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Barro & A. Sobrino. 5x Santiago de Compostela: Universidad de Santiago, 1993 5x 0pp. 115-22. 5x , ISBN 8460475107 5x 5xG=n/n/n/ 5x o ۟  ESES .,,. 6&&ein wittgensteiniana wittgensteiniano6&EstndarABAJO.PRSRSx  6&finitif@p@@FF MMx6&EstndarNELIA simple interl.DINA4 sans N)   # q| P7K6qP# dddd      <<  {## v    vv =ESUK# p.7n#X01Í Í\hX1*Í.  sF V A Chain of Fuzzy Strengthenings of Entailment Logic`n"#^ă   yidddy7 7  r &g A Chain of Fuzzy Strengthenings of  rF /~ԚEntailment Logic   W" 1` ÚLorenzo Pe9a (Instituto de Filosof1a del CSIC (H Pinar 25, E28006 Madrid, Spain .hFax # +3418030948  W '5 _mail: laurentius@pinar1.csic.es 7 7  U 1  1 !"%! r4 q dd% Q #Xh"^ vP)E!:XP#: Resumenă  K ESESEl sistema de l;gica del entailment de Anderson &  K Belnap, E , fue construido atendiendo a motivaciones muy alejadas de las que animaron a la puesta en pie de los sistemas fuzzy. Sin embargo, el presente trabajo muestra que cabe desarrollar un tratamiento fuzzy aprovechando la construcci;n  K de E . La idea central de E es que  pq  es verdadero si y s;lo  J si hay una deducci;n natural de  q  a partir de  p  y eso  Jy se nota en que en la deducci;n  p  no es una premisa ociosa. Reinterpretamos aqu1 ese v1nculo de relevancia en el sentido de que el grado de falsedad de la conclusi;n no exceda al de  K las premisas. Reforzando el sistema E con una serie de axiomas  J adicionales, obtenemos una l;gica conforme con tal lectura.ĵ$##88!8$ !8 !8 !8 !8 !8 !8 !8 !8 88##  W 1. Some anomalies of the logic of entailment  Wr! Entailmental logical system E has been publicized by its founders, Anderson and Belnap " A&B for short", as a logic of both relevance and necessity (see [And&Bel), pp.23ff). Admittedly the system has turned out to be less stable than it was supposed to be " a wide variety of strengthenings having been proposed in the literature, esp. by the authors of [RLR], pp. 242 ff. ", and in fact even A&B themselves realized  WV& that several axioms can be added to system E which do not run against the system's motivation [And&Bel], pp. 3401. (The fact has been highlighted and gone into from a variety of philosophical outlooks; see e.g. Orayen's discussion in [Orayen], n.14,  WF) p. 86.) However, system E has remained a paradigm of entailmental logic. Such strengthenings as have been proposed are either of no really weighty logical significance or else bring about the abandonment of at least a part of the initial motivation, relevance, as characterized by the standards of use-in-proof and variable-sharing " which06,=o.o.o.8rZ !0 nevertheless are different and separable constraints, as M)ndez (see [M)ndez]), Avron (see [Avron]) and other authors have shown.  Wj However, system E is afflicted with a number of shortcomings. Several of them have already been pointed at both by the deep relevantist school headed by Richard Sylvan in [RLR] and in many other places " as well as by the already mentioned M)ndez and Avron. Other among the shortcomings have not been emphasized yet. Let me enumerate some of those drawbacks. First and foremost, the Adjunction  ] rule " Adj for short " (namely  p ,  q    pUq ) is introduced as an additional primitive inferencerule, with no justification, but, worse than that, it is in fact crippled in any natural-deduction implementation of the system " in fact the rule applies only within the domain of logic, the system failing to allow a general application of the rule  U in virtue of which from any two premises  p  and  q  the conclusion  pUq  could  W be drawn. Any application of system E to a non-logical theory will have to fall back on accepting only one nonlogical axiom of the theory, which will be the conjunction  W of all its ordinary axioms. To that extent, system E is in a very appropriate sense a non-conjunctive or non-copulative system, even though it falls short of renouncing  ] Adj altogether as do other paraconsistent systems (Jakowski's discussive logic (see [Cos&Dub]) or Rescher & Brandom's modal approach (see [Res&Bra]). Moreover, the core motivation of entailmental logic was the Entailment Principle  U ([And&Bel], pp. 2778), viz. that an inference of  p  from premises  qN ,  q ,  U 8,  q  is valid iff  rp  is a logical theorem, where  r  is the conjunction of  qN ,  W  q , 8,  q . Thus, system E " unlike deep relevant logics " countenances  ] Conjunctive Assertion ( pqUpq ). But without a fullfledged or unshackled Adj  W we cannot in general conclude  pqUp  from  pq  and  p . Thus, E debars  W us from applying the Entailment Principle in ou inferring practice, i.e. to first draw  U from from { pq ,  p } the conclusion  pqUp  and then from that conjunction  U alone to infer  q .  WZ Furthermore, system E postulates a principle of distributivity which also causes serious trouble for the natural-deduction and Gentzen implementations of the system (see [Dunn]), and which anyway is sufficiently convoluted to deserve being a proved theorem rather than postulated as an axiom. No less surprising is the failure of Factor, a principle investigated by R. Sylvan. (On this point, my arguments borrow from [Syl&Urb], which is the main extant study  W" of Factor.) Factor is  pq.pUr.qUr . Due to lack thereof, system E does not allow us to conclude that heads of horses are heads of animals from the premise that horses are animals. For let `p' mean `x is a horse'; `q', `x is an animal'; `r', `z is x's head'; then the wanted inference would be: zx(pq) zx(yz(pUr)yz(qUr)); but  W& within a quantificational extension of E we can prove only this: zx,z(pqU.rr))  zx(yz(pUr)yz(qUr). (The axioms for introducing quantifiers are unproblematic.)  W( In other words, we cannot simply assume that what is a creature's head is its head; we need to state it as an additional premise, despite its being a logical theorem. Likewise, lack of Factor entails that set theory is going to become useless; for define inclusion, `xEz', as `zu(uxuz)'; then without Factor we'll lose the following~+o.,,AA  U settheoretical principles:  xEzzv(xvEzv) ;  x=zzu(ux=uz) , with equality defined as mutual inclusion; and even if we lay down the principle of ex U tensionality,  zx,z(x=zzv(xvzv)) , we'll be " without Factor " unable to  U conclude that  x=z.p[x]p[x/z]  (let `p' be, e.g., `xuUuy'). On the other  U hand, laying down extensionality in the stronger version of the schema  zx,z(x=z. U p[x]p[x/z])  yields  x=z.pp , even with no occurrence of `x' in `p'; which is precisely a settheoretical version of what A&B tried to avoid by banning Factor,  U namely  pq.rr .  WF Interestingly, Factor is " not just when added to E but even upon much weaker systems of deep relevant logic " tantamount to Inclusion, namely the principle  U<  pqI.pUqIp , where `I' is mutual implication. The rationale for Inclusion is that  U6  p  implies  q  iff the content of  q  is included in that of  p , i.e. iff  p   U0 is equivalent to  pUq  " so that by reaching  q  you are only expressing something  U* implicitly contained in  p . Such an idea has been widely invoked in support of relevant logic. And yet only extremely weak relevant logics can admit of Factor (or Inclusion);  ] for, if a logic has MP, Adj, the transitivity rule ( pq ,  qr    pr ), Com U  mutativiy of both disjunction and conjunction, Addition ( p.pVq ) and Simplifica W tion ( pUqp ), then, if it has either Idempotence of either conjunction  W Ԛ( p.pUp ) or disjunction ( pVpp ), or else mutual Distributivity, it has as a  U theorem  pp.qq . (See [Syl&Urb] again.)  W Most of all, system E allows suppression (or omission) of proved entailments in exported form only, not in imported form. That is very odd, since A&B ([And&Bel], p.262 top) allege that omission is what leads to the erroneous law of  Wr exportation in CL . E bestows a privileged status to implicational formulae and counte Un nances such principles as Suppression ( ppqq ) and exported transitivity ( pq. Uh qr.pr ), the only natural way of justifying which is to allow Exportation for  Wb implicational formulae, due to their special status " something E falls short of  U^ accepting. Thus although from  pNp.qrs  " provided  qr  is a theorem  UX " we can deduce  pNps , we cannot draw the conclusion from  pNpU( UR qr)s , even if  qr  is a logical theorem. So  pqU(rr)s.pqs  is not  WL a theorem of E . Accordingly a conjunction of two implicative formulae is not to count as an implicative formula at all! Although those different shortcomings may seem unrelated, in fact they arise from the same source: the weakened status of conjunction and the lack of what we can call interpolation axioms " axioms, that is, which are simpler and through which  W# the less obvious axioms of E would become provable theorems. 1   b' E.      ă  +o.,,AAԌ W  2. Strengthening system E into a logic of fuzziness  Taking as my starting point the considerations put forward above, I am going  Wj to propose a chain of strengthenings of system E which cure it from the anomalies. What is most interesting is that the emerging systems are clearly interpretable as logics  W` of fuzziness or graduality. Thus the E functor `', duly strengthened, receives an appealing construal as meaning `to the extent [at least] that'. Provided with such a reading, the functor has to be stronger than the original  W  E arrow, under natural assumptions concerning the structure of truth degrees.  W8 So, the resulting systems are closer to CL (Classical Logic). They are not relevant, in the sense that they do not comply with the relevantist constraints, or at least not with the one of variable-sharing. They comply with the constraint of use in proof, with some adaptations. But that natural-deduction implementation lies beyond the scope of the present paper. The resulting systems of fuzzy or gradualistic logic are intermediary between classi W cal and entailmental logic. The approximation to CL goes further than that. In fact,  W several of the systems contain all of CL , and are indeed conservative extensions of  W  CL . Only, the CL negation `' is given a different reading from what is customary. It is now read `not8at all'. Classical negation is complete negation " the functor which maps anything true, to whatever extent, into complete falsehood, and the utterly false  Wt into complete truth. In some important sense, those systems are to CL as relevant logic was meant to be to intuitionistic logic. All those systems have the laws of non-contradiction and excluded middle. They are paraconsistent copulative systems. I proceed to constructing the chain.  WJ System P0 = E System P1 = P0 + Factor  U, System P2 = P1 + Linearization ( pqV.qp )  U System P3 = P2 + the Self-Self Principle ( N(ppN(pp)).pp )  W  System P4 = P3 + IF [Implicational Funnel] ( pqV.pqr )  W~ System P4.5 = P4 + Mingle = CL . Mingle is  p.pp   U!  System P5 = P4 + one of these: Aristotle ( N(pNp) ), Boethius  U" Ԛ( pqN(pNq) ) or Contradiction (namely, for some particular sentential  U# constant j:  jUNj )#  UT%  System P6 = P4 + both these principles:  pq.NHNHpHq  and  UN& Ԛ HpqV.Npr  (`H' is read: `It is altogether true that')#  W'  System P3.5 = P3 + those two principles involving `H' = P6 minus IF #  U0) System P7 = P6 +  HpVNp   U*  System P8 = P3.5 + these two:    and  pV.pq  (`' is a sentential constant meaning the conjunction of all truths)#+o.,,AAԌ W System P8.5 = P8 +  H  = CL  Ur System P9 = P8 +  NH()   U System P10 = P8 +  HN(N)  It can be proved that P8 contains P7; that P10 contains P9; that P9 contains  WL P5; that P8 is a conservative extension of CL (if classical negation, `', is defined as `HN'), and that P5 and such systems as contain P5 are not just paraconsistent but  UB contradictorial (they have the theorem of Heracleitus:  N(pp) ). (In fact,  N(pp) W< (pp)(pp)N(pp)N(N(pp)(pp)(pp)N(pp))  is a theorem of E ; the apodosis is the negation of the protasis; when we add Aristotle or Boethius, the result is immediate: the negation of the theorem is also a theorem. But in P1 we prove  U,  pUNpN(qq) .) Among all those systems, P5 and P10 are by far the most important ones. They are the most stable. The main motivation in going from P4 to P6 and up is to  W encompass CL within fuzzy logic, as the extreme case wherein it is a question of either complete falseness or not (alternatively of either complete truth or not). System P8 and those above it allow to implement classical conditional (which can be defined in the usual way with strong negation, `', and disjunction, `V') in the relevantly recommended way: pDq iff there is some truth, namely , such that pUq. A number of Hilbert style axiomatizations are provided below, which are more elegant than the genetic presentation just described. Here is an axiomatization for P5 with ten axioms and one primitive inference rule:  WN  Primitive symbols : `U', `N', `'. `p', `q' etc are used as schematic letters. Notational conventions are ! la Church: all twoplace functors have the same weight and associate leftwards; a dot after a twoplace functor stands for a left parenthesis with its mate as far to the right as possible.  W B# Definitions :  pVq  abbr  N(NpUNq) ;  p  abbr  N(pNp)  P5a01 pqrU(qpr)r P5a02 pqU(qr).pr P5a03 pUqUr.rUpUq P5a04 pUqp P5a05 pqrsU((pp)(pq)s)s"o.,,AA P5a06 pq.rs.pqU.rs P5a07 pq.p.pUq P5a08 NpqN(pq) P5a09 pNq.qNp P5a10 NNpp8"o.,,AA# ,,AA8ԯ ]P$  Inference rule : DMP (i.e. disjunctive modus ponens): for n1: pNqV(pq)V8V.pq, pN, 8, p  q  ]8' MP [Modus Ponens] is a particular case of the rule " the one wherein n=1.  ]:( Adj is a derived inferencerule.  W) The significance of DMP is that to deduce  q  from a number of premises is  U* to show that  q  is deduced from at least one of them. Which does not mean that necessarily there is a proof from one of the premises alone, since the whole proof of+o.,,AA the conclusion from the premises consists in showing that it either follows from the first premise, or from the second premise, and so on. Those deductions are not  W complete proofs, but proofbranches or alternative subproofs. Five alternative axiomatizations of system P5 are now provided.  W  2d axiomatisation . P5b01 ppqq P5b02 pqU(qr).pro.,,AA P5b03 pUqUr.rUpUq P5b04 pUqp8o.,,AAH,,AA8ԯ U( P5b05 pqpp (provided  p  is an implicational formula)  U P5b06 p.q.pUq (provided  p  and  q  are implicational formulae) P5b07 pq.p.pUq P5b08 N(Npp) P5b09 pNq.qNpo.,,AA P5b10 NNpp P5b11 pqV.qp P5b12 (pp).pp8o.,,AA ,,AA8ԯ WX Same defintions and same inferencerule ( DMP ) as above.  W  3d axiomatisation . P5c01 pqr.qprr P5c02 pq.qr.pr P5c03 pUqUr.rUpUq P5c04 pUqpo.,,AA P5c05 (pp).pp P5c06 p(qr).pUqr P5c07 NpNp.pqpp P5c08 pq.p.pUq8o.,,AA<,,AA8ԯP5c09 Nppp (or, alternatively, pUNqN(pq)) P5c10 pNq.qNplo.,,AA P5c11 NNpp8lo.,,AAfl,,AA8ԯ P5c12 pNqN(pq) (or, alternatively, UN, with `' a primitive sentential constant)#  WF Same definitions and same inference rule ( DMP ) as above.  W  4th axiomatisation . P5d01 ppqq P5d02 pq.qr.pr P5d03 pUqUr.rUpUq P5d04 pUqp P5d05 pqrV.pq P5d06 pq(qp).qpZ(o.,,AA P5d07 pq.p.pUq P5d08 Nppp P5d09 Npq.Nqp P5d10 pNNp P5d11 pqN(Npq)) P5d12 N(pp).ppN(pp)8Z(o.,,AAT)*!,,AA8ԯ W) Same definitions; same inferencerule ( DMP ).  W<+  5th axiomatisation .<+o.,,AAԌP5e01 pq.qr.pr P5e02 pUqp P5e03 pq.rUp.qUr P5e04 NNp.pUp P5e05 pqrV.pqo.,,AA P5e06 pqrU(qpr)r P5e07 pNqN(pq) P5e08 pqN(pUNq) P5e09 pNq.qNp P5e10 (pp).pp8o.,,AA,,AA8ԯ W0 Same definitions; same inferencerule ( DMP ). (The rule can be weakened to n2  W, alone, thanks to P5e05, i.e. IF " taking as an instance thereof  pqqV.pq ; as much applies to any other formulation with the same disjunctive wording of the  ]" axiom, e.g. the 4th axiomatisation with P5d05. In that sense MP can be looked upon as enthymematic.)  W  6th axiomatisation  U Primitives: V, t , , N.  pIq  abbr  pqU.qp ;  pUq  abbr  N(NpVNq) . `t' is a new primitive (which is meant to be the conjunction of all implicative truths). `' is a sentential constant whose meaning does not concern us here. Two inference  ] rules: MP and Adj. P5f01 tpp P5f02 pq.qr.pr P5f03 p(pq).pq P5f04 p.pVq P5f05 q.pVq P5f06 pq.rq.pVrqo.,,AA P5f07 pNpNp P5f08 pNq.qNp P5f09 NNpp P5f10 NU..N P5f11 t.pqV.qpU.pqr8*o.,,AAj,,AA8ԯ U  We then prove that  It . In fact, we could use just one of those two primitives, except that for philosophical reasons it is better to prove their equivalence than to postulate it. Upon the basis of P4 all the follwing are equivalent (and all of them bring about  Wh a collapse into CL ):  pIqV.qIrV.pIr (i.e. the Dugundji formula in three schematic letters)# p.pqq (Exported Assertion) p(qr).q.pr (Permutation) pUqr.p.qr (Exportation) pUqr.pUNrNq (Antilogism) p$qrI.p.qr (The fusion principle, where `$' is a primitive fusion functor) p.q.pUq (the Adjunction Principle) p.pp (Mingle)*o.,,AAԌpUNpq (Cornubia) p.qp (VEQ) pqV.qr o.,,AA N(pp)q*o.,,AAԌN(pp).q.pp N(pq).pp p.qq8 o.,,AA *,,AA8ԯNow we present an axiomatisation for P10 by adding to P5 the following axiom  UJ schemata (`H' is a oneplace primitive functor, while  p  abbr  HNp  and  pDq   UD abbr  pVq ): P10a1 pUpq P10a2 pq.HpHq P10a3 Hpp P10a4 NHpHp o.,,AA P10a5  P10a6 pD.p P10a7 HN8 o.,,AA  ,,AA8ԯ Wt Since in P10 we can prove Funnel (namely  pqVp ), IF (as it stands in all the foregoing formulations of P5 " whether as such or as implicational Peirce, i.e.  Uj  pqpp  provided  p  is implicational) becomes redundant. Thus a more elegant axiomatisation of P10 has to be found. 1   b E.      ă   W9  3. Some outstanding results  Here is a short list of theoremschemata and rules which can be proved and derived, respectively, within P10: B#C|#҇pDq , p  q pqD.pDq pDq, q  p o.,,AA pUqI(pVq) pVqI(pUq) pDq  pVrD.qVr o.,,AA pD.qD.pUq p.qDp p.pDq> o.,,AA  ,,AA>ԯIt can be easily proved that the fragment of P10 in is exactly  W]"  CL " of which, therefore, P10 is a conservative extension. We had rather look upon  WY# P10 as the result of enlarging CL with new functors, a nonstrong negation, `N', and  UU$ an implication, `'. The intended sense is that  pq  is true iff the degree of truth  UO% of  p  is smaller than or equal to that of  q , whereas  Np  is as true {false} as  UI&  p  is false {true} " in other words, simple or natural negation reverses the order expressed by `,' and that is all it does. In the present context we'd better read classical  ]=( negation, `', as `not8at all'. The existence of two conditionals (the mere conditional `D' and implication, `') entails that there are in P10 two different deduction relations. We can keep `' as expressing mere inference, while using `*' as expressing a stronger inference relation,+ o.,,AA  U namely: pN, 8, p *P10q iff  pN8pq  is a theorem of P10; whereas for `' we  U do have the classical deduction metatheorem, viz. pN, 8, p, r P10q iff pN,8,p  U P10rDq " theorems being such formulae as are inferred from an empty antecedent. Clearly if p*q then pq. Our proof theory has to be implemented in such a way that those differences are taken into account. Entailment, duly strengthened, is a functor sensitive to degree differences, whereas the mere conditional, `D', only takes into account whether formulae are completely  WL false or not. CL is a poor system, not a wrong one. Its flaw consists in the fact that,  WH by lacking an implication and a negation which are degree sensitive, CL compels people  WD to think, legislate and act in terms of all or nothing " since for all CL knows `wholly' and `just a little' are merely stylistic variations, with no semantic difference between  ]: its being altogether true that and its being [just] true.  W Here is a semantic treatment of P5 and P10. By a P5matrix we mean an algebra  ^  A =<A,O,D> where A is a wellordered set of entities, DEA, and O is an ordered set of operations , N being unary and the other binary and such that there are three elements , 0, 1 satisfying these postulates: 0 is the minimum; 1,  ^ the maximum; xVz = max (x,z); xUz = min (x,z); xz iff NxNz; N=; D is  X a proper filter such that D; xz =:  if xz, and else 0; N0=1; NNx=x. Let  } T be an extension of P5. A valuation, v is a mapping from T into a P5matrix, A , such  } that for any two formulae,  p ,  q , the usual conditions are met: v (pUq) =  }  v (p)U v (q), and the like for the remaining functors and operations. A formula  p   }  of T is valid iff every valuation, v , is such that v (p)D (the respective set of designated  }- entities within its range). An inference (i.e. an ordered pair < C , p > where C is a  }O set of formulae) is valid iff every valuation v such that, for every  q  C , v (q)D  }q is such that v (p)D. Clearly, P5matrices are Kleene algebras (i.e. QuasiBoolean algebras satisfying the Kleene postulate that xUNx  zVNz). In fact it is easily proved that each finite P5matrix is isomorphic to one whose carrier is a finite subset of the set of the integers such that for every positive integer therein its negative also belongs, and conversely: the algebraic  will be the numeric zero, while the algebraic 1 will be the greatest number in the set. Soundness and completeness are straightforwardly proved. Notice, though, that none of those [finite] matrices is characteristic. But let  ^o us form an infinite P5matrix, A , whose carrier comprises any infinite subset of the natural numbers, including zero, and their respective negatives, plus  and -. Let  ^l" D be [0,]. Then A  is characteristic in the sense that only all P5 theorems are map Wo# ped into designated elements, but it is not strongly characteristic in that there are  ^k$ valid inferences (with respect to A ) which are not derivable in the system P5 (e.g.  Un% { p , Np ,Ó q , Nq }   pq ). On the other hand, if we let the set of designated values to comprise all elements except , there will be valid formulae  Ub' which are not theorems of P5, e.g.  pqVp  (Funnel).  W( Let us form a P10matrix by adding to a P5matrix an additional unary operation, H, and additional elements, 3 and , such that: 3=N; x if x/0; Hx=: 1 if x=1, and else 0 (1 and 0 are the algebraic 1 and 0 respectively). A valuation* o.,,AA  } is a mapping from a P10 theory T into a P10matrix (we lay down that v ()= and so on). Now, not only are soundness and completeness attained but, more importantly, a strongly characteristic matrix becomes available: the set of all integers plus four additional elements, > 3  every integer; and of course every integer  -3 =  >  X -, with D comprising all elements except . That matrix is the canonic P10 matrix. Thus P10 has the finite model property, namely that every [nondeliquescent, i.e. Postconsistent] nonconservative extension thereof has a finite characteristic matrix.  Xw For let L be such an extension of P10. There is a formula or a schema  p  within  }t the vocabulary of P10 which is a theorem in L but not in P10. Let B and v be a P10  } algebra and a valuation, respectively, such that v (p) is designated. The reason cannot be that the set of designated elements has been enlarged, of course. If the carrier of  ^  B is infinite, then B is isomorphic to the canonic matrix of P10. Therefore, B is finite.  X Which entails that for some natural number n L is P10 plus the Dugundji formula in  U n disjuncts, namely:  pNIpV.8V.pNIpV.8V.p-NIp . None of those formulae is a theorem of P10. Most interestingly, when propositional quantifiers are used with the usual postulates for them (see [And_Bel_Dun], pp. 19ff), " using `j' as a propositional variable "  U we define  ~p  as  pzjj , and  pq  as  yj(jUpqUj) ; the fragment of the  W  ensuing system, P10yzj, in is exactly CL . In that sense, P10 is to CL  W precisely as E is to intuitionistic logic.*%The research leading to this paper was developed mainly during my stay as a Visitor to the Australian National University in Canberra (from 01111992 to 30041993), thanks to a scholarship granted by the Spanish Ministry of Education. I owe my heartfelt appreciation to Richard Sylvan, whose friendly and supportive advice has a greater value than I can express here. My gratefulness, too, to Graham Priest, R.K. Meyer and J. Slaney for their helpful suggestions and comments. An ancestor of this paper was delivered at the MiniConference on Finite and Infinite Models and Related Issues, Canberra, April 4 1993. 1   bX E.      ă   W  4. References   ]M  [And&Bel] Alan R. Anderson, Nuel D. Belnap, Jr., et alii, Entailment: The Logic  ]O of Relevance and Necessity, vol. I. Princeton U.P., 1975.#O ` o.,,AAԌ [And_Bel_Dun] Alan R. Anderson, Nuel D. Belnap, Jr. and J. Michael Dunn (with  ] the collaboration of others) Entailment: The Logic of Relevance and Necessity, vol. II. Princeton U.P., 1992.#  ]l  [Avron] Arnon Avron, Whither Relevant Logic? , Journal of Philosophical Logic 21/3 (Aug 1992), pp. 24382.#  [Cos&Dub] N.C.A. da Costa & L. Dubikajtis, On Jakowski's Discussive Logic ,  ] NonClassical Logic, Model Theory and Computability, ed. by Arruda, da Costa & Chuaqui. NorthHolland, 1977, pp. 3756.#  ]J  [Dunn] J. Michael Dunn, Relevance Logic and Entailment , Handbook of Philosophi ]L Ԛcal Logic, vol III, ed. by D. Gabbay & F. Guenthner, Reidel, 1986, pp. 11724.#  ]  [M)ndez] Jos) M. M)ndez, The Compatibility of Relevance and Mingle , Journal  ] of Philosophical Logic 17/3 (Aug 1988), pp. 27988.#  [Orayen] RaCl Orayen, Una evaluaci;n de las cr1ticas relevantistas a la l;gica clsica ,  ]8 L;gica y filosof1a del lenguaje, ed. by S. Alvarez, F. Broncano & M.A. Quintanilla. Universidad de Salamanca, 1986, pp. 7388.#  ]  [Res&Bra] Nicholas Rescher & R. Brandom, The Logic of Inconsistency, Blackwell, 1979.#  ]  [RLR] Richard Routley, Val Plumwood, R.K. Meyer & R.T. Brady, Relevant Logics  ] and Their Rivals. Atascadero (California): Ridgeview, 1982.#  ]  [Syl&Urb] Richard Sylvan & Igor Urbas, Factorisation Logics. Canberra: Australian National University, 1989.#