(and)f(\(v\).)41 b(This)29 b(is)g(axiom)f(KA11,)25 2922 y(the)g(desired)f(result.)p 3862 2922 4 57 v 3866 2870 50 4 v 3866 2922 V 3915 2922 4 57 v 191 3113 a(Lo)r(oking)22 b(closer)f(at)i(KA11,)g(w)n(e)g(\014nd)g(another)f(surprise:)33 b(its)23 b(mapping)g(has)f(nothing)h(to)g(do)g(with)g(the)g(orthomo)r (dular)25 3238 y(la)n(w.)25 3479 y Fu(Theorem)30 b(2.7.)40 b Fv(The)31 b(mapping)g(for)g Ft(KA11)p Fv(,)361 3628 y Fg(a)18 b Fh([)h Ft(\()p Fg(a)573 3598 y Fn(?)647 3628 y Fh(\\)g Ft(\()p Fg(a)g Fh([)g Fg(b)p Ft(\)\))k Fh(\021)g Fg(a)18 b Fh([)h Fg(b)j Ft(=)h(1)25 3778 y Fv(is)30 b(true)f(in)h(al)t (l)h(ortholattic)l(es.)25 3968 y(Pr)l(o)l(of.)43 b Ft(In)26 b(a)f(manner)g(exactly)g(analogous)f(to)h(sections)g(\(iii\))i(and)e (\(iv\))h(of)g(the)g(previous)f(pro)r(of,)g(w)n(e)h(obtain)f(for)g (ortholat-)25 4093 y(tices)357 4242 y Fg(a)18 b Fh([)h Ft(\()p Fg(a)569 4212 y Fn(?)644 4242 y Fh(\\)g Ft(\()p Fg(a)f Fh([)h Fg(b)p Ft(\)\))k Fh(\024)g Fg(a)18 b Fh([)h Fg(b)25 4392 y Ft(and)357 4541 y(\()p Fg(a)g Fh([)f Fg(b)p Ft(\))593 4511 y Fn(?)668 4541 y Fh([)h Ft(\()p Fg(a)f Fh([)h Ft(\()p Fg(a)986 4511 y Fn(?)1061 4541 y Fh(\\)f Ft(\()p Fg(a)h Fh([)g Fg(b)p Ft(\)\)\))k(=)g(1)25 4691 y(By)30 b(analogy)e(to)i(section)g(\(v\))h(of)f(the)h(previous)e(pro)r (of,)h(but)h(omitting)f(the)h(\014nal)f(detac)n(hmen)n(t)g(of)g (excluded)h(middle,)g(w)n(e)25 4815 y(obtain)c(that)357 4965 y Fg(a)c Fh(\024)g Fg(b)k Ft(and)g Fg(b)772 4935 y Fn(?)846 4965 y Fh([)19 b Fg(a)k Ft(=)g(1)k(implies)h Fg(a)23 b Fh(\021)f Fg(b)h Ft(=)g(1)25 5114 y(in)28 b(all)f (ortholattices.)36 b(The)27 b(desired)g(result)h(follo)n(ws)e (immediately)i(from)f(these.)p 3862 5114 V 3866 5061 50 4 v 3866 5114 V 3915 5114 4 57 v eop %%Page: 6 6 6 5 bop 25 -257 a Ft(M.)28 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c)24 b(and)j(N.)h(Megill,)55 b Fv(Binary)31 b(Ortholo)l(gic)g(with)g(Mo)l (dus)f(Ponens)1480 b Ft(6)191 33 y(Giv)n(en)36 b(this)g(result,)i(it)f (is)f(natural)f(to)h(ask)f(whether)h(the)h(orthomo)r(dular)d(la)n(w)i (is)g(needed)g(at)g(all)g(in)g(the)g(algebra)25 158 y(underlying)27 b(K0.)36 b(The)28 b(answ)n(er)e(is)h(in)h(the)g(follo)n(wing)f (theorem.)25 399 y Fu(Theorem)j(2.8.)40 b Fv(\(i\))27 b(The)h(algebr)l(aic)h(mappings)f(for)g Ft(KA1)p Fv(,)f Ft(KA3)p Fv(,)g Ft(KA5)p Fv({)p Ft(KA13)p Fv(,)f(and)h Ft(KMP0)f Fv(ar)l(e)h(true)f(or)h(sound)g(in)f(al)t(l)25 523 y(ortholattic)l(es.)40 b(\(ii\))30 b(The)h(algebr)l(aic)h(mappings) f(for)g Ft(KA2)e Fv(and)h Ft(KA4)f Fv(do)i(not)e(hold)i(in)f(al)t(l)h (ortholattic)l(es.)25 714 y(Pr)l(o)l(of.)43 b Ft(KMP0:)36 b(This)27 b(rule)h(is)f(sound)g(in)h(all)f(ortholattices.)g([12)o(])191 839 y(KA1,)g(KA5{KA10,)e(KA12:)36 b(Eac)n(h)26 b(theorem)g(has)h(an)g (ob)n(vious)f(equational)g(analog)g(in)h(ortholattices.)36 b(Since)27 b Fg(a)c Ft(=)g Fg(b)25 963 y Ft(implies)30 b Fg(a)e Fh(\021)f Fg(b)g Ft(=)g(1)j(in)g(all)g(ortholattices)f(\(if)j Fg(a)27 b Ft(=)g Fg(b)p Ft(,)k(then)f Fg(a)e Fh(\021)f Fg(b)g Ft(=)g Fg(b)g Fh(\021)g Fg(b)g Ft(=)h(1\),)i(w)n(e)g(obtain)g (the)h(desired)f(result.)44 b(F)-7 b(or)25 1088 y(example,)31 b(for)e(KA5)h(w)n(e)g(ha)n(v)n(e)f(the)i(analog)e Fg(a)20 b Fh([)h Fg(b)27 b Ft(=)g Fg(b)20 b Fh([)h Fg(a)p Ft(,)31 b(from)f(whic)n(h)g(w)n(e)g(deduce)h(its)f(mapping)g Fg(a)20 b Fh([)h Fg(b)27 b Fh(\021)h Fg(b)20 b Fh([)g Fg(a)28 b Ft(=)f(1,)25 1212 y(whic)n(h)g(is)h(true)f(in)h(all)g (ortholattices.)191 1337 y(KA3,)37 b(KA13:)52 b(W)-7 b(e)36 b(use)f(the)h(fact)f(that)h Fg(a)g Fh(\024)g Fg(b)f Ft(implies)h Fg(a)2104 1307 y Fn(?)2184 1337 y Fh([)24 b Fg(b)36 b Ft(=)g(1)f(in)h(all)f(ortholattices)f(\(if)i Fg(a)24 b Fh([)g Fg(b)36 b Ft(=)g Fg(b)p Ft(,)h(then)25 1461 y Fg(a)69 1431 y Fn(?)143 1461 y Fh([)19 b Fg(a)f Fh([)h Fg(b)k Ft(=)g Fg(a)544 1431 y Fn(?)618 1461 y Fh([)c Fg(b)p Ft(\).)37 b(F)-7 b(or)27 b(KA13,)f Fg(a)18 b Fh(\\)h Fg(b)k Fh(\024)g Fg(b)k Ft(and)g Fg(a)1780 1431 y Fn(?)1855 1461 y Fh(\\)19 b Fg(b)1965 1431 y Fn(?)2043 1461 y Fh(\024)k Fg(a)2175 1431 y Fn(?)2231 1461 y Ft(,)28 b(so)e Fg(a)d Fh(\021)g Fg(b)g Fh(\024)f Fg(a)2728 1431 y Fn(?)2803 1461 y Fh([)d Fg(b)p Ft(.)191 1586 y(KA11:)36 b(Theorem)27 b(2.7.)191 1710 y(KA2,)g(KA4:)36 b(Eac)n(h)27 b(of)g(their)h(mappings)357 1860 y(\()p Fg(a)23 b Fh(\021)g Fg(b)p Ft(\))612 1830 y Fn(?)686 1860 y Fh([)c Ft(\()p Fg(b)k Fh(\021)g Fg(c)p Ft(\))1007 1830 y Fn(?)1081 1860 y Fh([)c Ft(\()p Fg(a)k Fh(\021)g Fg(c)p Ft(\))g(=)g(1)357 2009 y(\()p Fg(a)g Fh(\021)g Fg(b)p Ft(\))612 1979 y Fn(?)686 2009 y Fh([)c Ft(\()p Fg(a)g Fh(\\)f Fg(c)23 b Fh(\021)g Fg(b)18 b Fh(\\)h Fg(c)p Ft(\))k(=)g(1)25 2159 y(violates)28 b(the)i(non-orthomo)r(dular)e(ortholattice)g(of)i (Fig.)f(1)g(with)h(the)g(assignmen)n(t)f Fg(a)d Ft(=)g Fg(x)p Ft(,)k Fg(b)c Ft(=)g Fg(z)t Ft(,)k Fg(c)c Ft(=)g Fg(y)32 b Ft(sho)n(wing)c(that)25 2283 y(they)g(do)f(not)h(hold)f(in)h (all)f(ortholattices.)1974 3146 y Fc(@)1891 3063 y(@)1808 2980 y(@)1725 2897 y(@)p 2055 3146 4 167 v 2057 3146 a(\000)2140 3063 y(\000)2057 2980 y(\000)2140 2897 y(\000)p 2221 2980 V 1723 2814 V 1725 2814 a(\000)1808 2731 y(\000)2140 2814 y(@)2057 2731 y(@)1974 2648 y(@)1891 2565 y(@)1725 2648 y(\000)1808 2565 y(\000)p 1889 2648 V 2036 3241 a Ft(0)1984 2980 y Fg(x)2264 2990 y(y)1585 2849 y(z)1628 2819 y Fn(?)2264 2832 y Fg(z)1583 2675 y(y)1627 2645 y Fn(?)1916 2718 y Fg(x)1963 2688 y Fn(?)1870 2440 y Ft(1)2057 3146 y Fb(r)2057 2980 y(r)141 b(r)1725 2814 y(r)473 b(r)1725 2648 y(r)141 b(r)1891 2482 y(r)913 3412 y Ft(Figure)27 b(1:)36 b(Ortholattice)27 b(violated)g(b)n(y)g(mappings) g(of)h(KA2,)f(KA4)p 3862 3636 4 57 v 3866 3583 50 4 v 3866 3636 V 3915 3636 4 57 v 191 3877 a(Finally)-7 b(,)24 b(w)n(e)e(ask)f(whether)i(the)g(full)g(strength)f(of)g(the)h(orthomo)r (dular)e(la)n(w)h(is)g(em)n(b)r(o)r(died)h(in)g(the)g(algebraic)d (seman)n(tics)25 4001 y(for)27 b(system)g(K0.)36 b(The)28 b(answ)n(er)e(is)i(no.)25 4192 y Fu(De\014nition)j(2.9.)180 b Fv(We)29 b(de\014ne)h(a)g Ft(w)n(eakly)d(orthomo)r(dular)f(\(W)n (OM\))i(lattice)h Fv(as)i(an)e(ortholattic)l(e)i(extende)l(d)f(with)g (the)25 4317 y(fol)t(lowing)i(two)e(laws:)191 4466 y Fu(W1.)169 b Ft(\()p Fg(a)23 b Fh(\021)g Fg(b)p Ft(\))789 4436 y Fn(?)863 4466 y Fh([)c Ft(\()p Fg(a)f Fh(\\)h Fg(c)k Fh(\021)g Fg(b)18 b Fh(\\)h Fg(c)p Ft(\))k(=)g(1)191 4616 y Fu(W2.)169 b Ft(\()p Fg(a)23 b Fh(\021)g Fg(b)p Ft(\))789 4585 y Fn(?)863 4616 y Fh([)c Ft(\()p Fg(b)k Fh(\021)f Fg(c)p Ft(\))1183 4585 y Fn(?)1258 4616 y Fh([)d Ft(\()p Fg(a)k Fh(\021)g Fg(c)p Ft(\))g(=)g(1)25 4831 y(Because)k(these)i(are)e(the)i(mappings)f(of)g(KA4)g(and)g(KA2,)g(and) g(since)h(all)f(other)f(mappings)h(of)h(K0)e(are)h(true)g(or)f(sound)h (in)25 4956 y(ortholattices,)34 b(W)n(OM)f(lattice)g(algebra)f(is)h (su\016cien)n(t)g(to)h(mo)r(del)f(K0.)53 b(\(Later)33 b(w)n(e)g(shall)g(sho)n(w)g(W2)g(is)g(redundan)n(t;)j(see)25 5080 y(Theorem)27 b(2.14)f(b)r(elo)n(w.\))25 5321 y Fu(Theorem)k(2.10.) 40 b Fv(\(i\))24 b(Every)h(orthomo)l(dular)h(lattic)l(e)e(is)g(a)g(WOM) g(lattic)l(e.)37 b(\(ii\))25 b(Every)g(WOM)e(lattic)l(e)i(is)f(an)g (ortholattic)l(e.)p eop %%Page: 7 7 7 6 bop 25 -257 a Ft(M.)28 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c)24 b(and)j(N.)h(Megill,)55 b Fv(Binary)31 b(Ortholo)l(gic)g(with)g(Mo)l (dus)f(Ponens)1480 b Ft(7)25 33 y Fv(Pr)l(o)l(of.)43 b Ft(\(i\))25 b(W1)e(and)h(W2)g(are)e(true)i(in)g(ev)n(ery)f(orthomo)r (dular)f(lattice.)35 b(See)24 b(the)g(pro)r(ofs)f(of)h(the)g(KA4)f(and) h(KA2)f(mappings)25 158 y(in)28 b(Ref.)g([12)o(].)37 b(\(ii\))28 b(By)g(de\014nition.)p 3862 158 4 57 v 3866 105 50 4 v 3866 158 V 3915 158 4 57 v 191 399 a(Ho)n(w)n(ev)n(er,)j(no) h(theorem)g(of)g(W)n(OM)g(lattice)g(algebra)e(is)i(as)g(strong)f(as)g (the)i(orthomo)r(dular)d(la)n(w,)j(as)e(sho)n(wn)g(b)n(y)h(the)25 523 y(follo)n(wing)f(theorem.)52 b(This)32 b(sho)n(ws)g(that)h(K0)e (can)h(b)r(e)h(mo)r(deled)g(b)n(y)g(an)f(algebra)f(strictly)h(w)n(eak)n (er)f(than)h(orthomo)r(dular)25 648 y(lattice)c(algebra.)25 888 y Fu(Theorem)i(2.11.)40 b Fv(\(i\))28 b(Ther)l(e)h(exist)e(WOM)h (lattic)l(es)g(that)g(ar)l(e)g(not)f(orthomo)l(dular.)40 b(\(ii\))28 b(Ther)l(e)h(exist)f(ortholattic)l(es)h(that)25 1013 y(ar)l(e)h(not)f(WOM)h(lattic)l(es.)25 1204 y(Pr)l(o)l(of.)43 b Ft(\(i\))21 b(The)f(non-orthomo)r(dular)e(ortholattice)i(O6)f(\(Fig.) i(2\))f(is)g(not)g(violated)g(b)n(y)g(W1)g(and)g(W2.)34 b(\(ii\))21 b(Theorem)f(2.8\(ii\).)1891 1905 y Fc(@)1808 1822 y(@)1974 1905 y(\000)2057 1822 y(\000)p 1806 1739 4 167 v 2138 1739 V 1808 1573 a(\000)1891 1490 y(\000)2057 1573 y(@)1974 1490 y(@)1953 2000 y Ft(0)1719 1757 y Fg(x)2181 1766 y(y)2225 1736 y Fn(?)1722 1583 y Fg(y)2181 1609 y(x)2228 1578 y Fn(?)1953 1366 y Ft(1)1974 1905 y Fb(r)1808 1739 y(r)307 b(r)1808 1573 y(r)g(r)1974 1407 y(r)1504 2171 y Ft(Figure)27 b(2:)36 b(Ortholattice)27 b(O6)p 3862 2391 4 57 v 3866 2338 50 4 v 3866 2391 V 3915 2391 4 57 v 25 2698 a Fu(Theorem)j(2.12.)40 b Fv(No)30 b(c)l(ol)t(le)l (ction)h(of)g(the)l(or)l(ems)e(of)i(orthomo)l(dular)g(lattic)l(es)f(of) h(the)f(form)361 2847 y Fg(t)23 b Ft(=)f(1)25 2997 y Fv(\(wher)l(e)30 b Fg(t)g Fv(is)g(a)g(term,)g(such)g(as)g Fg(a)18 b Fh([)h Fg(a)1213 2967 y Fn(?)1269 2997 y Fv(\),)30 b(when)g(adde)l(d)i(to)d(an)h(ortholattic)l(e,)i(determines)e(an)g (orthomo)l(dular)h(lattic)l(e.)25 3188 y(Pr)l(o)l(of.)43 b Ft(By)27 b(completeness)f(of)g(K0)g([12],)h(an)n(y)f(theorem)g(of)h (orthomo)r(dular)e(lattices)h(of)h(the)g(form)g Fg(t)c Ft(=)f(1)27 b(is)f(pro)n(v)-5 b(able)26 b(from)25 3312 y(the)k(mappings)g(of)g(the)g(axioms)f(and)h(inference)f(rule)h(of)g (K0.)43 b(Th)n(us)30 b(W)n(OM)f(lattice)h(algebra,)f(whic)n(h)h(is)g (strictly)g(w)n(eak)n(er)25 3437 y(than)e(orthomo)r(dular)e(lattice)h (algebra)f(b)n(y)h(Theorem)g(2.11)f(\(i\),)j(is)e(su\016cien)n(t)h(to)f (pro)n(v)n(e)f(an)n(y)h(suc)n(h)g(theorem.)p 3862 3437 V 3866 3384 50 4 v 3866 3437 V 3915 3437 4 57 v 25 3677 a(Indeed,)36 b(W)n(OM)d(lattice)h(algebra)f(is)g(equiv)-5 b(alen)n(t)34 b(to)g(ortholattice)f(algebra)f(extended)i(with)h(all)e (orthomo)r(dular)g(lattice)25 3802 y(theorems)27 b(of)g(the)h(form)g Fg(t)23 b Ft(=)f(1)27 b([where)h Fg(t)f Ft(is)h(a)f(term)g(suc)n(h)h (as)f Fg(a)18 b Fh([)h Ft(\()p Fg(a)2198 3772 y Fn(?)2272 3802 y Fh(\\)g Ft(\()p Fg(a)g Fh([)g Fg(b)p Ft(\)\))k Fh(\021)g Fg(a)18 b Fh([)h Fg(b)p Ft(].)191 4026 y(Kalm)n(bac)n(h)e (de\014nes)i(another)f(system,)j(whic)n(h)e(w)n(e)f(shall)h(call)f(K1,) i(with)g(a)e(mo)r(dus)h(p)r(onens)g(rule)g(using)f(the)i(implication)25 4151 y Fg(A)j Fh(!)193 4163 y Ff(3)254 4151 y Fg(B)t Ft(.)37 b(K1)26 b(is)i(de\014ned)g(as)f(K0)g(min)n(us)g(KMP0,)g(plus)g (the)h(follo)n(wing)f(axioms)f(and)i(inference)f(rule:)h([12)o(])25 4342 y Fu(Axioms.)191 4516 y(KA14.)254 b Fh(`)22 b Ft(\()p Fg(A)i Fh(!)989 4528 y Ff(0)1049 4516 y Fg(B)t Ft(\))g Fh(!)1255 4528 y Ff(3)1315 4516 y Ft(\()p Fg(A)g Fh(!)1516 4528 y Ff(3)1576 4516 y Ft(\()p Fg(A)g Fh(!)1777 4528 y Ff(3)1837 4516 y Fg(B)t Ft(\)\))191 4665 y Fu(KA15.)254 b Fh(`)22 b Ft(\()p Fg(A)i Fh(!)989 4677 y Ff(3)1049 4665 y Fg(B)t Ft(\))g Fh(!)1255 4677 y Ff(0)1315 4665 y Ft(\()p Fg(A)g Fh(!)1516 4677 y Ff(0)1576 4665 y Fg(B)t Ft(\))25 4973 y Fu(Rule)31 b(of)g(Inference.)191 5147 y(KMP3.)169 b Fh(`)22 b Fg(A)86 b Ft(&)107 b Fh(`)23 b Fg(A)g Fh(!)1301 5159 y Ff(3)1362 5147 y Fg(B)112 b Fh(\))85 b(`)22 b Fg(B)p eop %%Page: 8 8 8 7 bop 25 -257 a Ft(M.)28 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c)24 b(and)j(N.)h(Megill,)55 b Fv(Binary)31 b(Ortholo)l(gic)g(with)g(Mo)l (dus)f(Ponens)1480 b Ft(8)191 33 y(Kalm)n(bac)n(h)34 b(sho)n(ws)h(that)i(KA14,)g(KA15,)g(and)f(KMP3)f(can)h(b)r(e)g(deriv)n (ed)g(in)g(K0,)h(and)f(also)f(that)i(KMP0)e(can)g(b)r(e)25 158 y(deriv)n(ed)29 b(in)i(K1.)43 b(Th)n(us)30 b(K0)f(and)h(K1)f(are)g (logically)g(equiv)-5 b(alen)n(t,)30 b(and)g(the)h(ab)r(o)n(v)n(e)d (results)i(apply)g(to)g(K1)f(as)g(w)n(ell)h(as)f(K0.)25 282 y(W)-7 b(e)28 b(men)n(tion)f(that)h(KA1,)f(KA11,)g(and)g(KA15)g (are)g(redundan)n(t)g(in)h(K1.)191 407 y(The)g(mapping)f(for)g(KA14)g (is)g(not)h(a)f(theorem)g(of)g(ortholattice)g(algebra.)35 b(In)28 b(particular)e(the)i(mapping)g(for)f(KA14)191 554 y Fg(a)235 524 y Fn(?)309 554 y Fh([)19 b Fg(b)k Fh(!)525 566 y Ff(3)585 554 y Ft(\()p Fg(a)g Fh(!)767 566 y Ff(3)828 554 y Ft(\()p Fg(a)g Fh(!)1010 566 y Ff(3)1070 554 y Fg(b)p Ft(\)\))h(=)e(1)25 701 y(violates)28 b(the)h(non-orthomo)r (dular)e(ortholattice)g(of)i(Fig.)g(3)f(with)i(the)f(assignmen)n(t)f Fg(a)d Ft(=)f Fg(y)s Ft(,)29 b Fg(b)c Ft(=)g Fg(w)r Ft(.)3198 671 y Ff(3)3276 701 y Ft(\(The)30 b(mapping)e(for)25 826 y(KA14)e(is)i(of)f(course)g(true)g(in)h(all)g(W)n(OM)f(lattices.\)) 1891 2021 y Fc(@)1808 1938 y(@)1725 1855 y(@)1642 1772 y(@)p 1972 2021 4 333 v 1974 2021 a(\000)2057 1938 y(\000)2140 1855 y(\000)2223 1772 y(\000)p 1640 1689 4 167 v 1642 1689 a(\000)1725 1606 y(\000)1808 1523 y(\000)1891 1440 y(\000)1891 1689 y(@)1808 1606 y(@)1725 1523 y(@)1642 1440 y(@)1974 1689 y(\000)2057 1606 y(\000)2140 1523 y(\000)2223 1440 y(\000)2223 1689 y(@)2140 1606 y(@)2057 1523 y(@)1974 1440 y(@)p 2304 1689 V 1640 1523 V 2304 1523 V 1642 1357 a(\000)1725 1274 y(\000)1808 1191 y(\000)1891 1108 y(\000)p 1972 1357 4 333 v 2223 1357 a(@)2140 1274 y(@)2057 1191 y(@)1974 1108 y(@)1953 2116 y Ft(0)1553 1707 y Fg(x)1943 1648 y(w)2347 1724 y(z)2390 1694 y Fn(?)1556 1533 y Fg(y)2347 1550 y(y)2391 1520 y Fn(?)1558 1375 y Fg(z)1915 1486 y(w)1976 1455 y Fn(?)2347 1392 y Fg(x)2394 1362 y Fn(?)1953 983 y Ft(1)1974 2021 y Fb(r)1642 1689 y(r)307 b(r)g(r)1642 1523 y(r)639 b(r)1642 1357 y(r)307 b(r)g(r)1974 1025 y(r)1018 2287 y Ft(Figure)27 b(3:)37 b(Ortholattice)26 b(violated)h(b)n(y)h(mapping)f(of)g(KA14)191 2502 y(The)37 b(mapping)f(for)g(KA15)f(is)i(true)f(in)h(all)f (ortholattices:)54 b(from)36 b Fg(a)2390 2472 y Fn(?)2470 2502 y Fh(\\)25 b Fg(b)38 b Fh(\024)f Fg(b)p Ft(,)i Fg(a)2868 2472 y Fn(?)2948 2502 y Fh(\\)25 b Fg(b)3064 2472 y Fn(?)3158 2502 y Fh(\024)37 b Fg(a)3304 2472 y Fn(?)3397 2502 y Ft(it)g(follo)n(ws)e(that)25 2626 y(\()p Fg(a)101 2596 y Fn(?)160 2626 y Fh(\\)s Fg(b)p Ft(\))s Fh([)s Ft(\()p Fg(a)423 2596 y Fn(?)483 2626 y Fh(\\)s Fg(b)577 2596 y Fn(?)633 2626 y Ft(\))23 b Fh(\024)g Fg(a)820 2596 y Fn(?)878 2626 y Fh([)s Fg(b)p Ft(;)g(this)d(and)g Fg(a)s Fh(\\)s Ft(\()p Fg(a)1507 2596 y Fn(?)1566 2626 y Fh([)s Fg(b)p Ft(\))j Fh(\024)g Fg(a)1847 2596 y Fn(?)1906 2626 y Fh([)s Fg(b)d Ft(imply)g Fg(a)j Fh(!)2395 2638 y Ff(3)2455 2626 y Fg(b)g Fh(\024)f Fg(a)2645 2596 y Fn(?)2704 2626 y Fh([)s Fg(b)p Ft(.)35 b(Th)n(us)19 b(\()p Fg(a)k Fh(!)3240 2638 y Ff(3)3301 2626 y Fg(b)p Ft(\))3369 2596 y Fn(?)3428 2626 y Fh([)s Ft(\()p Fg(a)3562 2596 y Fn(?)3621 2626 y Fh([)s Fg(b)p Ft(\))g(=)g(1.)25 2751 y(Since)29 b(KMP3)f(follo)n(ws)g (from)g(KMP0)g(applied)h(t)n(wice)f(to)h(KA15,)f(and)h(KMP0)f(is)h (sound)f(in)h(all)g(ortholattices,)f(KMP3)g(is)25 2876 y(also)e(sound)i(in)g(all)f(ortholattices.)191 3091 y(Giv)n(en)g(our)f (result)g(that)h(the)h(orthomo)r(dular)d(la)n(w)h(is)h(needed)g(only)f (for)h(KA2)f(and)h(KA4)f(in)h(the)g(algebra)f(for)g(system)25 3215 y(K0,)j(w)n(e)g(next)h(sho)n(w)f(that)h(it)g(can)f(b)r(e)h (eliminated)g(from)f(one)g(of)h(them.)43 b(W)-7 b(e)30 b(de\014ne)g(a)f(new)h(system,)g(K0)3387 3185 y Fn(0)3409 3215 y Ft(,)h(as)d(consisting)25 3340 y(of)f(KA3{KA10,)f(KA12,)h(KA13,) f(and)i(KMP0,)e(along)g(with)i(t)n(w)n(o)f(axioms)25 3522 y Fu(Axioms.)191 3692 y(KA2a.)254 b Fh(`)23 b Ft(\()p Fg(A)c Fh(_)g Fg(C)29 b Fh(\021)23 b Fg(B)f Fh(_)d Fg(C)6 b Ft(\))24 b Fh(\021)e Ft(\()p Fg(C)j Fh(_)19 b Fg(A)24 b Fh(\021)e Fg(C)j Fh(_)19 b Fg(B)t Ft(\))191 3839 y Fu(KA2b.)254 b Fh(`)23 b Ft(\()p Fg(A)c Fh(_)f Fg(C)30 b Fh(\021)22 b Fg(B)h Fh(_)c Fg(C)6 b Ft(\))23 b Fh(\021)g Ft(\()p Fh(:)p Ft(\()p Fh(:)p Fg(A)d Fh(^)f(:)p Fg(C)6 b Ft(\))24 b Fh(\021)e(:)p Ft(\()p Fh(:)p Fg(B)i Fh(^)19 b(:)p Fg(C)6 b Ft(\)\))25 4022 y(The)24 b(mappings)g(of)g(all)g(axioms) f(of)h(K0)1240 3991 y Fn(0)1262 4022 y Ft(,)h(with)g(the)f(exception)g (of)g(KA4,)g(are)f(true)h(in)h(all)f(ortholattices.)34 b(Th)n(us)24 b(KA4)f(alone)25 4146 y(con)n(tains)f(the)h(essen)n(tial)f (non-ortholattice)f(c)n(haracter)g(of)i(the)g(algebraic)e(seman)n(tics) h(for)g(K0)2947 4116 y Fn(0)2970 4146 y Ft(,)i(and)e(b)n(y)h(the)g (next)g(theorem,)25 4271 y(also)j(for)h(K0.)25 4498 y Fu(Theorem)j(2.13.)40 b Fv(Systems)29 b Ft(K0)g Fv(and)h Ft(K0)1390 4468 y Fn(0)1442 4498 y Fv(ar)l(e)h(lo)l(gic)l(al)t(ly)h(e)l (quivalent.)25 4680 y(Pr)l(o)l(of.)43 b Ft(The)29 b(pro)r(ofs)f(of)g (KA2a)g(and)h(KA2b)f(in)h(system)g(K0)e(are)h(straigh)n(tforw)n(ard.)38 b(The)28 b(deriv)-5 b(ation)29 b(of)f(K0)g(from)h(K0)3816 4650 y Fn(0)3867 4680 y Ft(is)25 4805 y(more)e(in)n(v)n(olv)n(ed.)191 4929 y(\(i\))h(F)-7 b(rom)27 b(KA4)g(and)h(KMP0)e(w)n(e)h(ha)n(v)n(e)g (that)h Fh(`)23 b Fg(A)g Fh(\021)g Fg(B)31 b Ft(implies)d Fh(`)23 b Fg(A)c Fh(^)f Fg(C)30 b Fh(\021)22 b Fg(B)h Fh(^)c Fg(C)6 b Ft(.)191 5054 y(\(ii\))21 b(F)-7 b(rom)20 b(\(i\),)i(KA2a,)f(KA2b,)g(KA3,)g(KA13,)g(and)f(KMP0,)g(w)n(e)g(ha)n(v) n(e)f(that)i Fh(`)h Fg(A)i Fh(\021)e Fg(B)j Ft(implies)20 b(b)r(oth)h Fh(`)i Fg(A)t Fh(_)t Fg(C)29 b Fh(\021)23 b Fg(B)8 b Fh(_)t Fg(C)25 5179 y Ft(and)27 b Fh(`)c Fg(C)i Fh(_)19 b Fg(A)k Fh(\021)g Fg(C)h Fh(_)19 b Fg(B)t Ft(.)p 25 5227 1560 4 v 117 5281 a Fi(3)152 5304 y Fo(The)28 b(authors)g(are)f(grateful)g(to)h(William)d(McCune)j(of)f(Argonne)h (National)g(Lab)r(oratory)-6 b(,)29 b(Argonne)f(IL,)f(USA)g(for)g (\014nding)h(this)f(lattice,)25 5383 y(using)d(the)g(matrix-\014nding)f (program)f(MA)n(CE)h(\()p 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Fh(\021)39 b Fg(C)6 b Ft(\).)68 b(F)-7 b(rom)38 b(this,)i Fh(`)g Fg(B)k Fh(\021)c Fg(C)6 b Ft(,)40 b(KA12,)g(KA13,)f(and)e(KMP0,)j(w)n (e)d(obtain)25 567 y Fh(`)23 b Ft(\()p Fg(A)6 b Fh(^)g Fg(C)g Ft(\))g Fh(_)g Ft(\()p Fh(:)p Fg(B)k Fh(^)c(:)p Fg(C)g Ft(\).)36 b(F)-7 b(rom)20 b Fh(`)j Fg(A)h Fh(\021)e Fg(B)t Ft(,)h(KA3,)f(KMP0,)f(\(i\),)i(and)e(\(ii\))h(w)n(e)f(obtain)g Fh(`)i Ft(\()p Fg(A)g Fh(\021)g Fg(C)6 b Ft(\))24 b Fh(\021)e Ft(\()p Fg(A)6 b Fh(^)g Fg(C)g Ft(\))g Fh(_)g Ft(\()p Fh(:)p Fg(B)k Fh(^)c(:)p Fg(C)g Ft(\).)25 692 y(F)-7 b(rom)27 b(these)h(t)n(w)n(o,)f(KA12,)f(KA13,)h(and)g(KMP0,)g(w)n(e)g (obtain)g Fh(`)c Fg(A)g Fh(\021)g Fg(C)6 b Ft(.)191 816 y(\(iv\))28 b(KA1)f(follo)n(ws)g(from)g(\(iii\),)h(KA9,)g(KA12,)e (KA13,)h(and)g(KMP0.)191 941 y(\(v\))32 b(W)-7 b(e)32 b(can)g(use)g(\(iii\))g(in)g(place)g(of)g(KA2)f(in)h(the)g(w)n(ork)f (of)g(Theorem)g(2.6)g(to)h(construct)g(pro)r(ofs)f(b)n(y)g (\\ortholattice)25 1066 y(analogy)-7 b(.")35 b(In)27 b(particular,)g(w)n(e)g(can)g(pro)n(v)n(e)f(KA11.)191 1190 y(\(vi\))k(Ha)n(ving)f(KA11)f(a)n(v)-5 b(ailable)28 b(as)h(an)h(analog)e(to)h(the)h(orthomo)r(dular)e(la)n(w,)h(w)n(e)g (can)h(further)f(construct)g(pro)r(ofs)g(b)n(y)25 1315 y(\\orthomo)r(dular)c(lattice)j(analogy)-7 b(.")35 b(In)27 b(particular)g(w)n(e)g(can)g(pro)n(v)n(e)f(the)i(analog)e(of)i(the)g (algebra)d(mapping)j(for)f(KA2,)440 1457 y Fh(`)c Ft(\()p Fg(A)g Fh(\021)g Fg(B)k Fh(!)892 1469 y Ff(0)953 1457 y Ft(\()p Fg(B)g Fh(\021)c Fg(C)29 b Fh(!)1334 1469 y Ff(0)1394 1457 y Fg(A)24 b Fh(\021)e Fg(C)6 b Ft(\)\))24 b Fh(\021)f Ft(1)25 1600 y(b)n(y)k(analogy)f(to)h(the)h(pro)r(of)f(in)h (Ref.)g([12)o(].)191 1724 y(\(vii\))g(W)-7 b(e)28 b(pro)n(v)n(e)e Fh(`)d Ft(1)k(as)g(in)h(Theorem)f(2.6.)36 b(Detac)n(hing)27 b(it)h(from)f(\(vi\),)h(w)n(e)g(deriv)n(e)e(KA2.)p 3862 1724 4 57 v 3866 1672 50 4 v 3866 1724 V 3915 1724 4 57 v 25 1983 a Fu(Theorem)k(2.14.)40 b Fv(Axiom)30 b Ft(W2)f Fv(for)i(WOM)f(lattic)l(es)g(is)g(r)l(e)l(dundant.)25 2147 y(Pr)l(o)l(of.)43 b Ft(W2)29 b(is)g(the)g(mapping)g(for)f(KA2.)41 b(By)28 b(analogy)f(to)i(the)h(pro)r(of)e(of)h(Theorem)f(2.13)g ([through)g(step)h(\(vi\)])h(in)f(ortho-)25 2271 y(lattices,)e(w)n(e)h (can)f(pro)n(v)n(e)f(W2)h(from)h(W1)f(\(the)h(mapping)g(for)f(KA4\).)p 3862 2271 V 3866 2219 50 4 v 3866 2271 V 3915 2271 4 57 v 191 2456 a(W)n(OM)g(lattices)h(are)e(c)n(haracterized)g(b)n(y)h(a) g(somewhat)g(simpler)h(la)n(w,)e(as)h(the)h(follo)n(wing)f(theorem)g (sho)n(ws.)25 2655 y Fu(Theorem)j(2.15.)40 b Fv(A)n(n)29 b(ortholattic)l(e)i(in)f(which)h(the)f(fol)t(lowing)i(law)191 2798 y Fu(WR1.)168 b Fg(a)23 b Fh(\021)g Fg(b)g Ft(=)f(1)29 b Fv(implies)j Fg(a)18 b Fh([)h Fg(c)k Fh(\021)f Fg(b)d Fh([)f Fg(c)23 b Ft(=)g(1)25 2940 y Fv(holds)31 b(is)f(a)g(WOM)g (lattic)l(e)g(and)g(vic)l(e-versa.)25 3104 y(Pr)l(o)l(of.)43 b Ft(This)31 b(la)n(w)g(follo)n(ws)f(directly)h(from)g(W1)g(\(w)n(e)h (obtain)f(the)g Fh([)h 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Ft(=)f(1)30 b Fv(implies)h Fg(a)18 b Fh([)h Fg(c)k Fh(\021)g Fg(b)18 b Fh([)h Fg(c)k Fh(\025)f Fg(a)25 4271 y Fv(holds)31 b(is)f(a)g(WOM)g(lattic)l(e)g(and) g(vic)l(e-versa.)25 4434 y(Pr)l(o)l(of.)43 b Ft(This)29 b(la)n(w)f(follo)n(ws)f(immediately)i(from)g(WR1.)40 b(Con)n(v)n(ersely)-7 b(,)27 b(using)h(this)h(la)n(w)f(t)n(wice)h(w)n (e)f(obtain)h(that)g Fg(a)24 b Fh(\021)h Fg(b)g Ft(=)f(1)25 4559 y(implies)32 b Fg(a)20 b Fh([)i Fg(b)29 b Fh(\024)g Fg(a)21 b Fh([)g Fg(c)30 b Fh(\021)f Fg(b)20 b Fh([)i Fg(c)p Ft(.)48 b(WR1)32 b(follo)n(ws)e(from)h(this)h(and)f(the)h(follo) n(wing)f(equation,)h(true)f(in)h(all)f(ortholattices,)25 4683 y(whose)c(pro)r(of)g(w)n(e)g(lea)n(v)n(e)f(to)i(the)g(reader:)357 4826 y(\()p Fg(a)19 b Fh([)f Fg(b)p Ft(\))h Fh([)g Ft(\()p Fg(a)f Fh([)h Fg(c)k Fh(\021)g Fg(b)18 b Fh([)g Fg(c)p Ft(\))24 b(=)e(1)p 3862 4826 V 3866 4773 50 4 v 3866 4826 V 3915 4826 4 57 v 25 5010 a(By)27 b(a)g(similar)g(pro)r(of)g(w)n (e)g(can)h(also)e(sho)n(w)h(that)191 5152 y Fu(W1)338 5122 y Fn(0)361 5152 y Fu(.)165 b Ft(\()p Fg(a)23 b Fh(\021)g Fg(b)p Ft(\))808 5122 y Fn(?)883 5152 y Fh([)18 b Ft(\()p Fg(a)h Fh([)g Fg(c)k Fh(\021)f Fg(b)c Fh([)h Fg(c)p Ft(\))24 b Fh(\025)e Fg(a)25 5295 y Ft(can)27 b(replace)g(W1.)p eop %%Page: 10 10 10 9 bop 25 -257 a Ft(M.)28 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c)24 b(and)j(N.)h(Megill,)55 b Fv(Binary)31 b(Ortholo)l(gic)g(with)g(Mo)l (dus)f(Ponens)1439 b Ft(10)25 33 y Fs(3)135 b Fj(Binary)37 b(Orthologic:)47 b(Mo)s(dus)39 b(P)m(onens)f(Rule)e(Determines)g (Logics)25 240 y Ft(In)20 b(this)g(section)g(w)n(e)g(use)g(binary)f (orthologic,)h(as)f(an)h(e\016cien)n(t)g(form)n(ulation)f(of)h (implicational)g(logic,)g(in)h(order)d(to)i(form)n(ulate)25 364 y(the)g(appropriate)e(mo)r(dus)h(p)r(onens)g(rule.)34 b(W)-7 b(e)20 b(then)g(sho)n(w)f(that)h(this)f(mo)r(dus)h(p)r(onens)f (rule)h(turns)f(orthologic)e(in)n(to)j(quan)n(tum)25 489 y(logic.)35 b(Thereup)r(on)26 b(w)n(e)g(sho)n(w)f(that)i(binding)f 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b(aluations)29 b Fg(v)k Ft(on)d(all)g(orthomo)r (dular)e(\(distributiv)n(e\))i(lattices.)44 b(By)30 b(means)f(of)h (this)g(c)n(haracterization)e(quan)n(tum)25 1236 y(implications)f (determine)h(quan)n(tum)f(logic)g(and)h(the)g(classical)e(implication)h (classical)f(logic,)h(in)h(the)g(follo)n(wing)f(w)n(a)n(y:)25 1465 y Fu(Theorem)j(3.1)h(\(P)m(a)m(vi)m(\024)-45 b(ci)m(\023)g(c,)33 b(1987\).)40 b Fv(A)n(n)29 b(ortholo)l(gic)j(in)e(which)191 1612 y Fh(`)23 b Fg(A)g Fh(!)433 1624 y Fe(i)484 1612 y Fg(B)197 b Fh(,)c Fg(A)23 b Fh(`)g Fg(B)25 1760 y Fv(holds)31 b(is)f(a)g(classic)l(al)i(lo)l(gic)f(for)f Fg(i)23 b Ft(=)g(0)29 b Fv(and)h(a)g(quantum)f(lo)l(gic)i(for)f Fg(i)23 b Ft(=)g(1)p Fg(;)14 b(:)g(:)g(:)f(;)h Ft(5)p Fv(.)38 b(Her)l(e,)30 b Fh(`)23 b Fg(A)30 b Fv(me)l(ans)f Fg(C)c Fh(_)19 b(:)p Fg(C)29 b Fh(`)23 b Fg(A)p Fv(.)191 2034 y Ft(W)-7 b(e)28 b(also)e(ha)n(v)n(e)h(a)g(corresp)r(onding)f (result)h(for)g(lattices:)25 2263 y Fu(Theorem)j(3.2)h(\(P)m(a)m(vi)m (\024)-45 b(ci)m(\023)g(c,)33 b(1993\).)40 b Fv(A)n(n)29 b(ortholattic)l(e)i(in)f(which)191 2411 y Fg(a)23 b Fh(!)341 2423 y Fe(i)392 2411 y Fg(b)41 b Ft(=)g(1)192 b Fh(,)h Fg(a)23 b Fh(\024)g Fg(b)25 2558 y Fv(holds)31 b(is)f(a)g(distributive) h(lattic)l(e)f(for)h Fg(i)22 b Ft(=)h(0)29 b Fv(and)i(an)e(orthomo)l (dular)j(lattic)l(e)e(for)g Fg(i)23 b Ft(=)g(1)p Fg(;)14 b(:)g(:)g(:)f(;)h Ft(5)p Fv(.)191 2741 y Ft(Theorem)27 b(3.1)f(enables)i(us)f(to)h(axiomatize)e(classical)g(and)i(quan)n(tum)f (logic)g(in)h(the)g(follo)n(wing)e(w)n(a)n(y:)25 2924 y Fu(Axioms.)191 3072 y(A1.)254 b Fg(A)23 b Fh(`)g(::)p Fg(A)191 3219 y Fu(A2.)254 b Fh(::)p Fg(A)24 b Fh(`)f Fg(A)191 3366 y Fu(A3.)254 b Fg(A)23 b Fh(`)g Fg(A)c Fh(_)g Fg(B)191 3514 y Fu(A4.)254 b Fg(B)27 b Fh(`)c Fg(A)c Fh(_)f Fg(B)191 3661 y Fu(A5.)254 b Fg(B)27 b Fh(`)c Fg(A)c Fh(_)f(:)p Fg(A)25 3936 y Fu(Rules)30 b(of)i(Inference.) 191 4106 y(R1.)253 b Fg(A)24 b Fh(`)f Fg(B)160 b Fh(\))108 b(:)p Fg(B)27 b Fh(`)c(:)p Fg(A)191 4254 y Fu(R2.)253 b 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b(and)i(S.)g(Stenholm,)37 b(Ph)n(ys.)26 b(Rev.)i(A)g Fu(54)p Ft(,)f(4701)f(\(1996\).)66 1660 y([4])42 b(A.)28 b(Barenco)e(et)h(al.,)37 b(Ph)n(ys.)26 b(Rev.)i(A)g Fu(52)p Ft(,)f(3457)f(\(1995\).)66 1851 y([5])42 b(E.)27 b(Knill)g(and)h(R.)g(La\015amme,)36 b(Ph)n(ys.)26 b(Rev.)i(A)g Fu(55)p Ft(,)f(900)f(\(1997\).)66 2042 y([6])42 b(A.)28 b(Ek)n(ert)e(and)h(R.)h(Jozsa,)35 b(Rev.)28 b(Mo)r(d.)g(Ph)n(ys.)e Fu(68)p Ft(,)i(733)e(\(1996\).)66 2233 y([7])42 b(H.)29 b(Leblanc,)41 b(Alternativ)n(es)29 b(to)g(standard)f(\014rst-order)f(seman)n(tics,)41 b(in)29 b Fv(Handb)l(o)l(ok)j(of)g(Philosophic)l(al)k(L)l(o)l(gic)p Ft(,)30 b(edited)196 2358 y(b)n(y)j(D.)h(Gabba)n(y)f(and)g(F.)h(Guen)n (thner,)h(v)n(olume)e(I:)h(Elemen)n(ts)f(of)g(Classical)f(Logic)h(of)g Fv(Synthese)j(Libr)l(ary;)j(v.)d(164)p Ft(,)196 2482 y(c)n(hapter)26 b(I.3,)i(pages)e(189{274,)e(D.)29 b(Reidel,)e(Dordrec)n (h)n(t,)g(Holland,)g(1983.)66 2673 y([8])42 b(C.)27 b(G.)h(Morgan,)35 b(Logique)27 b(et)h(Analyse)f Fu(26)p Ft(,)g(323)f(\(1983\).)66 2864 y([9])42 b(S.)27 b(S.)h(Holland,)g(JR.,)36 b(Bull.)28 b(Am.)g(Math.)g(So)r(c.)g Fu(32)p Ft(,)f(205)f(\(1995\).)25 3055 y([10])41 b(M.)30 b(Fitting,)45 b Fv(First-Or)l(der)32 b(L)l(o)l(gic)h(and)g(A)n(utomate)l(d)e(The)l(or)l(em)i(Pr)l(oving)p Ft(,)46 b(T)-7 b(exts)30 b(and)g(Monographs)e(in)i(Computer)196 3180 y(Science)d(XIV,)h(Springer-V)-7 b(erlag,)26 b(New)h(Y)-7 b(ork,)27 b(1990.)25 3370 y([11])41 b(M.)27 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c,)33 b(In)n(t.)28 b(J.)g(Theor.)e(Ph)n(ys.)h Fu(32)p Ft(,)g(1481)f(\(1993\).)25 3561 y([12])41 b(G.)27 b(Kalm)n(bac)n(h,)36 b Fv(Orthomo)l(dular)30 b(L)l(attic)l(es)p Ft(,)37 b(L.)28 b(M.)g(S.)f(Monographs;)f(v.)h(18,)g(Academic)h(Press,) e(London,)h(1983.)25 3752 y([13])41 b(G.)27 b(M.)h(Hardegree,)35 b(Notre)27 b(Dame)h(J.)g(F)-7 b(ormal)26 b(Logic)h Fu(22)p Ft(,)g(163)f(\(1981\).)25 3943 y([14])41 b(J.)27 b(Kotas,)35 b(Studia)28 b(Logica)e Fu(27)p Ft(,)i(73)e(\(1971\).)25 4134 y([15])41 b(R.)27 b(I.)h(Goldblatt,)37 b(J.)28 b(Phil.)f(Logic)g Fu(3)p Ft(,)g(19)g(\(1974\).)25 4325 y([16])41 b(M.)27 b(P)n(a)n(vi)n(\024)-39 b(ci)n(\023)g(c,)33 b(In)n(t.)28 b(J.)g(Theor.)e(Ph)n(ys.)h Fu(26)p Ft(,)g(845)f(\(1987\).)25 4516 y([17])41 b(G.)27 b(M.)h(Hardegree,)35 b(J.)28 b(Phil.)f(Logic)g Fu(4)p Ft(,)g(399)f(\(1975\).)25 4707 y([18])41 b(H.)28 b(Dishk)-5 b(an)n(t,)37 b(Rep.)27 b(Math.)h(Logic)f Fu(3)p Ft(,)g(9)g(\(1974\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF