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~)N-&o~mNzdxCX|%R-&RLb:xЀC2vl {a^gn΍=|J8U|\  PCPDS?뮝\  PCP<BS? *f9 xCXDS?pU9 xICUhNB\  PCP<RhN wB*f9 xCXUhN6B9 xIC7oC2o\  PCXP2f=.f\  PC&P9tE4-t\  PCqP !i(P0$_P\  PCPy.]8*C]\  PCP 9tE4-t\  PCqP 9yE47-y4  p(ACqy.a8*7 a4  p(AC 9yE47-y4  p(ACqGWA\  PC PGWA74  p(AC ~)N-&H'NxzPCPX7H`~)R-&YE>R"zpC>d!A$Y|A"zpC~)R-&YE>R"zpC3b8/HRbxzPCqP>3f8/YRf"zpCq3f8/YRf"zpCq GWA74  p(AC ~)N-&o~mNzdxCX|%R-&RLb:xЀCD~)[8.i[2PAP<7sE4 -s*f9 xCqXD3rE:oRr2PAqP2n #]Flt=|J8U|\  PCPDS?뮝\  PCP<BS? *f9 xCXDS?pU9 xICUhNB\  PCP<RhN wB*f9 xCXUhN6B9 xIC7oC2o\  PCXP2f=.f\  PC&P9tE4-t\  PCqP # !!!!i(P0$_P\  PCPy.]8*C]\  PCP 9tE4-t\  PCqP 9yE47-y4  p(ACqy.a8*7 a4  p(AC 9yE47-y4  p(ACqGWA\  PC PGWA74  p(AC ~)N-&H'NxzPCPX7H`~)R-&YE>R"zpC>d!A$Y|A"zpC~)R-&YE>R"zpC3b8/HRbxzPCqP>3f8/YRf"zpCq3f8/YRf"zpCq GWA74  p(AC ~)N-&o~mNzdxCX|%R-&RLb:xЀCD~)[8.i[2PAP<7sE4 -s*f9 xCqX$D3rE:oRr2PAqP"D?WH[2PA PCgE.{g*6j Hxg#CqX8uE4-u9 xICq2p H=|J8U|\  PCPDS?뮝\  PCP<BS? *f9 xCXDS?pU9 xICUhNB\  PCP<RhN wB*f9 xCXUhN6B9 xIC7oC2o\  PCXP2f=.f\  PC&P9tE4-t\  PCqP% # !!!!i(P0$_P\  PCPy.]8*C]\  PCP 9tE4-t\  PCqP 9yE47-y4  p(ACqy.a8*7 a4  p(AC 9yE47-y4  p(ACqGWA\  PC PGWA74  p(AC ~)N-&H'NxzPCPX7H`~)R-&YE>R"zpC>d!A$Y|A"zpC~)R-&YE>R"zpC3b8/HRbxzPCqP>3f8/YRf"zpCq3f8/YRf"zpCq GWA74  p(AC ~)N-&o~mNzdxCX|%R-&RLb:xЀCD~)[8.i[2PAP<7sE4 -s*f9 xCqX$D3rE:oRr2PAqP"D?WH[2PA PCgE.{g*6j Hxg#CqX8uE4-u9 xICq1b;,b\  PCP0*x4UUUUUUUUUUUUU Vss  Y(Y( ' '))''kY(kY(ksks737B(B(373=Њ dddd #\  PCP#    (x (x (x (x (x (x (x (x (x (xy!i 0*xddBORDE-8.WPG8!&y (x (x "Lorenzo Pe9a (x Q  Partial Truth, Fringes and Motion:  Three Applications of a Contradictorial Logic  (x (x  v   Studies in Soviet Thought  (x # vol 37 (Dordrecht: Kluwer, 1990), pp. 83122. (x (x !"ISSN 00393797=n/n/n/ (x {So ۟ ESUK ,.,. 6&6&StandardII.PS)LAURENTI.PRSXh46&finitif@p@@FF MMx6&StandardII.PS)LAURENTI.PRSXh4   #XpiP;rEXP#     << X` hp x (#%'0*,.8135@8: Wolff, Der Begriff des Widerspruchs: Eine Studie zur Dialektik Kants und Hegels, K?nigstein: Hain, 1981; RGdiger Bubner, Zur  ~J Sache der Dialektik, Stuttgart: Philipp Reclam jun., 1980; Franz Gr)goire, Etudes h)g)liennes, LouvainParis: Nauwelaerts, 1958. My own (traditional) more literal construal coincides with those of Nicolai Hartmann, Heinz Heimsoeth, or, among marxists, Georg Lukcs (at least as I understand him) and Jacques D'Hondt. A similar hermeneutic controversy concerns the thought of the founders of Marxism, esp. Engels's and Lenin's. Both of them seem to maintain the thesis that there are contradictory truths. Still, a number of interpreters argue that what they call `contradiction' is no contradiction in the logical sense of the word. It has been alleged that mistaking dialectical contradictions for logical ones was a confusion pertaining to the vulgarized DIAMAT of the Stalinist era. I refrain from going into those exegetical issues in this paper. Frankly, my belief is that, as in most if not all cases, the traditional construal is the best one. Charity is more often than not misapplied or overstretched in philosophical interpretations. This is likely to  ZK% be a matter of opinion, but I think it better to construe any philosophical corpus as literally as possible, if only because by so doing we grant it a greater interest. Charitable interpretations succeed in rendering history of philosophy dull, with a Parmenides who would not have denied-'x=o.o.o. the existence of plurality, a Berkeley who would not have denied that of bodies and a Hegel who would have said nothing falling afoul of Aristotelian logic. Be it as it may, I have already, in a number of previous papers, dealt with problems  ZI of Hegelian and Engelsian interpretation. I( ~J ԍMy contribution to Hegelian interpretation is put forward in Dial)ctica, l;gica y formalizaci;n: de Hegel a la filosof1a