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Logique Analy s e 141-142 (1993), 25-38 MODA L IIIE S IN SUBSTRUCTURAL LOGICS Greg RESTALL 1. Logics Logic is about valid deduction. One central result in logic is the deduction theorem. A, I I This Bresult i f ties a fact from the metalanguage (that B follows from A, in the context a nof 1) to a fact in the object language ( A ---> B , in the context of 1). This dconnection is very important, because it shows how the properties o f o the conditional in a logic depend on the way premises are collected ton l y gether — represented here by the comma. Standard brands o f logics, like i f E intuitionistic and classical logic, allow fo r all sorts o f rearrangements in l - A They have structural premises. rules like these: — > B F(A, A) I- B 1 Contraction (WI ) W e a k e n i n g (K ) -T (( A ) 11 A Mingle (M) C o m m u t e d Associativity (CB) C-) 11A, A) I- B 1 1 ( 2 1 , C), MI - D 1 ( 1A (B) C o m m u t a t i v i t y (CI) -T)BO , B), O F D Associativity 1 1 B , A)I-- C F E ( A(In , Ba substructural logic, one or more of these rules are rejected. This means that , our premise collections have more structure than would otherwise be (F the(Acase. The number of premises, their order, their bracketing, or what isn't B a premise, matters. Relevant logics, linear logics, and logics like BCK and , )A logics are all substructural logics, because these logics keep track of fuzzy O,I premises in ways that logics with all the structural rules don't. (FHowever, even if we reject certain structural rules in general, by adoptC DB ing a substructural logic, we may be interested in cases when substructural F, (0 A) ,1 B)D I 26 G R E G RESTALL rules do apply. We may not think that all premises commute — but there may be a small class of premises which do commute, and this may be important. Similarly, we may not be able to contract all premises — but some might contract validly. So, it is interesting to see how we can keep track of structural rules with a limited applicability. That is the topic of this note. We will take our cue from Girard, and add modalities to our logics to regain structural rules [2]. It is easiest to give our account of modalities in substructural logic if we use an algebraic presentation of the formal systems. This will make the theorems easier to prove than would otherwise be the case. So it is to this that we will turn. Substructural logics (like any) can be modelled by propositional structures. Think of the elements of these structures as propositions, and the operations as ways to form new propositions out of old. Definition 1 A basic propositional structure is a 5-tuple ( P; • , e, where • P is a nonempty set of propositions. • i s a partial order on P, representing entailment between propositions. • The binary operation • on P, called fusion. This represents premise combination. Fusion preserves the entailment ordering. I f a a ' and b b ' then a • b • b r • The element e E P is a right identity for fusion. This means that for all a e P, a •e = a Because of this, e represents logical truth. •The binary operation o n P residuates fusion. For all a, b, c E P, a • b c if and only if b 5. a c This means that i s our conditional operation on P. (Recall the deduction theorem, tying the conditional to valid deduction. This condition is the deduction theorem in algebraic form.) If a b then by residuation and identity, e a b So, if a entails b, then the conditional a b is a logical truth. This is a desirable connection between entailment, logical truth, and the conditional. If we wish to model conjunction and disjunction, we require that 5_ be a lattice order. In these structures, conjunction is the greatest lower bound, and disjunction, the least upper bound. Definition 2 A basic propositional structure with conjunction and disjunction is a basic propositional structure where the underlying entailment ordering is a lattice. Definition 3 A basic propositional structure with a distributing conjunction and disjunction is a basic propositional structure where the underlying en- MODALITIES IN SUBSTRUCTURAL LOGICS 2 7 tailment ordering is a distributive lattice. I f we wish to model negation, we can add a unary operator — on P, with suitable conditions. Definition 4 A basic propositional structure with an intuition istic negation is a basic propositional structure equipped with a unary operation — such that for each a, b , c e P, a • b – c if and only if c • b 5.–a.. This c o n d it io n p ro v id e s u s w i t h co n tra p o sitio n i n t h e f o r m a – b = b – a , and double negation introduction, a – – a. Definition 5 A basic propositional structure with a de Morgan negation is a basic propositional structure with an intuitionistic negation — such that, in addition, for each a E P, a = – – a. We use propositional structures to model valid deduction in the usual way. Take a propositional language with connectives – ), 0 and t and any o f A, y and –, as desired. We define the interpretation h(A) o f a formula A in a propositional structure P recursively, f ro m arbitrary assignments t o atoms, using these clauses: /40 = e I( , 40 B)= h(A)• h(B) h( A–) B)= /1(A) h ( B ) k i t A B) = h(A) n h(B) h(A\./ B)= h(A)u h(B) 1 ( Take- a1 class , 1 ) =X of– propositional h ( A ) structures. We say that a formula A is X-valid if for each structure P E X and for each interpretation h into P, we have e h ( A ) . We write this ' I -I A . ' For largely uninteresting historical reasons, the class of all propositional structures with a de Morgan negation, conjunction and disjunction is called LDW, and its cousin which contains only those structures in which disjunction and conjunction distribute is called DW. One interesting propositional structure which illustrates our definition is 13N4. Belnap [I I, Meyer, Giambrone and Brady [3] and Slaney [4] all extol its virtues. The structure is simple. 28 GREG RESTALL T e= B F F B N F T B N F F T T N F T T F N T B T B N F B N N F F N T N T N N F F T F B A I T' F F F T T T T F Conjunction and disjunction are given by greatest lower and least upper bounds on the lattice ordering — which is indicated here by the Hasse diagram. This structure has a number of notable features: not least being the fact that negation has two fixed points, B and N. This shows that a de Morgan negation need not satisfy the usual classical conditions. Furthermore, premise combination is associative and commutative, but not idempotent. So BN4 satisfies some of the structural rules we've seen, but not all. I t is helpful to catalogue structural rules in algebraic form, like this: WI b•aa M a • a a B ( a • b) • c a • (b • c) CB ( a • c) • b a • (b • c) CI a • l , b • a We can tack selections from these onto our basic logic to make all sorts of interesting formal systems. Given a class of propositional structures X, the class X + Y is the subclass of X of all structures which satisfy condition Y. Girard's original linear logic (without exponentials) is then given by LDW + B + CI. So, in linear logic premise combination is associative and commutative. The contraction-free relevant logic C adds distribution to linear logic, so it is given by DW + B + CI The relevant logic R is given by DW + B + CI + WI. We elide the distinction between classes of propositional structures and logics, so we will say things like R = C + WI from time to time. Now to round this section off we'll prove an important result which shows that we can restrict our attention to propositional structures of a particular kind, when using them to model a logic. Definition 6 A propositional structure P is said to be complete if for every X ç P , the join and meet VX and AK exist. Fact 1 Any propositional structure can be embedded as a substructure in MODALITIES IN SUBSTRUCTURAL LOGICS 2 9 a complete propositional structure. Proof Take a propositional structure P. We take the elements of the completion of P, P* to be the ideals of P. A subset X g P is an ideal if for each x e X, if y x then y E X too. If P is equipped with disjunction, we require that x, y E X only if XE X too. Conjunction in this structure is simply intersection of ideals (the intersection of any set of ideals is an ideal). So, arbitrary meets exist in this structure. This means that containment, c , is the ordering in the new structure. If disjunction is present in P, then the disjunction X u Y is the set t tion z as union of ideals. (If ideals are merely downwardly closed sets, then the union of two ideals is an ideal.) Arbitrary joins exist in this structure : I f disjunction is simply union, then ideals are closed under infinite too. 3 unions. If disjunction is defined in terms of disjunction in P, the definition isx a little more complex. 3 y V I x , : i E / 1 = l z : f o r so m e i E a n_ X ii E ( t, Z U . d U 1 X x 4 , E E Fusion, the residual, e* and negation ( if present) are defined in similar ways X , y X• Y=tz :3x 3y ( x EX,y EYandz x oy ) } EX = , Yt z : e a3 x- 3*Xy=1.z:3x(x e X and z n( x x x) 1 ItdEis simple e to check that the conditions on propositional structures are satzX }in, the completion P*, so P* is a propositional structure. The original isfied structure P lives in P* in the guise of its principle ideals. The map xy fu: l y : y 5_ x l is an injection of P into P*. EIf our original structure P satisfied a structural rule, then it is clear that P* a a satisfies that rule (check the definition of fusion). also ln .d result shows us that complete propositional structures are enough, This O when z it comes to modelling logics. The class of validities in all 91 propositional structures is the same as the class of validities in all complete tx 91 propositional structures, because any counterexample to validity in an h> e r w i s e , 30 G R E G RESTALL incomplete structure survives as a counterexample in the completion of that structure. Th is fact is important, because we w i l l need to assume that structures are complete in one result coming up. Note that in complete propositional structures P, A P x for any x E P So, complete propositional structures have a least element. In what follows, we will use '0' to denote the least element in a propositional structure. Excursus This embedding result also motivates a short polemical point about conjunction and disjunction. The result has a simple corollary that any structure without conjunction and disjunction can be extended to include conjunction and disjunction in a natural, painless way. Furthermore, in this extended structure, conjunction distributes over disjunction. This is especially important for devotees o f linear logic. In linear logic, additive conjunction does not distribute over additive disjunction. Our result shows that this is not a feature of the substructural nature of the logic — rather, it is a bare fact about the additive connectives. Since it is quite hard to interpret additive disjunction in linear logic, it is quite hard to see what a failure of distribution amounts to. Unless some interpretation for disjunction can be found, in such a way that motivates the failure of distribution, there is little reason to favour linear logic over its distribution-added cousin C. E l Leaving polemics about distribution aside, we will assume from now, that all logics come equipped with disjunction and conjunction (whether they distribute or not) because this will make the modal conditions simpler to state. 2. Modalities There are a number of ways to introduce modalities to substructural logics. This note will focus on extremely popular modalities in the vicinity o f S4 and S5. In our context, an S4 necessity satisfies the following conditions. T EA 4 A A o 4 D(A B ) —› (DA —› EB) E I f h A then D A A 0 A E D B I f a de C and if the S4 necessity also satisfies A Morgan negation is present, O D A B 5 o ) 0 A A to be an S5 necessity. These necessities have interesting properties it is said — > n — c — A MODALITIES IN SUBSTRUCTURAL LOGICS 3 1 in substructural logics. Fact 2 All S4 necessities satisfy 1- * A Y oB) 44 DA y DB Proof 1- * A Y oB) n A y oB by T oI- DA -4 oA v DB, so by N we have I- oB ° .I-oDA F —} o D(oA r ty DB) h Similarly e result, using the lattice properties of y . ( o, Ft ho e( or A D — Ai call d r* eva proposition c t[ D i oB Nn) a, ,necessitive i f and only i f for some A, We'll s o w e o N 44AoA. Iet is simple to show w that if o is an S4 necessity, that N is a nea aif and v only cessitive v hh ve if I-eN o N . This fact means that necessitives are o u )disjunction r, closed under (where present). Necessitives are also closed unD B der fusion. a n d — > Fact 3 I n lo g ic s w i t h f u s io n , a l l S 4 necessities s atis fy o D(EA0 oB)<—>IDA0DB. w i t h Proof Clearly, I- * A 0 DB) —* oA 0 cif F or the converse, w e have I-4 DB—*(DA—*DA DB). So, I- DDB o ( D A —*DA0 DB). This quickly gives g oB --> DA i —*(DA v 0 DB) and hence I-DA 0 DB—* o(DA 0 oB) as desired. < e s , In addition, if the modality is S5ish, then necessitives are closed under negation. Fact 4 All S5 necessities satisfy I- D—IA 44 Proof Left to right is T negation elimination. o structures we can model a necessity by adding a unary , Inapropositional n d operator on the structure. r i g h t t o 7 In a propositional structure P, a function I : P P is an S4 Definition l e operator interior f t if it satisfies these conditions. i s ae n g i 1(a)5_ v i 1 4 n 1 ( ya b e n b ) 1 4 l If, in addition, it satisfies ( = , 1 a 5 ( 1 a ) ( o e ) = a ) nn I d = (e o d u •I b l ( e 1 Ia 32 G R E G RESTALL it is said to be an 55-interior operator. A propositional structure with an interior operator (whether S4 or S5) is said to be a modal propositional structure. These conditions on I are forced by the axioms for S4 and S5 necessities. (For the fusion and disjunction clauses, see Facts 1 and 2). / is called an 'interior' operator on a set of subspaces of a space because of conditions like 4 modalities, and the observation provokes an alternate way of presenting the / semantics of modal operators. Instead of defining an 'interior' operator /, a n we can restrict ourselves to the class C of 'open' elements. The objects a d such that 1(a)= a.. That is, the necessitives of our earlier discussion. T r Fact 5 In any complete modal propositional structure the class 0' of open T elements contains e and 0, and is closed under fusion, conjunction and disjunction. h i Proof This is simple. T s by 1 eg i v e s h conjunction, disjunction and fusion. lu s a T 4 0call a ) class U on a propositional structure a potential open class if it Let's s h 5 contains e and 0, and is closed under fusion, conjunction and disjunction. n e Then we 0 , have another fact. o cs o t o Fact 6( A propositional structure P with a class potential open class C is a 1 0 ) g modal propositional structure with interior operator I if we define n = o d O . n 1( a) =V x 5 a and x 0 1 iWt e ie provided that the infinite joins exist. u o h a n proviso is unimportant in practice, because we can assume that This n v e n propositional structures are complete, because of Fact 1. s1 o n ( e Proof t It is simple to show that I a) E C for each a e P, d that 1(a)= a i) and I(a)n5 a for each a E P. ifor each a EC. It follows that I(I(a)S= 1(a) a ,= We have 1(e)= e as e EC U . Similarly, as C is closed under disjunction ( if c u e present) a n d f u s io n , w e h a v e 1( .1( a) u 1(b))= 1(a)u 1(b) and e /41(a). 1(b)). 1(a)• 1(b). d a Finally, if conjunction is present, we can note that 1(a nI)) 1 ( a ) and i nI( anb) . 0 ) gives 1(a n17)5_ 1(a)nl(b). Conv ers ely , 1 ( a ) n l ( b ) n d t h / e s a h l o g w MODALITIES IN SUBSTRUCTURAL LOGICS 3 3 a nb and I( a) n 1(b) E 0 gives I ( a) n 1 ( a n b), as desired. This is a useful fact, because we can define a modal operator on a propositional structure merely by specifying its set of open elements. This way of specifying a modal operator in a propositional structure is much simpler than defining a function / from scratch. So, in what follows we will define our interpretations of modalities by specifying the set of open elements. This set of open elements provides an interesting structure inside the larger propositional structure. We can pick out these elements in our language by using the modal operator o Whatever the formula A is evaluated as, we knowe that DA will be evaluated as an open element. This is where adding modalised structural rules becomes interesting. We can add a number of axioms or conditions to our logic, saying that the open elements satisfy structural conditions which are not shared by all other propositions. Here are some axioms and conditions. WI 0 E I A —› EA 0 EA a a • a f o r each a e C K ( D B 0DA—› EA h i • a a f o r each a, b E C M cB n O A 0 DB)0 DC—>EA 0 (DB0 EC) ( a•b) •c a•( b•c ) for each a,b,c E C o E CB0( D A 0 o A Ed3 o EA a • b 1 3 • a f o r each a,bc e CI A o0 C )D These axioms say that necessitives satisfy conditions which propositions in A general, in our logics, fail. The first axiom, WI , ensures that necessitives 0 — contract. I n logics like CK, not all propositidis contract. We 'regain' a E measure )I of contraction by adding a modality which contracts. Similarly, we may E be interested in modalities which give us weakening, or mingling for B necessitives. -A -a Definition 8 Given a non-modal logic X, we'll call XS4 C n • E „logic t h you e get when you add an S4 modality satisfying the conditions a C A loa Adding structural rules to the open elements of a propositional structure provokes two questions. Firstly, does it do anything to the non-modal struco (f of the logic? That is, is the extention conservative? ture o tD r o Definition B 9 A logic X is a conservative extension of Y if it extends Ys voe C cabulary, but nothing that is X valid in the old vocabulary of Y isn't also Y 0 valid. .oa c S C i)h m (a iaE l•0 a c r) 34 G R E G RESTALL Secondly, does the extension do us any good? We will answer these two questions in the next sections. 3. Conservativeness Fact 7 XS4 and XS5 are conservative extensions of X. Proof Suppose A is not X valid. It has a counterexample in the propositional structure P. P can be made into an S4 or S5 modal structure by defining the class 0 of opens on P to be the class of all propositions. < Fact 8 Any logic X is conservatively extended by XS4 + C rules C istructural o .I .D. Ct o t Proof C , , ,This time, on any propositional structure P, take the set of opens to n beo10, el. This is a sublattice, and it satisfies any added structural rule ,:qp.f o r a n y < . Fact 9 A n y logic X satisfying K is conservatively extended by XS4+ C lo Proof . . . C In logics satisfying K, 10, el is closed under negation. Firstly, note that — 0 = Tand — T = 0 by contraposition, where T is the top element no of the lattice. Then it is sufficient to show that given K, e = T . But this is , easy. Given K,b=b•e5_e for all b. So, we have our result, as 10, el is a f of o opens satisfying all structural rules. set < r However, the conservative extension results end here. In logics where — a e 0 , the class of opens must at least contain e,—e, 0, —O. And in many logics, —e is not guaranteed to satisfy structural rules. Let f = —it Then we n can collate together a number of facts. y s t 10r Any logic X in which le f —› fo f is not conservatively extended Fact u XS5 c + WI.. by t u r Fact a l i l Any logic X in which i i l o f — ) f is not conservatively extended by r XS5 + M o o r u Fact X S 125 Any logic X in which k f o f --> fo f is not conservatively exl tended + Kby XS5+ CI o .oe Fact 13 Any logic X in which k f 0(f 0 f) - 4 ( f 0 f)—) f is not conserva.s C l o t o C n MODALITIES IN SUBSTRUCTURAL LOGICS 3 5 tively extended by XS5+ B o These facts have bite, because they apply to interesting logics. It is not . difficult to show that in linear logic or C, f - - - >f o f . So, these logics are not conservatively extended by their cousins with a contracting S5 modality. Similarly, in R, k f o f —> f . So, R is not conservatively extended upon the addition of an S5 modality with weakening or mingling. What is the significance of these results? It is hard to tell, beyond the fact that it is another reason to prefer S4 over SS. S4 gives us a sublattice of 'opens', closed under fusion. We have sublattices like these at hand which satisfy every structural rule, in every propositional structure. S5 forces the sublattices to be closed under negation, and this seems to be a Bad Thing in general. Only the case of logics satisfying K stands out as an exception. So much for tracking the effects of adding modalities on the nonmodal fragment of logics. Now we will see what these modalities can do for us. 4. Embeddings The substructure of opens in a modal propositional structure has a number of interesting properties. It is a sublattice, and it is closed under fusion. If the modal operator is S5ish, it is also closed under negation. However, in general, it is not closed under implication, and if the modality is S4ish, it is not, in general, closed under negation. However, we have a number of interesting results in approximating implication and negation in the substructure of opens. Fact 14 If 0 is a collection of opens in a propositional structure P, then for each a, b, c E 0, a- b c if and only if b 1 ( a c ) . Proof I f b 1 ( a c ) then b a c and hence, a. b c . Conversely, if a -1 , From now, let a D b stand for I(a b ) . This result means that in the c context of 0 , D residuates fusion. This is an important fact. a n 15 If 0 is a collection of opens in a propositional structure P equipped Fact d a de Morgan negation —, then for each a, b, c E 0, a • b 1(—c) if and with only if c - b I(—a), / ( Proof: I f a • b .. I(—c) then — /(—c)• b5_—a, and hence I(-4-c)• b) 1(—a). b However, c —/(—c), so c • b --1(—c)• b. But /(c • b)= c • b (as b and c are ) 1 ( a c ) . H 36 G R E G RESTALL opens), so c • b I ( c • b) I (-1 (— c )• b) 1 (—a ) as desired. From now, let — a stand for I(—a). This result means that in the context of 0, — is an intuitionistic negation. This, also, is an important fact. Now we get to the payoffs. Let's show how we can embed logics with structural rules in logics which don't have those structural rules (which, in case you've forgotten, was the motivating thought behind this work). We need to define the translation A ' o f a formula A. It is defined recursively: 13 (A A B) 3 n v= B) (A = A D(A 0'= = n A (Av —> = B p n (— A c• n(A° —› BE) )° Fact 16 D In -any modal propositional structure P, a formula of the form A ' is always , interpreted as an open element. ( 1 Proof: Observe the construction of A ° I t s atomic parts p ' are interpreted V ) as open elements, and any of the original connectives in A are modified in A ' to always map into open elements too. Fact 1 7 A i s X S 5 + q , . . . c C v„ a—lvalid.. nXo+ — i d i f a n d Proof: o nI f Al ' i sy X S 5 + C „ , — valid t h e n i t i s t r u e i n a l l iXS5+ C f an XS5 + C Al E 1. . elements. all So, A ' will come out true in this structure by our assumption ithat. C s ,n. o. . itC is „ , XS5 + C terpretation, n ss tot rr u — c because I is the identity operation (as 0 = P). This means that A X + C vtu is a l i uc rt e d IH No le t A b e x + c , . . . c iu r ow e w e v _e XS5+ n C v , . valid B u t this is simple. Take a XS5 + C fs . r ,a Cl i „ d — ture in which A " is invalid. Let h be an interpretation which sends A ' to — lA o W e w L element x e in P. The class of open elements e in P is a X i van . Cl i in s h a e.e.wa structure, as we have seen, where we take D to be the residual o f fusion. dd n t E s t o r u tt c a a n d e f i n ec a- n in t e rp re t a t io n h ' i n t o 0 b y s e t t i n g We aA s h o khP' (p )= l(h (p ))= / sh w a e(bc dvt t e s eh a e ithp t)r. e I n ta h i s dh ein An . t e r p r e seX' t a t i o n se+i , a thCs ' ( m o MODALITIES IN SUBSTRUCTURAL LOGICS 3 7 tion, and so, we have a counterexample to A in this structure. But we assumed that A has no counterexamples in X + C couldn't have a counterexample to A ' i n P. This means that A i s I XS5 + C _ C , , s t r u c t u r e s , sn 1 8o A i s X S 4 w+ c eno—valid i f a n d o n l y i f A ' i s Fact , X + C „ — valid, provided X is negation free. 3 — Just like the proof in the previous fact. We can get away with using Proof. v S4 a style l modality because we have no need to show that the structure C an of i opens d is closed under negation. What about negation and S4 modalities? Is a there anything else we can say? There is, but it is incomplete. For now, if Xs is a logic, let JX be the corresponding logic given by 'liberalising' the negation laws to allow an intuitionistic negation. Then, we have the followd result. e ing s i r e19 A is XS4 + c Fact if X + C , , — valid nd A a .' is f— v a l i d Proof Just as before, but we note that the negation operator—on C satisfies ( w h e r the conditions for an intuitionistic negation, by Fact 15. So, the resulting esubstructure C inside an original modal stucture P is a JX + C „ structure. X h It is aalot harder s to prove the converse, because we would have to show that a every J X + q... C„ structure can be embedded within a XS4 C This 1 0 dstructure. e is difficult, because we need to construct a de Morgan negation and a modality from 'thin air' while ensuring that they interact in the .M. . C o r desired way (satisfying —a = I (—a) for each a E g a n and and for other techniques. C ) .oneT I must h i sleave ifor sanother occasion, a n d i fe f i gc u l t a t5. Conclusion at s i k o , n ) We have learned a number of things in these few pages. We have shown how to model modalities in a general substructural setting, and we have seen the way that adding structural rules to the necessitives or 'opens' may leave the underlying non-modal structure unscathed — or it may change it. Finally, we've seen how adding structural rules gives us a translation from systems with structural rules in general, into systems which contain those structural rules in a qualified form. This has opened a way to see Girard's results in modalities in linear logic in a much wider setting.* 38 G R E G RESTALL 6. Note *Thariks to Pragati Jain, Allegra Bencivenni and an anonymous referee for comments helping me to clarify my presentation. After writing and submitting this paper, I came across Kosta Dosen [2], which covers similar ground (but not exactly the same ground) to this paper. Anyone interested in these issues should carefully study that paper. Australian National University REFERENCES [1] Nuel D. Belnap, Jr. "A Useful Four-Valued Logic," pages 8-37 in J.M. Dunn and G. Epstein, editors Modern Uses of Multiple-Valued Logics, D. Reidel, D o [2]vKosta Dosen. "Modal Translations in Substructural Logics," Journal of Philosophical r d r e c h t Logic 21 (1992) 283-336. [3], Jean-Yves Girard. "Linear Logic," Theoretical Computer Science 50 (1987)1-101. 9 7 K. Meyer, Steve Giambrone and Ross T. Brady. "Where [4]1Robert 7 . Gamma Fails," Studia Logica 43 (1984) 247-256. [5] John K. Slaney. n'he Implications of Paraconsistency," Proceedings of the 12th International Joint Conference on Artificial Intelligence, Vol. 2 (1991) 1052-1057.