WPCjB 2k BNm ZTimes Roman3|x ierHelvetica  PCXP"^2CRddCCCdq2C28dddddddddd88qqqYzoCNzoozzC8C^dCYdYdYCdd88d8ddddCN8ddddY`(`lC2CC!CCCCCCCCCCd8YYYYYYzYzYzYzYC8C8C8C8ddddddddddYddddYYYYYYYdzYzYzYzYddddddddC8C8C8C8Ndz8z8z8z8z8ddddddCCCoNoNoNoNz8z8z8dddddddzYzYzYdz8dCoNz8dddddLaurentius_PostScript_(HP_LJ_III.PS)LAURENTI.PRSo\  PChhhh2.HXP2M ?3'3'Standard6&&ein wittgensteiniana wittgensteiniano6&Standardius_PostScript_( #x  @U X@# dddd X` hp x (#%'0*,.8135@8:_}}_ 99_99__UU&9xx0Xxx@2% F)RRz""m^88Goo,CCNu8C88oooooooooo88uuuo˅z8dozz888^o,oodoo8oo,,d,ooooCd8oddddC4CuC8CC!CCCCCCCCCCz8oooooȲdoooo88888888ooooooooodoozodoooddddooooooooooooo88888,88ddo,o,o,o,o,ooooooȽCCCddddz8z8z8oooooodzdzdzdoo,oCdz8ddoooKF8koCzoooooJIoo&CCoCCoodd,CS$ W2PkCPFdSW2p}wCHfx`Mx0NxxHTr~~xf`x`~~,NN xH~xx8,fTimes RomanCourierHelveticaHelvetica BoldHelvetica ObliqueSymbolheZNjb**TQ~hjTjTLWeW~QLL5F5F"IQLQL$QL""I$pLNQQ2@'LChFFCFFFFxFFxxFFxFxFFLWIWIWIWIWIshLZLZLZLZL*"*"*"*"hLjNjNjNjNeLeLeLeLLFWIeQjNjNLFxxTQWIWIWIhLhLhLhLeQZLZLZLZLjQjQjQjQjQjQbL***"*x*TIQ$Q$Q$xxT$hLhxhLhLjNjNT2T2T2L@L@L@L@W'W'eLeLeLeLeLeL~hLFLCLCLCeQQ$hLT2L@W'LFLFeQjNeLxxxN_xx"FF'@>>FNxF5xppF-xssTp->>F''FFs_sNssLLxx">xx-"^SSk CdduSdSSSS1SSSSCSCCCdSdNddSdd,2ddddddddddS, SSSSSSSSSSSSSCSSCCCCC,dddSSSCdSKiSdon9dd,dd,,Cd, xTxxD,r"^Sd SdduSdSSdd%SdSdSdSSS uduTudSdd,2ddddddddddS, SSSSSSSSSSSSSSSSSSSSS,uuudddSudKiSdonG,dd,,S,xk<ixxU20],& ' (^R,Times RomanCourierHelveticaHelvetica BoldHelvetica ObliqueSymbolHelvetica Bold Obliquerd (Bold) (Glyphix)Oxford (Italic) (Glyphix)Rockland (Bold) (Glyphix)Greek (Glyphix)Oxford (Bold Italic) (Glyphix)"m^8C_oo8CCNu8C88ooooooooooCCuuuzÐz8ozzzC8Cuo8ozozoCzz88o8zzzzNoCzooodN8NuC8CC!CCCCCCCCCCz8oooooȲooooo88888888zzzzzzzzzoozzzoooooooozoooozzzzzzzz88888888ooz8z8z8z8z8zzzzzzȽNNNoooozCzCzCzzzzzzozdzdzdzz8zNozCoozzzKF8ooCzoooooJIoo0ddoCCoozz8dxV8hxxH"9 ^8CRddCCCdn2n28dddddddddd88nnnYzoCNzoozzC8C^dCYdYdYCdd88d8ddddCN8ddddY`(`lK\2[dCYddddd7>dd$YYdCCddooCYqnnn!8nBBnnnyyPn7c1RyyXyycnnnndccccccccMMMMMMMMMMMMы~nyRzcXcyhFBnnshcnntnvyX~Xsyn~XyBBnss~y~~~~~~~~~~~~~~~~~~~XXXXXXXyyyyyyyyyyyyyyyyyyyyBBBBBBBBBBBBnnnnnnnssssssssssssFFnb9/b2PkCPkF}xx_ 7oC2o\  PCXPT?xxx  x6X@KX@L 1sC88:s2PkCXP   L j"Q/'ˏQ2PkCP /xC8:x2p}wCXLIdS$ W2PkCP FdSW2p}wCL=SFw2PkCP:SF2p}wCL)b9/b2PkCP'1sC8-:s2x(CXX D21mC8?K:m2PAXP/xC8FzF:x2x4vCXH@@@2s6`1@8l *|'n==,n4 PI>'P2xxx,#x  @UX@}?hCh,4iXhvPEXP  ~sP1,,FzP4 PcUP?lCp,YXl ^Sp{EX_d,4cvPEP^d,Y ^Sp{EOS,43vPEPNS,Y\ ^Sp{E@hCh,qXhiSxnEXX |LlC,Xls p7EX W5L/b,k3Ls p7E G|C,X|\ PK@XPB@pCp,XpiSxFEX2;6s9::o)cr.ANELIA simple interl.DINA4 sans NpageESES ,.,. 6&6&EstndarHPLASII0.PRSXh46&finitif@p@@FF MMx6&EstndarrJet IIIDl,*\    #q| P7>qP# ``dd      K< 3{## {##Њ v Bibliogr.yBibliograf1a'S33 #3 separaci;n entre un cap. y su bibliogr.e¦ xك  n al Coloqd" XBك 2;=@hBb mESES .,,. 6&&ein wittgensteiniana wittgensteiniano6&EstndarABAJO.PRSRSx  6&finitif@p@@FF MMx6&Estndaros_HP_LJ4Si)h  #?sjp P7?P# h   =d)cr.DINA4 simple interl.&Npage,ANELIAge ESES ,.,. 6&6&EstndarBRUDGLYP.PRSXh46&finitif@p@@FF MMx6&EstndarBRUDGLYP.PRSXh4,; ````  Xh4 PEqS    =     K< X` hp x (#%'0*,.8135@8: 8,  q , the flags or subscripts assigned to the premises must be passed over to the conclusion, thus keeping track of the reasoning thread. We could then expect that within  R that natural deduction framework  pUq  would be deduced from {p,q} provided the  R subscripts of both premises are inherited by the conclusion, that is to say provided  pUq   R is assigned all the subscripts of  p  and all the subscripts of  q . But no, things cannot  R be like that. For then, within system R we could prove  p.q.pUq , and therefore  R  p.qp  " which, in the colourful expression of Dunn, one of the champions of the school, would be equivalent to washing dirty money through the third world. Within  R system E that dismal outcome is averted, even with Adjunction formulated as we have  Rs just done, but a different if less damaging result ensues: we could then prove  pq. R\ p.pUq , which is the principle of reduced factor, RF for short. From which the whole  RG! Factor would be easily deduced:  pq.pUr.qUr . Factor is a principle which has been studied by the Australian relevantists, specially by Sylvan and by Sylvan & Urbas in a joint publication. Those authors have  Rx$ shown that Factor leads to irrelevance when joined to the axioms of system E but not necessarily so when several of those axioms are sufficiently watered down.  R& Within the framework of system E of entailment, Factor (or equivalently RF ) immediately lead to validating as a theorem the principle of implied selfimplications, PII,  R(  pq.rr : any selfimplication is implied by any implication. True, that result is very very far from the banned VEQ. A system with PII may nonetheless comply with some of the relevantist strictures. For instance it may fail to have a strongest formula, one,  RQ+  p , that is, such that for all formulae  q :  pq . It may have the Ackermann property,Q+ o.,,55  R which means that for no implicative  q  is  pq  a theorem, when  p  is a sentential  R variable. In fact a system with PII (or equivalently " within the framework of E " with Factor) may remain miles away from classical logic. However, such a system is no longer a relevant system. For a minimal, necessary (although not sufficient) condition  R of relevance is that no formula  pq  is a theorem if  p  and  q  share no common variable.  ^  3." Three Ways Out " or the Reasons for a Gradualistic  [  Appropriation # It is a widespread tendency in human behaviour to be content with nothing less than grand thoroughgoing principles, adherence to which can afford stable situations. Most of the time, things are less straightforward and more complicated than we had fancied. The all or nothing criterion is likely to lead us astray. That seems to me the case as regards the idea of relevance. It was a nice idea. It appealed to some qualms a number of students have felt over the years when first becoming acquainted with classical logic and the apparently whimsical turns of classical inference. Yet implementing the idea up to the hilt leads to such amazing results that you ought to wonder if the price is right. There are several ways out. One is offered by what I have called radical relevantism, that of M)ndez and Avron. The word may not be apposite, since in some important (and relevant) respects it amounts to jettisoning a nuclear part of the original relevantist plan, namely having systems with the Ackermann property, and avoiding that a logical axiom follows from a contingent sentence; in other words, claiming that from  RG  p  alone nothing follows except  p  (and  NNp  and  pUp  " and also  pVq  and  qVp  etc?); in particular nothing follows which we can also know from other sources " as logical theorems can be, through the study of logic presumably. (Radical relevantists fall  R back on the converse Ackermann property: no theorem of the form  pqr , where  r  is a sentential variable. In some important sense, as will emerge below, that kind of approach is the dual of the one I shall be suggesting towards the end of the paper.)  R3 The second way out is offered by deep relevantism , socalled, whose main champion is Richard Sylvan. Its general plan " except perhaps as concerns Factor and  R some other isolated principles " is to further weaken system E . There are a number of features separating Sylvan's philosophical enterprise from the original relevantist idea, not all of which are directly related to his weakening proposal. Such a proposal can be independently defended on the ground that it is more cautious to assert less: if we can implement logical systems sufficiently useful for reasoning keeping clear of the more controversial principles, it is reasonable to refrain from asserting those principles: since  R# logic is a priori and acquired through (considered or reflective) intuition, the more contro Rh$ vertible a principle is, the less likely it turns out to count as a genuine a priori, analytic, nonfactual truth. The other main idea in Sylvan's own enterprise seems to me a quite different and even, to some extent, opposite idea. The founding fathers loathed contradictions as much as the classicist, and never for a minute thought that a contradiction could be true at all; their objection to classical logic was not that, in virtue of the Cornubia rule, it could lead from a true contradiction to an utterly false conclusion, but that it leads from statements taken as premises to a statement taken as [pseudo]conclusion which in fact has nothing to do with the premises. As against that point of view, Sylvan has been led little by little to the idea that there can