anal1tica ,  ~J Cuadernos Salmantinos de Filosof1a, vol. XII (1986), pp. 14971. As for problems related to how to understand Engels on dialectical  ~JZ contradictions, see Engels y las nuevas perspectivas de la l;gica dial)ctica , ap. Estudios sobre Filosof1a moderna y contempornea  ~J" (ed. by Isabel Lafuente). Le;n: CEMI (University of Le;n), 1984, pp. 163218. See also: Formalizaci;n y l;gica dial)ctica,  ~J prepublished, Quito: Pontifical University, 1980.; Negaci;n dial)ctica y l;gica transitiva , Cr1tica [M)xico], N 43 (april 1983),  ~J pp. 5177.; Significaci;n filos;fica de la l;gica transitiva , Ideas y Valores [Bogot], N 63 (dec. 1983), pp. 59101. About the  ~Jz evolution of the notion of dialectics and the relationship between traditional dialectics and the kind of approach proposed in [Sects.  JB 2&3 of] this paper, see the article Dialectics and Inconsistency , apud  Handbook of Metaphysics and Ontology , ed. by H. Burkhardt & Barry Smith, Munich: Philosophia Verlag, 1991, pp 2168. Thus I hope I can afford to skirt those controversial issues in this paper. Anyway I want to point out that, if by `dialectical' we mean any conception asserting the existence of contradictory truths, then the dialectical tradition can be cogently argued to be much, much wider than is usually supposed, comprising " as I've tried to show  Z  elsewhere ( ~Jb ԍOn Plato, see my paper Dos sentidos de la preposici;n `!#o'' en algunos pasajes de Plat;n , Estudios Human1sticos Filolog1a,  ~J* N 8 (1986), pp. 3958, and El tratamiento de los comparativos en el FEDON  scheduled to appear in Nova Tellus. That St. Augustine, too, is at least committed to espouse the existence of contradictory truths and the relative falseness of the Aristotelian principle of noncontradiction I've tried to show in my two papers El significado de `nihil' en diversos escritos de S. Agust1n ,  ~J Estudios Human1sticos, N 9 (1987), pp. 15568, and La identificaci;n agustiniana de verdad y existencia: una defensa filos;fica ,  ~JJ La ciudad de Dios CCII/1 (januaryapril 1989), pp. 14972. On Nicholas de Cusa, see my 3 papers: Audel! de la co5ncidence  ~J des oppos)s: Remarques sur la th)ologie coupative chez Nicholas de Cuse , Revue de Th)ologie et de Philosophie, 121 (1989),  ~J pp. 5778; La superaci;n de la l;gica aristot)lica en el pensamiento del Cusano , La ciudad de Dios (1988), pp. 57398; La  ~J concepci;n de Dios en la filosof1a del Cardenal Nicols de Cusa , Revista de la Universidad Cat;lica (Quito, Ecuador), N 47 (1987), pp. 30128. " thinkers such as Heracleitus, Plato himself (not just in the Parmenides and  Z Sophist dialogues but even in the Phaedo, Republic etc.), $nesidemus, most " if not all " neoPlatonists including St Augustine and, most of all, Nicholas de Cusa, with his copulative theology. And nowadays there are outside Marxism at least two philosophical conceptions which claim to be dialectical in that very same sense, namely the neoenergetism proposed  Z by Stephane Lupasco and Marc BeigbederN 4( ~J ԍSee: Stephane Lupasco, Logique et contradiction, Paris: PUF, 1947, and Du devenir logique et de l'affectivit), Paris: Vrin, 1973  ~J{ (2 vols., 2d. edit.: the 1st edit. appeared in 1935); and Marc Beigbeder, Contradiction et nouvel entendement, Paris: Bordas, 1972.N and the ontophantic system of metaphysics and  Z epistemology proposed by the present writer ( X ( ~J! ԍThat metaphysical system, which also comprises a treatment of some main issues of philosophical theology, has been proposed  ~J" in my 3 books: La coincidencia de los opuestos en Dios, Quito: Educ, 1981; El ente y su ser: un estudio l;gicometaf1sico. Le;n:  ~J# Universidad de Le;n, 1985; Fundamentos de ontolog1a dial)ctica. Madrid: Siglo XXI, 1987; and in a number of papers dealing  ~JT$ with epistemological issues, among which are my Belgian Ph.D. diss. Contradiction et v)rit): )tude sur les fondements et la port)e  ~J% )pist)mologique d'une logique contradictorielle (Li/ge, 1979), Conocimiento y justificaci;n epist)mica , Revista de la Universidad  ~J% Cat;lica, N 28 (nov. 1980), pp. 3567, and Naturalized Epistemology and Degrees of Knowledge  (paper delivered to an international Conference held in Tepoztln, M)xico, in aug. 1988). More accessible presentations of some parts and motivations of the system are to be found in Verum et ens conuertuntur: The Identity between Truth and Existence within the Framework  ~J<( of a Contradictorial Modal SetTheory , ap. Paraconsistent Logic (ed. by G. Priest, R. Routley & J. Norman), Munich: Philosophia  ~J) Verlag, 1989, pp. 563602; Identity, Fuzziness and Noncontradiction , NoEs, XVIII/2 (may 1984), pp. 22759, Dialectical Arguments,  ~J) Matters of Degree, and Paraconsistent Logic , ap. Argumentation: Perspectives and Approaches (ed. by F.H. van Eemeren et  ~J* al, Dordrecht: Foris Publications, 1987, pp. 42633. The ontophantic approach is now being extended to other philosophical fields,  ~J\+ as ethics (see, e.g., my papers El conflicto de valores: Reflexi;n desde una perspectiva l;gicofilos;fica , ap. Crisis de valores\+o.,,  ~J (ed. by J. Gonzlez L;pez). Quito: Educ, 1982, pp. 13362, and Un enfoque no clsico de varias antinomias de;nticas , Theoria (S. Sebastin) N 789 (1988), pp. 6794), so as to become a comprehensive account of all major problems of philosophy, which  ~J  is going to be offered in a new book, viz. Hallazgos filos;ficos (Philosophical Findings). . Furthermore, some philosophers who, needless o.,,:: to say, would have balk at the idea of contradictory truths can nonetheless be cogently argued to be committed to it all the same. Thus, even if Leibniz held the principle of noncontradiction  Z to be undeniable, he espoused a principle of continuity which in fact yields contradictions.( J ԍSee my paper Armon1a y continuidad en el pensamiento de Leibniz: Una ontolog1a barroca ,  Cuadernos Salmantinos de  JO Filosof1a vol. XVI (1989), pp. 1955. And, of course, outside philosophy proper there are lots of writers, esp. poets, who literally assert contradictions and say that the world is contradictory. I have argued elsewhere that more often than not they can be taken to mean what they say since not every contradiction  Z must perforce be absurdH( ~J ԍSee my paper La ruptura del sistema l;gico en la teor1a po)tica de Carlos Bouso9o  Anthropos, N 73 (junio 1987), pp. 4350.. Hence, there is no shortage of philosophical and nonphilosophical approaches which seem to countenance the existence of true contradictions, i.e. which can be termed `dialectical' in a straightforward sense of the word which can be claimed to stem from Hegel's work. Whether each of them does in fact commit itself to a contradictory reality or not is a matter for interpreters to discuss, but anyway there are reasonable grounds for assigning them such a commitment. Yet my present concern is not an exegetical one but only that of showing that we can make good sense of the idea of contradictory truths in at least three fields, namely those of partial truth, fringes of application of sundry predicates, and motion. All those three fields have of course been claimed to be amenable to contradictorial treatments by the dialectical tradition, but, since I want to keep clear of interpretive matters here, I'll abstain from commenting on any previous attempt at a dialectical treatment of those issues. To my own proposal on them is the remaining of this paper to be given over. Let me conclude this Section by pointing that the formal system which is sketched out below has already been presented in similar if not quite identical ways in some previous  Z papers, which however are not easily accessible( ~JB ԍSee, for instance, Tres enfoques en l;gica paraconsistente  I & II, Contextos Ns 3 & 4, pp. 81130 & 4972, resp.;  ~J   (Quasi)Transitive Algebras , Proceedings of the XIII International Symposium on MultipleValued Logic (Kyoto, may 1983), Los  J Angeles, Ca: IEEE Computer Society, pp. 12935.; Consideraciones filos;ficas sobre la teor1a de conjuntos ,  Contextos 11 & 12 (University de Le;n, 1988), pp. 3362 &743; Algunos desarrollos recientes en la articulaci;n de l;gicas temporales , apud  Jf!  Lenguajes naturales y lenguajes formales IV.1 , ed. by Carlos Mart1n Vide. Barcelona: Universitat de Barcelona, 1989, pp. 41339..   Z  2. An infinitevalued paraconsistent approach  Z 2.1. Paraconsistent logics ) Paraconsistent logics are by now gaining recognition in the community. There are a number of alternative ways of laying down requirements for acceptable paraconsistency. The discrepancy among the several paraconsistent schools hinges mainly upon what is regarded as required for a functor to qualify as a negation. My own requirements are not uncontroversial. Still, I think they are highly plausible. o.,,::ԌIntuitively, a paraconsistent logic is one which allows for two theorems of a theory to be such that one is a negation of the other without thereby everything being a theorem  Z of that theory. Following da Costa ( ~Jc ԍNewton C.A. da Costa has laid down those condition in his paper On the Theory of Inconsistent Formal Systems  (Notre Dame  ~J+ Journal of Formal Logic, 15/4 (1974), pp. 497510). See also a paper by N.C.A. da Costa and E.H. Alves, Relations between  ~J Paraconsistent and ManyValued Logics , Bulletin of the Section of Logic 10/4 (1981), 185191. An interesting discussion of those  ~J conditions from a technical viewpoint is to be found in Igor Urbas's Ph.d. diss., On Brazilian Paraconsistent Logics, Canberra: Australian National University, 1987, pp. 17ff, 73ff. Notice that da Costa also lays down that for a system to be paraconsistent it must not have noncontradiction as a theorem. But there da Costa and I part company. On the relationship between those general requirements for paraconsistency and the particular kind of logical framework I am going to develop in this section, see another  ~J paper, also by Newton da Costa: Aspectos de la filosof1a de la l;gica de Lorenzo Pe9a , Arbor N 520 (Madrid, april 1989), pp. 932., I also add some further requirements for a paraconsistent logic to be acceptable: (1) It must possess an intuitive appeal; (2) it must recognize the validity of as many usual ways of reasoning as possible; (3) it must contain as much of classical logic as possible keeping short of counternancing the Scotus rule (p, notp  q). The 2d. requirement leads me to also demand that our chosen logic should be a fuzzy one, since only logics of fuzziness can capture reasonings with words such as `up to a point', `more8 than' and so on. Thus, from a prooftheoretic viewpoint my requirements for correct  paraconsistency are the following ones. (In what follows, a dot written immediately on the right of a functor stands for a lefthand parenthesis with its mate as far to the right as possible. Remaining  d  ambiguities are dispelled by associating leftwards.) A theory T is a 2tuple < T , R >, where  d  T is a set included in a set F of formulae, formulae being such finite sequences of members  d of some set V of symbols as comply with a number of stipulations (formation rules) and R  d is a set of operations on F (inferencerules). (Henceforth, while speaking about a theory whose  d set of inferencerules is R I'll mean the fact that R contains a rule of the form {p1,8,pn}   Z q simply by saying that in that theory p1,8,pn  q.) A healthy theory is a theory T whose  d set of symbols, V , comprises two twoplace functors, V, U, and oneplace functor, ~, such  d that any p, q, r  F , T is closed for every operation (inferencerule) belonging to R , and (writing pq whenever s  s', where s' is like s except for containing occurrences of p at zero or more places where s contains respective occurrences of q, or conversely): (1) T is nondeliques d cent (i.e. T / F ); (2) pVqVr  qVrVp; pUqUr  qUrUp; (4) pVqUr  pUrV.qUr; pUqVr  pVrU.qVr; (5) pVqUq  q  pUqVq; (6) p  pVq; (7) pUq  p; (8) ~(pVq)  ~pU~q; (9) pV~p  d is a theorem (i.e.  T ); (10) ~(pU~p)  T ; (11) ~~p  p; (12) p  ~~pUp. Let T be a healthy theory, ~ being a oneplace functor thereof satisfying the above  d requirements: then T is also said to be negationally right as regards ~. A theory < T,R > negatio d nally right is said to be negationally inconsistent for ~ iff T comprises two theorems p, q  d such that q = ~p. If T = < T,R > is a theory, a theory T' = < T',R' > is said to be a taut extension  d of T iff T' includes T and R' includes R . A theory which is negationally right for one of its  Z symbols, ~, is said to be paraconsistent as regards ~ iff there is a taut extension thereof which  Z is both nondeliquescent (i.e. it's not closed for the rule p  q, for any p and q) and yet negationally inconsistent for ~. A theory is paraconsistent iff it is paraconsistent for at least one of its symbols.  o.,,::Ԍ d A healthy theory T = < T,R > will be said to be proficuous iff its vocabulary comprises  d a symbol ~ such that T is negationally right for ~ and the following rule belongs to R : pVq, ~p  q. (Any proficuous theory is a taut extension of classical logic.) Nedless to say, if a theory is negationally inconsistent as regards a symbol N, then that very same symbol cannot be one which renders it proficuous (i.e. disjunctive syllogism cannot hold for N). Still a theory can be both proficuous and negationally inconsistent for different negations, one strong negation ~ and another, simple, negation N. Such is the case  Z' as regards the system Aq I'm now going to set forth.   Z  2.2. System Aq : Syntactic approach  d^ )  Aq = < T,R > where T is the smallest subset of F containing all instances of each axiom  dY schema A01 through A08 below and closed for every rule belonging to R ; where F is a set  dT comprising an element, , and closed for this formationrule: whenever p, q  F , then so do Bp, Hp, pq, pIq, pq, zxp (for any variable instead of x). DEFINITIONS: Np  abbr. pp  pVq  abbr. N(pq)  pUq  abbr. NpNq  p  abbr. HNp    abbr. I  Lp  abbr. Np  0  abbr. IV(IN)  Xp  abbr. pp  1  abbr. N0  pDq  abbr. pVq  Sp  abbr. pUNp  np  abbr. pN  mp  abbr. NnNp  pq  abbr. qUpIq  pq  abbr. pDqU.qDp  Yp  abbr. pIUp  fp  abbr. YpUp  p&q  abbr. LpUq  p\q  abbr. pqU(qp) * o.,,::Ԍ  p  abbr. np\p&fSp  yxp  abbr. Nzx(1Np)  p8q  abbr. B(pq)  Jp  abbr. Bp  Kp  abbr. NXNp  AXIOM SCHEMATA: A01 pUqDp A02 rUsIpD(pqI.qsV.qr) A03 pIqD(rIqI.pIr)U.KXpIpU.YpVYqVY(pq)U.fSpUfSqD(pq\p)U.pUqD.pq A04 qUpVpIpU.HpUHqILH(pUq)U.pIqD(HpVHrIH(qVr))U.pqpU.p1Ip A05 pINqI(NpIq)U.pIpIU.p'UpIqD(qrsI.srpU.sp'r)U. pUfNqD N(pmq) A06 pIqD(qDp)U.mpmnpVHpU.mpnp(YpVYNp)U.qnpV(pImq)ULpV.pq A07 BpVBBLpU.BpIpVBpU.p8q&BpBq  A08 yx(zxqp)Izx(yxpq)U.zx(pq)(zxpq)U.zxs\rDyx(s\r)U.zxpUyxqyx(pUq)U.zxpyxpU.nr\rDyx(ryxp.rp)# (In A08 r  is to be a formula with no free occurrence of variable `x'.) INFERENCE RULES:  Z  rinf01 (modus ponens) p , pDq  q  Z  rinf02 p  Bp  Z}   rinf03 (Universal generalization) p  q if q is nothing else but the result of prefixing to p any finite string of universal quantifiers#  Z   rinf04 (alphabetic variation) p  q if q is nothing else but the result of replacing within p a formula r by another r', where r' is an alphabetic variant of r.#  Z-  rinf05 (shift of variables) p  q if q is nothing else but the result of uniformly replacing all free occurrences of some variable with respective free occurrences of another variable.# READINGS of those symbols: `': `The least true of truths (is true)' (or `That which is just infinitesimally true in all respects (is true)'; `1': `The wholly true (is true)'; `0': `The wholly false (is true)'; `': `That which is just as true as false (is true)'; `': `neither8 nor'; `': `not only8 but also'; `I': `as8 as' (or: `to the same extent as'); `H': `(It's) wholly (true that)'; `B': `It's truthfully assertable that' (or `It's in every way the case that' or `In all respects'); `N': `not'; `V': `or'; `U': `and'; `': `not at all' (or `by no means'); `L': `more or less' or `(at least) up to a point' (or `to some extent', `in some degree'); `X': `(It's) very (true that)'; `D': `only if'; `n': `It's overtrue that'; `m': `(It's) much like (true that)'; `': `insomuch only as' (or: `only to the extent that'); `': `iff'; `Y': `(It's) (just) infinitesimally (true that)' (or: `It's just in the smallest degree true that'); `f': `(It's) somewhat (true that)'; `K': `(It's) a little4* o.,,:: (true that)'; `\': `It's less true that8 than that'; `J': `(It's) (at least) relatively true that'; p&q : While p, q . `z' (`y') is the universal (existential) quantifier prefix. It seems safe to lay down a transformation rule  " or surfacestructure  generating rule " to the effect that whenever a oneplace functor, g, is prefixed to a formula, p, p being a predicative formula of the form x is soandso , an alternative reading is available, viz. x is g soandso .   Z -2.3 MODELS. Semantic treatment  Zh ) I am going to set forth the notion of quantificational transitive algebras . A quasi  ZY transitive algebra is an algebra where  = <1,N,H,n,V,,I> where 1 is a zeroary operation, N, H, n are unary operations, , V, I are binary operations, all of then satisfying the postulates (01) through (24) below. Let's first introduce some abbreviations. 0 is N1; xy  dD means that y = yVx; x such that  = BB}, being a q.t.a., and B being a unary operation which satisfies this further postulate: (25) For any xA, either xV=x=Bx, or else x/0=Bx (the 2d. disjunct means that, while x/0, 0=Bx).  Z7 @A quantificational transitive algebra , tQa, is a t.a. enlarged sith two (infinitary) operations, , , carrying nonempty subsets of A (where A is the carrier of the tQa under consideration) into members of A, and such that for any elements x,z,y of A and any nonempty  Z! sets A1, A2 included in A:  Z# (26) {A1z:zA2} = {A2y:yA1}  ZM% (27) {xz:xA1, zA2}  A1y (if yA2)  Z' (28) {xIz:xA1, zA2}  A1IA2  Z( (29) A1UA2  {yUx:yA1, xA2}  Z* (30) nz\z  {zA1(zy):yA1}* o.,,::Ԍ Z (31) A1 = N{Nx:xA1} Let's now go about finding out a tQa as follows. Let's take the set of standard reals; for each of them, x, let's take x, x+ and x,  being some given (nonstandard) infinitely  Z) small number, with `+' standing for addition and `' standing for subtraction . All those  ZV numbers will be called hyperreals . An alethic number is a hyperreal h such that 0h1. We  Z now define these algebraic operations. If x is a standard alethic number, then: (1) Nx = 2logx2 if 0 provided that n is one of three [given, fixed] numbers greater than 1 and 0r1; (3) A set D defined as follows: Let R be the set nonnegative reals and D' = RB{}. Now D = {x:for some yD':x=y; or x = y+ and yR; or x = y and y>0}, where `+' stands for addition and `' stands for some (arbitrarily chosen) infinitesimal. Now we define these operations: nx =: y+, if yR is such that y=x or x = y; x, else mx =: y, if yD'{0} is such that y=x or y+ = x; x, else Nx =: 0, if x=; , if x=0; 1/x, if xR"{0}; mNy, if x=ny; nNy, if x=my xy = max(Nx, Ny)&  o.,,::Ԍxy = x+y, if both x,yR x = x = ; mxy = ymx = m(xy) if y / ny nxy = ynx = n(xy) