Isomorphic Formulae in Classical Propositional Logic

Isomorphic F orm ulae in Classical Prop ositional Logic Ko st a Do ˇ sen and Zo ran Petri ´ c Mathematical Institute, SANU Knez Mihailov a 36, p.f. 367, 11001 Belgrade, S erbia email: { ko sta, zp etric } @mi.san u.ac.rs Abstract Isomorphism betw een form ulae is defined with respect to categories for- malizing equ ality of deductions in classical p ropositional logic and in the multipli cative fragmen t of classica l linear prop ositional logic caugh t by proof nets. This equality is motiv ated b y generality of deductions. Char- acterizations are given for p airs of isomorphic form ulae, which lead to decision pro cedures for this isomorphism. Mathematics Subje ct Classific ation (2010): 03F03, 03F07, 03F52, 03G30 Keywor ds: isomorphic formulae, classical propositional logic, classical linear prop osi- tional logic, categories, equality o f dedu ctions, iden tity of proofs, categorial coherence 1 In tro duction Isomorphism betw een formulae should b e an equiv alence r elation stronge r tha n m utual implication. This is presumably the r elation underly ing the relation tha t holds betw een prop ositions that have the same meaning just b ecause o f their logical form. Any prop ositions that are instances, with the same substitution, of isomorphic form ulae w ould hav e the same meaning, whic h presumably need not b e the ca se for formulae that ar e just equiv alen t, i.e. which just imply ea c h other. One ma y try to characterize isomor phic formulae b y lo o king only into the in- ner s tr ucture o f formulae . This is the wa y envisaged by Carna p and Ch urch (see [4], Sections 14-1 5 , wher e r elated w ork by Quine and C.I. Lewis is men tioned, [5], [1], Section 2 , and references therein). Another wa y is to tr y to characterize isomor phis m be t ween formulae by lo oking a lso at the outer structure in which formulae ar e to b e found. This outer str ucture may b e a deductive structure , which is ch ar a cterized in terms of ca tegories in categor ial pro of theo r y . The categor ies we need a r e syntactical: their ob jects are formulae a nd their arrows are deductio ns . 1 Isomorphism b etw een fo r m ulae ma y then b e understo o d exactly as iso mo r- phism b et ween ob jects is understo o d in catego ry theo ry . The formulae A and B ar e isomor phic when there is a deduction f , i.e. a rrow, from A to B , a nd another deduction g fr o m B to A , suc h that f comp osed with g is eq ua l to the ident ity deduction from A to A , while g comp osed with f is equal to the iden tity deduction from B to B . This analys is of isomorphism presuppo ses a notion of equality b etw een deductions, which is formalized in our syntactical ca tegories. ( Equality b etwe en de ductions s tands here for what we and other author s hav e called els ewhere identity of pr o ofs ; see [8 ], [13], Sections 1 .3-1.4, and references therein.) Characterizing this notion of e qualit y betw een deductions is the main task of categorial pr oo f theo r y , and of g e neral pro of theor y . That A and B are is o morphic means here in tuitively that they function in the s ame manner in deductions. In a deduction one can replace one by the other, either as premise o r a s conclusio n, so that nothing is lost, nor gained. The re- placements, which are made by comp osing our deduction with the deductions f and g , ar e such that they enable us to r eturn to our o riginal deduction by further comp osing with g a nd f , s ince f comp osed with g a nd g compos ed with f a re identit y deductions, and hence may be cance lled. (F or a view concern- ing isomorphic formulae like the one presented here, a nd its relations hip with prop ositional identit y , see [6], Section 9, and [7], Section 5.) The study of isomorphic formulae first started in intuitionistic lo gic, for which it is widely believed that w e have a solid nontrivial notion o f equality of deductions. This notion is characterized either in terms of the typed lambda calculus (via the Curry-Howard corresp ondence), or in terms o f categories based on ca rtesian clo sed categ o ries (these characterizatio ns may be equiv alen t; see [19]). A result exists in this area for the implication-conjunction- ⊤ fragment of int uitionistic logic (se e [2 1]). As far as we know, the latest a dv a nce s concerning the still op en pro blem of c hara cterizing for mulae isomor phic in the whole of in- tuitionistic propo sitional logic (whic h is rela ted to T arski’s high-s c ho ol algebra problem; see [3]) were made in [2 ] and [16]. Ther e is a rela ted r esult characteriz- ing iso mo rphic fo r m ulae in the analogo us multiplicativ e frag men t of linear logic, which co rresp onds to symmetric monoidal clo s ed categor ies, and is common to classical and in tuitionistic linear logic (see [22] and [9]). The problem o f characterizing iso morphic formulae w as not a pproached up to now in classical prop ositional log ic, and the results we ar e going to present here cover this logic. They cover als o a fragment of classic a l linear prop ositional logic. T o b e able to a pproach our problem, and to obtain significa n t r esults, we need for the logic s we wan t to cov er a plausible and nontrivial notion of equality of ar r o ws in catego ries for malizing equality of deductions in these log- ics. A co ns ensus for class ical linear prop ositional logic may b e found aro und the mult iplicative fra gmen t of that logic c a ugh t by pro of nets, which leads to notions of ca tegory clo sely r elated to star-a utonomous categor y (see [20], [14] and references therein). 2 F or class ical prop ositional logic, it is on the contrary widely b elieved that no nontrivial notion o f ca teg ory would do the job. It is b elieved that no nontrivial notion o f Bo olean categor y may b e found. This is indeed the case if one wan ts these Bo olean c a tegories to b e cartesian close d (s e e [8], Section 5, [13], Section 14.3, and references therein). But, whereas on the level of theorems classical logic is an extension of intuitionistic logic , it is not clear that this should b e so at the level of deductions and of their equa lit y . If one do es not require that Bo olean ca tegories be cartes ian clo sed cate- gories, and bases equality of deductio ns in Bo olean categor ies on coher ence results analogous to those av ailable for classical linea r prop ositional log ic, a nontrivial notion may a rise. The coher ence res ults in ques tion a r e c a tegorial results analog ous to the clas s ical coher ence result o f Ke lly and Mac Lane for symmetric mono idal closed catego r ies (s e e [17]). They reduce equa lit y of arrows in the s yn tactical categor y to equalit y of ar rows in a graphical mo del catego ry . Such a nontrivial no tio n of Bo olean category may b e found in [13] (Chapter 14), and Section 4 of the pr esen t pap er deals with isomo rphism o f formulae engendered by that par ticular notion. Section 3 of the pap er deals with isomor - phism of formulae in class ical and classic a l linear prop ositional logic different from the notion of Section 4 . That other notion, which in volv es graphical mo del categorie s quite like those of Kelly and Ma c L a ne, is motiv ated by g enerality of deductions (as sug gested by [18]). The notion of Section 4 may also be un- dersto o d as inv olving g e ne r alit y up to a p oin t, but the notion of Sectio n 3 do es so more cons isten tly . Both notions are howev er analo g ous in that they base equality of deductions on equality o f arrows in so me g raphical mo del categor y . The ma in results of the pap er in Sectio ns 3 and 4 are based on some pre- liminary elementary re s ults co ncerning classical prop ositional logic, which are established in Section 2, and o ccasionally la ter in the pap er. Although these results are no t difficult to reach, when they are com bined with mor e adv anced results, s uc h as those that may be found in [13], they give a complete c hara cteri- zation of isomorphic formu lae in classica l and classica l linear prop ositional logic. These characteriza tions a re such that they easily lea d to decision pro cedures for the isomor phisms in question. This pap er is devoted to the problem o f c hara cterizing pair s of isomo rphic formulae. A rela ted, but different, problem in volving isomor phism is to charac- terize arr o ws that are isomorphisms, in categor ies for malizing equalit y of deduc- tions. (This problem for the conjunction- ⊤ frag men t of classical or in tuitionistic logic is dealt with in [10].) Although we hav e not dealt with that s econd prob- lem explicitly in this pap er, a solution for it may easily b e inferred from o ur results. W e will how ever not dwell on that, in order not to overburden the text with categoria l matters. In the whole pap er w e try to keep the presence of c ategories to a minim um, a nd give more pro minence to elementary , easily under standable, logical facts. T o appre ciate the full imp ort of o ur results the reader should how ever b e a c quain ted up to a point with certa in notions covered in par ticular 3 by [13] (and whic h we cannot p ossibly exp ose all her e). 2 ∧∨ and ¬ ∧ ∨ -equiv ale nces Let L b e a prop ositional language genera ted out of an infinite set o f pr opos itional letters, which w e call s imply letters , with the nullary connectives ⊤ and ⊥ , the unary connective ¬ , and the binary connectives ∧ a nd ∨ . W e use p, q, r , . . . , sometimes with indices, fo r letters, and A, B , C , . . . , s ometimes with indices, for the fo rm ulae of L . Let L ∧ , ∨ , L ¬ , ∧ , ∨ and L ⊤ , ⊥ , ∧ , ∨ be the prop ositional langua ges defined as L but with only the connectives in the subscripts. W e envisage also prop ositiona l lang uages extending L , in which we hav e moreov er the binary connectives of equiv alence ↔ and implication → . W e may imagine that these t wo additional connectives are defined in L , and we shall not int ro duce sp ecial na mes for these e xtended language s . Let a ∧∨ -e quivalenc e b e a formula A ↔ B wher e A and B ar e form ulae of L ∧ , ∨ . Consider the formal system S ∧ , ∨ whose axioms are the ∧∨ -equiv alences of the follo wing forms: A ↔ A , (( A ∧ B ) ∧ C ) ↔ ( A ∧ ( B ∧ C )), (( A ∨ B ) ∨ C ) ↔ ( A ∨ ( B ∨ C )), ( A ∧ B ) ↔ ( B ∧ A ), ( A ∨ B ) ↔ ( B ∨ A ), and whose theorems are the ∧∨ -equiv alences obtained s tarting from these ax- ioms with the follo wing rule s : A ↔ B B ↔ A A ↔ B B ↔ C A ↔ C A ↔ B C ↔ D ( A ∧ C ) ↔ ( B ∧ D ) A ↔ B C ↔ D ( A ∨ C ) ↔ ( B ∨ D ) A formula is diversifie d when ev ery letter o ccurs in it not mo re tha n once. A ∧∨ -equiv alence is diversifie d when A and B ar e diversified. Assume that let A is the set of letters o ccurring in A , and let A p B be the r esult of substituting the formula B for every o ccurrence of p in A . W e can prov e the following lemmata . Lemma 1. Assu me that A is a diversifie d formula of L ∧ , ∨ , that B is a su b- formula of A , and t hat let A − let B = { p 1 , . . . , p n } (wher e { p 1 , . . . , p n } is empty if n = 0 ). Then ther e is a se quenc e S 1 , . . . , S n , wher e S i ∈ {⊤ , ⊥} , such t hat A p 1 ...p n S 1 ...S n ↔ B is a tautolo gy . Proof. W e pr oceed by induction o n n . If n = 0, then B is A , and A ↔ B is of cour se a tauto lo gy . If n = k +1 , and A is of the for m C ∧ D with B a subfor m ula of C , then for let C − let B = { q 1 , . . . , q m } , where { q 1 , . . . , q m } ∪ { r 1 , . . . , r l } = { p 1 , . . . , p n } for 4 m ≥ 0 and l ≥ 1 , w e have by the inductio n hypothesis that fo r so me S 1 , . . . , S m the formula C q 1 ...q m S 1 ...S m ↔ B is a tautology . Hence A q 1 ...q m r 1 ...r l S 1 ...S m ⊤ ... ⊤ ↔ B is a tautology to o. W e proce ed analo gously when B is a subformula of D , or when A is of the form C ∨ D . In the latter ca se w e substitute ⊥ , . . . , ⊥ fo r r 1 , . . . , r l . ⊣ Lemma 2. If B is a diversifie d formula of L ∧ , ∨ that has a letter q not in the formula A of L , t hen A ↔ B is not a tau t olo gy. Proof. By Lemma 1, for some S 1 , . . . , S n , where S i ∈ {⊤ , ⊥} , we hav e that B p 1 ...p n S 1 ...S n ↔ q is a tautolo gy . On the other hand, A p 1 ...p n S 1 ...S n ↔ q cannot b e a tautology . So A p 1 ...p n S 1 ...S n ↔ B p 1 ...p n S 1 ...S n is not a tautolo g y , and hence A ↔ B is not a tautology . ⊣ F or A a diversified formula of L ∧ , ∨ we say that the letters p and q are c onjunctively joine d in A when A has a subformula P ∧ Q or Q ∧ P suc h that p is in P and q is in Q , and we say that p a nd q are dir e ctly co njunctiv ely joined in A when A has a subformula P ∧ Q or Q ∧ P such tha t p is in P and q is in Q , no subfor m ula of P containing p is a disjunction, and no subfor mula o f Q containing q is a disjunction. W e define ana logously disjunctively and directly disjunctiv ely joined formulae (we just r eplace ∧ by ∨ and ∨ b y ∧ ). W e can prov e the following prop osition. Proposition 1. A diversifie d ∧∨ -e quivalenc e is a tautolo gy iff it is a the or em of S ∧ , ∨ . Proof. It is c le ar that every theorem of S ∧ , ∨ is a tautology . (This is es tablished by induction on the length of pro of in S ∧ , ∨ .) F or the co n verse, we supp ose that the diversified ∧∨ -equiv alence A ↔ B is a ta utology , and we pro ceed by induction o n the num ber n of co nnectiv es in A . By Lemma 2 , this num b er must also b e the n um b er of connectives in B . If n = 0, then, by Lemma 2 , the for m ulae A a nd B must b oth b e a letter p , and p ↔ p is an a xiom of S ∧ , ∨ . If n = k + 1, then A has a subformula either of the form p ∧ q or of the for m p ∨ q . Supp ose p ∧ q is a subfor m ula of A , and consider how p and q ar e jo ined in B . (1) It is imp ossible that p a nd q be disjunctively jo ined in B . Suppo se they are. By Le mma 1, for some S 1 , . . . , S k , where S i ∈ {⊤ , ⊥ } , w e hav e that A r 1 ...r k S 1 ...S k ↔ ( p ∧ q ) is a tautology . On the other hand, b y us ing Lemma 2 w e infer that B r 1 ...r k S 1 ...S k ↔ C c an be a tautolo gy o nly if either C ↔ ( p ∨ q ), o r C ↔ p , or C ↔ q , or C ↔ ⊤ , or C ↔ ⊥ , is a ta uto logy . So A r 1 ...r k S 1 ...S k ↔ B r 1 ...r k S 1 ...S k is not a tautology , which contradicts the assumption that A ↔ B is a tautolo gy . (2) It is also imp ossible that p and q b e conjunctiv ely joined in B , but not directly . O therwise, w e would ha ve in B a subfor m ula P ∧ Q or Q ∧ P with p in P and q in Q , and a subformula C ∨ D o f P with p e ither in C or in D ; we need not consider s eparately the ana logous case when C ∨ D is a subfor mula 5 of Q with q either in C or in D . Then A p ⊥ ↔ A ′ and B p ⊥ ↔ B ′ are tautologies for A ′ a formula of L without q a nd b ′ a diversified formula of L ∧ , ∨ with q . By Lemma 2, A ′ ↔ B ′ is not a tautology , which contradicts the ass umption that A ↔ B is a ta utology . So p and q are directly conjunctively joined in B . So in S ∧ , ∨ we hav e as a theorem B ↔ D for a diversified formula D of L ∧ , ∨ with a subformula p ∧ q . Since A ↔ B is a tautology , it is clear that A ↔ D is a ta utology too . If A p ∧ q q and D p ∧ q q are obtained fro m resp ectively A and D by replacing the sing le o ccur r ence of p ∧ q by q , then A p ∧ q q ↔ D p ∧ q q is a tautology , bec a use A p ⊤ ↔ D p ⊤ is a tautology . In A p ∧ q q we hav e k c o nnectiv es, a nd so b y the induction hypothesis we obtain that A p ∧ q q ↔ D p ∧ q q is a theor em of S ∧ , ∨ . Hence ( A p ∧ q q ) q p ∧ q ↔ ( D p ∧ q q ) q p ∧ q is a theorem o f S ∧ , ∨ ; in other words, A ↔ D is a theorem of S ∧ , ∨ , and sinc e B ↔ D is such a theor em too , w e o btain that A ↔ B is a theorem o f S ∧ , ∨ . W e pro ceed analogo usly when p ∨ q is a subform ula of A . ⊣ A ¬ ∧ ∨ -e quivalenc e is defined a s a ∧∨ -equiv alence with L ∧∨ replaced by L ¬ , ∧ , ∨ , and, as b efore, a ¬ ∧ ∨ -equiv a lence A ↔ B is dive rsifie d when A and B ar e diversified. The for mal system S ¬ , ∧ , ∨ is defined as S ∧ , ∨ sav e that the axioms a nd theor ems are ¬∧ ∨ -equiv alences instead of ∧∨ -equiv alences, and we hav e the additional axioms o f the following forms: ¬¬ A ↔ A , ¬ ( A ∧ B ) ↔ ( ¬ A ∨ ¬ B ), ¬ ( A ∨ B ) ↔ ( ¬ A ∧ ¬ B ), and the following additional r ule : A ↔ B ¬ A ↔ ¬ B A fo r m ula of L is called ¬ - r e d uc e d when ¬ o ccurs in it o nly b efore letters (i.e. only in subform ulae of the form ¬ p ). A letter o ccurs p ositive ly in a ¬ -reduced formula when it is not in the sco p e of ¬ ; otherwise it o ccurs n e ga tively . W e c a n prov e the following. Proposition 1 ¬ . A diversifie d ¬ ∧ ∨ -e quivalenc e is a tautolo gy iff it is a t he or em of S ¬ , ∧ , ∨ . Proof. It is clear that every theorem of S ¬ , ∧ , ∨ is a tautology . F or the conv erse, we supp ose that the diversified ¬ ∧ ∨ -equiv alence A ↔ B is a tautology . It is easy to s ee that in S ¬ , ∧ , ∨ we hav e as theor e ms A ↔ A ′ and B ↔ B ′ for ¬ -re duced formulae A ′ and B ′ of L ¬ , ∧ , ∨ (for that, the new a xioms of S ¬ , ∧ , ∨ are es s en tial), such that the div ersified ¬ ∧ ∨ -equiv alence A ′ ↔ B ′ is a tautolog y . It is impo s sible that a letter p occur s p ositively in A ′ and nega tiv ely in B ′ . Suppo se it do es. V ery muc h a s in the pro of of Lemma 1, we w ould obtain for some S 1 , . . . , S n , where S i ∈ {⊤ , ⊥ } , that ( A ′ ) q 1 ...q n S 1 ...S n ↔ p is a tautology . On the other hand, ( B ′ ) q 1 ...q n S 1 ...S n ↔ C can b e a tautology o nly if either C ↔ ¬ p , or 6 C ↔ ⊤ , or C ↔ ⊥ , is a ta utology . So ( A ′ ) q 1 ...q n S 1 ...S n ↔ ( B ′ ) q 1 ...q n S 1 ...S n is not a tautology , and hence A ′ ↔ B ′ is also not a tautology , contradicting what we have inferr ed ab o ve from the a ssumption that A ↔ B is a tautolo gy . It is thereby imp ossible to o that a letter occ ur s negatively in A ′ and p ositively in B ′ . So the diversified ¬ ∧ ∨ -equiv alence A ′ ↔ B ′ is an instance of a diversified ∧∨ -equiv alence A ′′ ↔ B ′′ , which is a tautology . (Just repla ce every letter p o ccurring nega tiv ely in A ′ and B ′ by ¬ p .) By P ropo sition 1, we hav e that A ′′ ↔ B ′′ is a theor em of S ∧ , ∨ . Hence A ′ ↔ B ′ is a theor em of S ¬ , ∧ , ∨ , and since A ↔ A ′ and B ↔ B ′ are theorems o f S ¬ , ∧ , ∨ , we obtain that A ↔ B is a theorem of S ¬ , ∧ , ∨ . ⊣ It is eas y to see that the formal s ystems S ∧ , ∨ and S ¬ , ∧ , ∨ are decidable formal sys tems. (F o r every formula of L ¬ , ∧ , ∨ we hav e a ¬ -r educed normal for m in L ¬ , ∧ , ∨ unique up to asso ciativity and commutativit y of ∧ a nd ∨ .) 3 Isomorphic form ulae with p erfect generaliz- abilit y Let K b e a categor y whose o b jects are the for m ulae of L ¬ , ∧ , ∨ , a nd whose arrows f : A → B a re intuitiv ely interpreted as deductions , or pr oo fs, fr om A to B . W e assume that K ha s isomo r phisms covering the theo rems of S ¬ , ∧ , ∨ . This mea ns, for example, that w e hav e an isomorphism o f the t yp e ( A ∧ B ) ∧ C → A ∧ ( B ∧ C ), whose inv erse is o f the t yp e A ∧ ( B ∧ C ) → ( A ∧ B ) ∧ C . These isomorphisms of K corresp ond to deductio ns in the multiplicativ e fragment of linear pr o positio nal logic. These a re the basic a rrows of K , but K can ha ve more arrows than that. W e a ssume how ever that K doe s not go b ey ond deductions of classical pro positio na l logic, which means that if f : A → B is an arrow of K , then A → B is a tautolog y , with the a rrow → interpreted a s the connective of mater ial implication. F or a formula A , let | A | b e the num b er of o ccurrences of letters in A . F o r every arr o w f : A → B of K , if x is an o ccurrence of a letter in A , then let o ( x ) = n − 1 iff x is the n -th o ccurrence of letter in A counting from the left, and if x is an o ccurr ence of a letter in B , then let o ( x ) = | A | + n − 1 iff x is the n -th o ccurrenc e of letter in B counting from the left. W e assume that every ar row f : A → B of K induces on the ordinal | A | + | B | an equiv alence re lation L f , ca lled the linking r elation of f , which satisfies the condition that ( o ( x ) , o ( y )) ∈ L f only if x and y ar e o ccurrences in A o r B of the same letter. A linking relation is p erfe ct when instead of only if in this condition we hav e if and only if . W e say that A and B are un iform instanc es of A 1 and B 1 when they are instances of A 1 and B 1 resp ectively with the same letter-for-letter substitution (i.e. substitution that replaces a letter by a letter). 7 W e s a y that an arr o w f : A → B is gener alize d to a n arr ow f 1 : A 1 → B 1 when A and B are uniform instances of A 1 and B 1 , and the linking rela tions L f and L f 1 are the same. W e sa y that a category K is p erfe ctly gener alizable when each o f its arr o ws can b e generalized to a n arrow of K with a p erfect linking rela tion. Exa mples of per fectly gener alizable categories with syntactically defined arr ows will b e g iven below after Lemma 3 . A ca teg ory K will b e ca lled p ermutational when for every isomorphis m f : A → B of K the linking relatio n L f corres p onds to a bijection b et ween | A | and | B | , and if g : B → A is the inv erse of f , then L g corres p onds to the inv erse of this bijection. Perm utational categories may b e b oth per fectly generaliza ble and no t perfectly g eneralizable. In this section we consider thos e of the firs t kind, while in the next one w e deal with the second kind. Perm utational catego ries aris e natura lly whenever we hav e a ce r tain mo d- elling of categ ories that we ar e no w going to descr ibe. The category SplPr e is the ca tegory whose arrows a r e split pr e or ders b et ween finite ordinals; Gen is a sub c ategory of S plPr e whos e arrows ar e split e quivalenc es b e tween finite ordi- nals, while R el is a c ategory who s e arrows are rela tions b et ween finite ordina ls. (The category R el has an isomorphic image within SplPr e by a ma p that pre- serves comp osition, but not identit y arrows.) A split preo rder is a pr eordering relation on the disjoint union of tw o sets c onceived as source and targe t, and analogo usly for split equiv alences, which are equiv alence relations. W e have inv estigated SplPr e and Gen sys tematically in [1 5], [11] and [12], and w e hav e used Gen and R el as model categories for equality o f deductions in [13] and [14], and in other papers related to these tw o b o oks. W e may use these mo del ca tegories to pr oduce the linking relation of K in the following manner . F or a functor G from K into a mo del catego ry such as SplPr e , Gen o r Re l , we take that ( n, m ) ∈ L f iff the ordinals co rresp onding to n a nd m are linked b y the reflexive, symmetric and transitive closur e o f Gf , which coincides with Gf if Gf is an equiv alence relation of Gen . In this se c tio n we are interested in pa rticular in linking relations pro duced by Gen , while in the next s ection w e will encoun ter als o one pro duced b y R el . It may happ en that the same categ ory K pro duces differ e nt linking r elations with differe nt functor s G . The notions o f p erfectly g eneralizable and permuta- tional catego ry ar e relative to a c hosen kind of linking relations. Let us g iv e an example of linking r e la tions. W e may hav e in K an arrow f : p ∧ ( ¬ p ∨ p ) → p corres p onding to mo dus p onens, such that L f will give the following linking   p p ∧ ( ¬ p ∨ p ) 8 and another arrow g : p ∧ ( ¬ p ∨ p ) → p corr e sponding to the first pro jection, such that L f will give the following linking ❅ ❅ p p ∧ ( ¬ p ∨ p ) The arrow f can be generalized to an ar row of the type p ∧ ( ¬ p ∨ q ) → q , while g ca n be ge ne r alized to a n arrow o f the t yp e p ∧ ( ¬ q ∨ r ) → p . Bo th of these arrows to whic h f a nd g are g e neralized have a p erfect linking. If the linking relations of the arr ows of K ar e pr o duced b y a functor G into SplPr e (or a sub categor y thereof ), as explained ab ov e, then we may conclude that K is per m utational. This is bec ause the isomor phisms of SplPr e corr espond to bijections. This is so b oth for p erfectly generaliza ble and for not p erfectly generaliza ble catego ries K . W e can prov e the following lemma. Lemma 3. If K is a p ermutational p erfe ctly gener alizable c ate gory, and A and B ar e isomorphic in K , then ther e ar e diversifie d formulae A 1 and B 1 such that A and B ar e uniform instanc es of A 1 and B 1 and A 1 ↔ B 1 is a tautolo gy. Proof. Since K is p erfectly genera lizable, the arrows f : A → B and its inv erse g : B → A o f K a re generalize d to the arrows f 1 : A 1 → B 1 and g 2 : B 2 → A 2 with per fect linking relations. Since K is p erm utational, these relations cor resp ond to bijections inv erse to e a c h other. F rom tha t we c o nclude that A 1 , B 1 , A 2 and B 2 are diversified formulae. Since B is a letter-for- letter instance of b oth B 1 and B 2 , which are diversified, B 1 is a letter -for-letter instance o f B 2 , and since the linking relations of f 1 and g 2 corres p ond to bijections in verse to each other, our letter-for -letter substitution produces A 1 out of A 2 . Accor ding to what w e assumed at the b eginning of the section, A 1 → B 1 and B 2 → A 2 are tautologies, and hence B 1 → A 1 is a tautolog y to o. ⊣ As categories K cov ered by this lemma we hav e the free distributiv e lat- tice ca tegory of [13] (Sectio n 1 1.1), which axiomatizes equality of deductions in conjunctive-disjunctiv e cla s sical lo gic (the o b jects of this categ ory are in L ∧ , ∨ ), and the free pro of-net category of [14] (Section 2.2 ), which a xiomatizes equa lit y of deductions in the multiplicativ e frag ment of linear logic without prop ositional constants (its ob jects are in L ¬ , ∧ , ∨ ). One may e a sily conceive other such ex- amples, a nd in pa rticular the e xample o f a categ ory axioma tizing equality of deductions in the whole of classica l pro pos itio nal lo gic (whose ob jects are in L ¬ , ∧ , ∨ ). One would just take as equations of K those equations f = g such that Gf = Gg , where G is a nontrivial functor from K to Gen mak ing K p erfectly generaliza ble. The following propo sition c hara cterizes pa ir s of isomor phic form ulae for p er- m utational p erfectly generalizable catego ries. 9 Proposition 2 ¬ . The formulae A and B ar e isomorphic in a p ermutational p erfe ctly gener alizable c ate gory iff A ↔ B is a the o r em of S ¬ , ∧ , ∨ . Proof. F rom left to right , supp ose that A is is omorphic to B . So, by Lemma 3, there are diversified for m ulae A 1 and B 1 such tha t A and B are unifor m in- stances of A 1 and B 1 , and A 1 ↔ B 1 is a tautology . By Pr opo sition 1 ¬ , we obtain that A 1 ↔ B 1 is a theorem of S ¬ , ∧ , ∨ , and hence A ↔ B is that too . F or the other dir ection, we hav e ass umed that the equiv alences of S ¬ , ∧ , ∨ are cov ered b y the isomorphisms of our category . ⊣ W e hav e a Prop osition 2 ana logous to P rop osition 2 ¬ with S ¬ , ∧ , ∨ replaced by S ∧ , ∨ , for categ ories whose ob jects are the formulae o f L ∧ , ∨ . Since b oth S ∧ , ∨ and S ¬ , ∧ , ∨ are eas ily seen to be decidable systems, Prop ositions 2 a nd 2 ¬ lead to a decis ion pro cedure for isomorphic formulae. 4 Isomorphic form ulae without p erfect general- izabilit y F or f : A → B an a rrow, let the set of diagonal links of f b e the se t D f = { ( n, n ) | n ∈ | A | + | B |} . W e will here call an ar row f : A → B in a categ o ry a zer o arr ow when its link ing relation L f is equal to the s e t D f of its diago nal links. A zer o identity arr o w is an a rrow 0 A : A → A with L 0 A = D 0 A . The un ion of the arrows f , g : A → B will here b e a n arrow f ∪ g : A → B such that L f ∪ g is the trans itive closure of L f ∪ L g . (F or a genera l treatment o f ze r o a rrows and union of ar rows se e [13].) Consider a categ ory K such that the ob jects of K ar e the formulae of L and its arrows cor respo nd to deductions in classical prop ositional logic. W e assume that in K we hav e for ev ery A of L a zero iden tity arrow 0 A , a nd we hav e also closure under unio n of ar rows. As a pa rticular case of K we hav e the category B of [13] (Section 14 .2), but pos s ibly a lso a c a tegory that besides the zero arrows of the t yp es ⊤ → ¬ A ∨ A and A ∧ ¬ A → ⊥ , like those w e hav e in B , has also arrows of this t yp e with link ing relations given by the following pictures: ¬ A ∨ A ⊤ A ∧ ¬ A ⊥ The link ing relation L f of an arr ow f : A → B of B , which is an equiv alence relation, is not the relation Gf ⊆ A × B , where G is the functor into Re l with resp ect to which B is coherent; i.e ., the functor G s uc h that for f , g : A → B arrows of B w e have that f = g iff Gf = Gg (se e [13], Sec tio n 14.2). W e define L f as the r eflexiv e, sy mmetr ic and transitive clo sure of Gf . Note that for f , g : A → B arrows of B we do not have that L f = L g only if f = g in B , 10 but we still hav e that for particular a r rows f a nd g , as in the pro of of Lemma 7. What w e need e ssen tially is that this holds when one of f and g is an iden tit y arrow. F or the catego r ies K other than B we need the same assumption, which is guaranteed o f cour se if eq ualit y o f ar rows in K is defined via the linking relations—if we hav e namely that for f , g : A → B the equation f = g holds in K iff L f = L g . Our as sumptions ab out K must g ua rantee also that we hav e an analog ue of Lemma 6 b elow. Thes e a ssumptions can b e quite s tandard, like those s pelled out in [13] a nd [14]. If co mposition of these linking re la tions is defined naturally , like comp osition in the category Gen (see [15], Section 2, for a detailed study), then in B we do not hav e that L g ◦ f is equal to the co mp osition of L f with L g , but this ma y hold for other categor ies K . W e take that our a ssumptions imply that K is p ermutational in the sense of the prece ding section. How ever, K is not p erfectly generaliza ble, b ecause of the presence of z e r o identit y a rrows. The arr o w 0 p : p → p cannot b e gener alized to one with a p erfect linking rela tion. The catego ry B is p erm utational and is not per fectly genera lizable. The definition o f the linking rela tion for the arrows o f B giv en ab ov e, and our notio n of linking rela tion for K in general, is motiv ated b y the wish to have a unifor m notion base d, as in the pr e c eding section, on an equiv alence relatio n. This unifor m notio n is related to generality of deductions (mor e consistently in the pr eceding section than in the presen t one). If how ever we do not striv e for this uniformity , then w e may define L f for B just as Gf , and our pr oo fs would still go thr ough. F or K in gener al, we may a lso have a no tio n of linking relation like Gf . W e prove firs t a simple pr eliminary lemma. Lemma 4. F o r every formula A of L ⊤ , ⊥ , ∧ , ∨ in which the letter p o c curs ther e ar e formulae A 1 and A 2 of L ⊤ , ⊥ , ∧ , ∨ such that A ↔ (( p ∧ A 1 ) ∨ A 2 ) is a taut olo gy. Proof. W e pro ceed by induction on the num b e r n of o ccurrences of binary connectives in A . If n = 0, then A is p , and w e tak e A 1 and A 2 to be resp ectively ⊤ and ⊥ . Suppo se n > 0. If A is A ′ ∧ A ′′ with an o ccurr ence of p in A ′ , then by the induction h yp othesis we hav e fo r m ulae A ′ 1 and A ′ 2 of L ⊤ , ⊥ , ∧ , ∨ such that A ′ ↔ (( p ∧ A ′ 1 ) ∨ A ′ 2 ) is a tautolo gy . W e take A 1 and A 2 to b e r espectively A ′ 1 ∧ A ′′ and A ′ 2 ∧ A ′′ . If A is A ′ ∨ A ′′ with an o ccurrence of p in A ′ , then b y the induction h yp othesis we hav e formulae A ′ 1 and A ′ 2 as b e fore, and we take A 1 and A 2 to b e res pectively A ′ 1 and A ′ 2 ∨ A ′′ . If A is A ′′ ∧ A ′ or A ′′ ∨ A ′ with an o ccurrence of p in A ′ , then we hav e that A ↔ ( A ′ ∧ A ′′ ) or A ↔ ( A ′ ∨ A ′′ ) is a ta utology , and we pr ocee d a s befor e. ⊣ The tautolo g y of this lemma is prov able in every logic that algebraic a lly corre- sp onds to distr ibutiv e lattices with top and bo ttom. W e have next the following 11 lemmata. Lemma 5. If B ′ is obtaine d fr om a formula B of L ⊤ , ⊥ , ∧ , ∨ by r eplacing a single o c cu rr enc e of q by p ∧ q , then ( p ∧ B ) → B ′ is a tautolo gy. Proof. W e pro ceed by induction on he num b er n of o c currences of binary connectives in B . If n = 0 , then B is q , and ( p ∧ q ) → ( p ∧ q ) is a tautology . Suppo se n > 0 . If B is B 1 ∧ B 2 with the repla c ed occur rence of q in B 1 , then by the induction hypo thes is we have that ( p ∧ B 1 ) → B ′ 1 is a tautolo gy , and hence, by relying on the a sso ciativit y of ∧ , we have that ( p ∧ ( B 1 ∧ B 2 )) → B ′ , where B ′ is B ′ 1 ∧ B 2 , is a tautolog y too . If B is B 1 ∨ B 2 with the replaced o ccurrence of q in B 1 , b y the induction hypothes is we hav e aga in that ( p ∧ B 1 ) → B ′ 1 is a tautolog y , and since ( p ∧ ( B 1 ∨ B 2 )) → (( p ∧ B 1 ) ∨ B 2 ) (whic h corr espo nds to a diss ocia tivit y arrow; se e [1 3], Chapter 7) is a tautology , w e hav e that ( p ∧ ( B 1 ∨ B 2 )) → B ′ , wher e B ′ is B ′ 1 ∨ B 2 , is a tauto logy to o. If B is B 2 ∧ B 1 or B 2 ∨ B 1 with the repla c ed o ccur r ence o f q in B 1 , then we hav e tha t B → ( B 1 ∧ B 2 ) or B → ( B 1 ∨ B 2 ) is a tautology , and we pro ceed as b efore. ⊣ Lemma 6. If A and B ar e formulae of L ⊤ , ⊥ , ∧ , ∨ , with x and y o c curr enc es of the letter p in r esp e ctively A and B , and A → B is a t autolo gy, then ther e is an arr ow f : A → B of B such that in L f − D f we find only the p ai rs ( o ( x ) , o ( y )) and ( o ( y ) , o ( x )) . Proof. By Lemma 4 , we ha ve for m ulae A 1 and A 2 of L ⊤ , ⊥ , ∧ , ∨ such that A ↔ (( p ∧ A 1 ) ∨ A 2 ) is a tautology . The pro of of this lemma yields an arrow τ : A → ( p ∧ A 1 ) ∨ A 2 such that ( o ( x ) , | A | ) ∈ L τ (note that if z is the leftmost o ccurrence of p in ( p ∧ A 1 ) ∨ A 2 , i.e. the o ccurrence written do wn, then o ( z ) = | A | ). W e also ha ve arrows σ : ( p ∧ A 1 ) ∨ A 2 → A and g : A → B of B , b ecause their t yp es corr espo nd to tauto logies. If ι 1 : p ∧ A 1 → ( p ∧ A 1 ) ∨ A 2 and ι 2 : A 2 → ( p ∧ A 1 ) ∨ A 2 are injection arrows, then for ζ i = d f 0 B ◦ g ◦ σ ◦ ι i , where i ∈ { 1 , 2 } , we hav e L ζ i = D ζ i , b ecause the comp osition ends with 0 B . If π : p ∧ A 1 → p is a first pro jection a rrow, then for h π , ζ 1 i : p ∧ A 1 → p ∧ B we hav e L h π ,ζ 1 i − D h π ,ζ 1 i = { (0 , | A 1 | + 1) , ( | A 1 | + 1 , 0) } (this means that the tw o o ccurrences of p that are written down a re linked) . If B ′ is obtained fro m B by replacing the o ccurrence y o f p in B by p ∧ p , then the pro of o f Lemma 5 yields an arr o w η : p ∧ B → B ′ of B . Out o f the first pro jection arrow π ′ : p ∧ p → p w e obtain an arrow θ : B ′ → B suc h that for θ ◦ η : p ∧ B → B we hav e (0 , o ( y )) ∈ L θ ◦ η (i.e. the occur rence of p written down in the source p ∧ B and the o ccurrence y in the target B a re linked). F or µ = d f [ θ ◦ η ◦ h π , ζ 1 i , ζ 2 ] : ( p ∧ A 1 ) ∨ A 2 → B 12 we hav e L µ − D µ = { (0 , o ( y )) , ( o ( y ) , 0) } , and we take f : A → B to be µ ◦ τ . W e hav e L µ ◦ τ − D µ ◦ τ = { ( o ( x ) , o ( y )) , ( o ( y ) , o ( x )) } . ⊣ The formulae A and B of L ⊤ , ⊥ , ∧ , ∨ are letter-homo gene ous when every letter that o ccurs in A o ccurs in B the same num ber of times, and vice versa, every letter that o ccurs in B o ccurs in A the same n umber of times. Two ¬ -re duced formulae of L ar e letter-homoge ne o us when they a re uniform insta nces of tw o letter-homoge ne o us formulae of L ⊤ , ⊥ , ∧ , ∨ . This means that such fo rm ulae share bo th po sitiv e and negative o ccurrences of letters. W e ca n prov e the following. Lemma 7. F o r A and B formulae of L ⊤ , ⊥ , ∧ , ∨ , if A ↔ B is a tautolo gy and A and B ar e letter-homo g ene ous, then A and B ar e isomorphic in B . Proof. Supp ose A → B is a tautology and A and B a re letter-homo geneous. F or every bijection mapping the o ccurr ences o f letters in A to the o ccurrences of the s ame letters in B , there is an a rrow f : A → B of B such that L f corres p onds to this bijection. This is guara n teed by Lemma 6 a nd the op eration of union of arrows of B . Since the same holds for the tautology B → A a nd the in verse of our bijection, we co nclude that ther e is an arrow g : B → A of B such that L g corres p onds to this inv erse. With the help of Bo olean Coherence (see [1 3], Section 1 4.2; this is the assertion that the functor G , on which the definition of L f is based, is a faithful functor from B to R el ), w e infer from L g ◦ f = L 1 A and L f ◦ g = L 1 B that f and g ar e isomor phisms, inverse to each other. ⊣ W e can now prov e the following. Proposition 3. F or A and B ¬ -r e duc e d formulae of L we have that A and B ar e isomorp hic in B iff A ↔ B is a t autolo gy and A and B ar e letter- homo ge ne ous. Proof. The dir ection from left to r igh t is a n easy cons e quence of the fa ct that the a rrows of B corr espo nd to implications that are tautologie s , of the fact that no arrow of B links a p ositive o ccurrence of a letter in the sourc e with a neg ativ e o ccurrence o f a letter in the ta rget, and vice v ersa , and of the fact that B is a per m utational ca teg ory in the sense of Sectio n 3. F or the other direction we use Lemma 7 and a ppeal to the fact that if the right-hand side holds, then there ar e formulae A 1 and B 1 of L ⊤ , ⊥ , ∧ , ∨ such that A 1 and B 1 are letter-homo geneous, A a nd B are uniform instances of A 1 and B 1 and A 1 ↔ B 1 is a tautology . W e der iv e that A 1 ↔ B 1 is a tautology by appropria te substitutions w ithin A ↔ B . ⊣ Every formula of L may effectively be reduced to a ¬ -r educed form ula iso- morphic in B . 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