be true contra+ o.,,55Ԯdictions, and in fact that there are. Now, that may be the case, and I am in fact sure it is the case. Yet, in an important sense, this runs against relevantism as initially conceived. For if the Cornubia rule is to be rejected on the ground that a contradiction can be true after all, the classical view that what is [utterly] impossible implies everything is not challenged: you only displace the bounds of the impossible. Of course, you can be both a relevantist and a believer in true contradictions. Still, you are then bound carefully to sort out your grounds for each of your departures from classical logic, or from any logic you happen to take as your starting point. Finally, and most significantly for our present concerns, Sylvan has developed a very different approach to a logic of reasoning, which is at variance with the classical outlook in a much more radical way than even mainstream relevantism does. In so doing, he renounces the claim that RL is in general a logic of reasoning. Yet, canvassing the pros and cons of Sylvan's plan for a logic of reasoning goes beyond the scope of the present paper. A third way out is provided by a gradualistic appropriation of the relevantist plan. `Reappropriation' seems to me the right word, since gradualism and relevantism have not been bed fellows, their leanings taking them apart from one another. The main idea of gradualism has little to do with relevantist concerns. In fact it consists of recognizing that there are degrees of truth and, accordingly, since what is to some extent true is true, there are true contradictions, but that in so much as the general classical view of logic can be adapted to such an acceptance of degrees of truth, it can and must remain unchanged in all other respects. In particular, gradualism has been keen on keeping, alongside a weaker or natural negation, a strong, classical negation, endowed with the reading `not8at all', through which in fact systems of gradualistic logic are conservative extensions of CL, which maintain not only all classical theorems, but also all classical inference rules (provided the translation of classical negation is strong negation, of course). Thus, the gradualistic approach relinquishes the Cornubia rule for natural or weak negation but keeps it for strong or classical negation. However, if you have a strong negation, you are bound to countenance inferences which fall afoul of the relevantist constraints. You can no longer pride yourself on being relevant in that sense.  R Not because you avoid  pUNpq  if you accept  pUpq , `' being strong negation. Your logical choice may have a number of reasons to recommend it, but not the general unqualified principle of relevance. This state of affairs probably explains why thus far no bridges have been built between the two schools. Their original motivations kept them quite apart. Gradualism has remained adamant in its closeness to the classicist's main ideas and in fact it has been developed with forceful allegiance to an extensionalistic, Quinean approach to many philosophical subjects. The idea of degrees of truth is compatible with extensionalism, and in fact is the only ground on which Quine himself has contemplated  R# abandoning CL (in his What Price Bivalence? ,  JP , 78/2 [febr 1981], pp.90ff). However, logic has a number of surprises to offer. One of them is that gradualism is a not so distant relative of relevantism, which is going to become clear  R& through some mending (in fact a powerful strengthening) of system E . Needless to say, the kind of moderate, middle of the road approach I am going to sketch as the final part of this paper entails renouncing the unqualified main tenets of the founding fathers and probably of most relevantist thinkers. A number of inferences which do not conform to the relevantist constraints have to be accepted. To such extent, the main motivation of the relevantist movement " to capture a logic of reasoning, in a somehow puritanicalD+ o.,,55 sense of the word " seems to me hard to retrieve. Yet, something in the neighbourhood emerges, something through which the relevantist enterprise is vindicated all the same.  ^1  4." A Gradualistic Construal of SubscriptAssignment, and how  [O to Strengthen System E # The relevantist implementation of natural deduction techniques consists in assigning subscripts to the premises and thus keeping track of the thread of the argument. Since reasoning is putting forward grounds for getting some conclusion, the procedure seems quite reasonable. In fact the relevantist logicians didn't need to invent it, since it had already been designed even within the framework of CL as a didactic tool. It had only to be put to a more substantial use. Does the idea work? Well, within the relevantist program only after a fashion. The trouble comes with Adjunction, as we have seen. The natural way of countenancing  R Adjunction fails. It could be enacted as a strengthening of system E , but, as we have seen, that would entail acceptance of Factor and PII. The relevantist logicians offer us  Rr some makeshifts; for instance  pUq i can be inferred from the couple of premises  p i  R[ and  q i, that is to say both premises have to possess the same subscript. Which in  RD practical terms means that outside logic nothing can be inferred from premises  p  and  R-  q , given independently from one another. Any nonlogical theory has to be provided with only one axiom, which can be a conjunction of formulae, or else nothing can be inferred from the separate axioms, unless they are cast in terms allowing use of implicative MP. What in effect the relevantists are doing is to reduce Adjunction to a systemic rule, which is to be applied only to such premises as are logical theorems. For a logic of general reasoning such a step is a policy of despair. With Adjunction so hamstrung no bright prospect is opened for reasoning.  Ra Now, what if we strengthen E through Factor, thus unshackling Adjunction at the same time? We have seen that the general principle of relevance will no longer be in  R5 operation, since we'll have  pq.rr , the PII. But can something of the initial implementation of natural deduction techniques be rescued all the same? Yes, a lot of it can be saved. But an overhaul is necessary, and a different interpretation is to be put on the whole assigning of subscripts. The interpretation to be now considered is that keeping track of the use of the premises is a guarantee to the effect that the conclusion is not less true than the premises. That idea is closely connected with a program put forward by Guccione & Tortora, two Italian logicians working in the field of manyvalued and fuzzy logic. And  R# surprisingly once at least within the relevantist movement " apropos system RM , which, granted, is no longer a relevant system of logic " Robert K. Meyer developed similar ideas. With such an overhaul, the target is no longer that of keeping clear of irrelevancies, but that of avoiding an increase in the degree of falseness of your assertions. So, let us think of subscripts assigned to the premises as variables ranging over  R( degrees of truth or falsehood. The main idea in now that from  pq  and  p i you can  R) conclude  q j provided you jot down that ji (the degree of falseness of the conclusion  R* does not exceed that of the premise). What about the very same implication,  pq ? Doesn't it receive a subscript? All asserted implications receive the same subscript.+ o.,,55 Their degree of truth is immaterial. In fact there are grounds for regarding implication as twovalued " which does not mean that the two values must necessarily be the two classical extremes of complete truth and complete falsity. Thus implications are special. It seems to me this is as it should, even from the very same relevantist motivations. After all the founding fathers made much of the cleavage between facts and entailments. Not that I think they were quite right on that score either, since entailments are entailmental facts; their looking at entailments as  R nonfacts is perhaps connected with their acceptance of system R as a relevant logic: if an implication is, if true, a fact, then the permutation principle is hard to believe: even if p is relevant for the fact that q is relevant for r, it does not follow that q is relevant for the fact " if it is one " that p is relevant for r. Likewise, even if the authorities' carelessness causes that the earthquake causes many damages, it does not follow that the earthquake causes that the carelessness of the authorities causes many damages.  R Once we accept Factor " without relinquishing any other E principle or rule ",  R things begin to be straightened out, and a number of oddities in system E vanish. For  R instance, with system E you cannot infer  p.pUq  from  pq . Yet, in any theory  R wherein you have, for some formulae  p  and  q , both  pp  and  pq  you'll also  Rm have  p.pUq  " if Adjunction can be applied to those theorems. How is that possible? The answer is that inference, in the canonic relevant sense, is not the usual consequence operation. A theory's being closed for some operation - [on sets of of formulae] is neither a necessary nor a sufficient condition for it to be the case that within such a theory formulae in -S can be inferred from the set of formulae S. Yet, isn't it really odd that within a theory in which two implications are theorems which share all their atomic  R formulae and are in fact very similar,  pq  and  p.pUq , the latter cannot be inferred from the former? What else do we need in order to be able to draw the conclusion? Now, with our overhauled implementation, we have a different situation: any asserted implication in a system can be inferred from any implication (can be inferred provided it is asserted, of course). As regards implications, our ways are classical. That does not mean that an asserted implication can be inferred from nothing or from anything. The Ackermann property keeps sway. And proof theory becomes so much simple! Inference, so implemented, is still not necessarily the same as the consequence operation. A theory may contain theorems less true than true implications are; yet an implication, if true, does not imply those theorems. On the other hand, a true implication is not implied by all sentences; so, in particular it is not the case that for any two  R! theorems,  p ,  q , pq. But we can (and must) also recognize a different inference relation which coincides with the consequence operation in the usual, classical sense; let us say, *. From Sp it follows that S*p, but not conversely. (There are other links  Rv$ between * and : if S*p, there is some truth,  q  such that SB{q}p.) While failing to be identical to the consequence operation (*) outright,  as now conceived is much much closer to it, due to the particular status of implications. But it is not close enough yet. Even with Factor and PII implication is still too distant from our methodological maxim: Remain as close to the classical model as is compatible with carrying out your program of a logic of degrees of truth . We have made implications classical in some sense by rendering all true implications equally true. But what about false implications? We'll advance in our classicalizing enterprise by rendering allK+ o.,,55 implications which are not true enough to be assertible completely false. Which means  R that we countenance the principle of implicative funnel:  pqrV.pq : an implication is either true or else so false that it implies everything. From a prooftheoretical viewpoint that means that we split our proofs in two branches: one wherein we suppose that pq, the other wherein we suppose that pqr; if the same conclusion follows from both branches, we'll assert it. But there is a similar consideration as regards another classical principle, compatible with our overhaul of relevant implication as a connective expressing that the degree of falseness of the apodosis is at most as high as that of the protasis. I am  R referring to the principle of linearization,  pqV.qp . Same procedure: we split our proofs, and look out for the outcome.  ^  5." Proving (and Deriving) what in E has to be Taken as  [  Given # Two striking results follow. One, Adjunction may cease being a primitive rule. We can countenance this rule as the only inference rule in our Hilbertstyle system: for  Rp 1n,  pNqV.pqV.8V.pq ,  pN , 8,  p   q. When 1=n, it is MP. The rationale for the rule is that either p implies pandq or else q implies pandq. Second striking result: a number of interpolation principles may become axiomatic " in alternative presentations of the system " through which some principles ac R cepted, so to say, blindly and without justification in system E can be provided with rea Ru sonable warrant. For instance, E countenances distribution:  pVqUr.pUrV.qUr . Why is it true? Within the framework of the system we are now considering, its proof is obtained from linearization (and Factor): since the disjunction between p and q either implies p el else implies q, it is immediate that the conjunction between such disjunction and r implies either pandr or qandr. More importantly, such (widely challenged, and  R yet to my mind correct) principles of system E as conjoined assertion ( pqUpq ) and  R contraction ( p(pq).pq ) now become provable: the former is proved from  R implicative funnel: we have as a particular case of implicative funnel that  pqqV. R pq : each of the disjuncts implies the principle of conjoined assertion. Another formula  R which E countenances as an axiom (by force, so to speak "using A&B's own words)  R is  prU(qr).pVqr . It seems very clear to me that the axiom is not obvious. A  R~ natural link is missing, which is provided in our system by  pVqpV.pVqq . Another  Rg principle which is not altogether obvious is the principle of conjoined apodoses:  pqU RP (pr).p.qUr , which can be proved, too, as a theorem within our system. The most  R9! controversial principles of E thus become theorems and are endowed with enhanced evidence (although to be sure our new axioms are probably as controversial as those of  R #  E ).  Rn$ In the same way, some suppression principles implicitly accepted in E (as R.  RY% Sylvan has pointed out) through which exported syllogism could be justified (but in E it is not justified, just taken as an additional primitive and underivable evidence) now can be taken as axiomatic, thus rendering exported syllogism a proved theorem. For  R( instance an Adjunction principle for implications:  pq.rs.pqU.rs . The general Adjunction principle is wrong, but we are by now aware that implications are  R) special. That gives us a rationale for those suppressive principles of system E .)o.,,55Ԍ R What constitutes E 's weirdness in that connection is that it countenances suppression in exported form but not in imported form; else, Factor would become provable.  R In particular from  pqU(rr)s  the protasis's second conjunct cannot be suppressed.  R3 Moreover, some other anomalies of E are cured. For instance in E there is an  R asymmetry between disjunction and conjunction:  p(qUr)I.pqU.pr  (`I' is mutual  R implication), but not  p(qVr)I.pqV.pr . Likewise, in E we have  pVqrI.prU.qr   R but not  pUqrI.prV.qr . All those equivalences obtain in the system we are sketching.  ^: 6." Conclusion From an orthodox relevantist viewpoint " if there is such a thing " all this enterprise is pointless, for we are doomed to countenance irrelevances. That seems to me the usual all or nothing , a bad maxim we had better shy away from. Our middle course offers a view of reasoning (if you like, of some idealized sort of reasoning) which  Rr is in between that of classical logic and that of orthodox relevantism. I call it relevantoid  R] logic . It eschews VEQ ( p.qp ); it has the Ackermann property. It has (at least  RH within the domain of the sentential calculus) the entailment property ( pN , 8,  p    q   R1 iff  rr    pNU8Upq ). It avoids the Cornubia rule for nonstrong negation. It also  R avoids unqualified exportation ( pUqr.p.qr ). It also avoids the validity of the  R Dugundji formulae (for any finite n):  pNIpV.pIpV.8V.p-NIp . The system just sketched is close enough to the most widely publicized relevant  RK system of entailment logic, E , with which, despite the chasm our strengthening has opened, sufficient closeness remains to allow bridges to be built. (On the other hand, our system is much closer to classical logic than in fact almost any other nonclassical logic; to be more specific: we are very close to accepting what the classicist accepts, but at the same time we are far apart from the classicist attitude as rejection is concerned: we refrain from rejecting true contradictions, while the classicist wrongly equates rejecting something with asserting a negation thereof.) I think this system is a better logic of reasoning. Reasoning as thus implemented is of course somehow artificial. I do not deny that more natural systems can be found. But naturalness has its price, too. I wonder if some part of the task can be afforded by a pragmatic rounding out of purely inferential logic. But those are matters for a further inquiry.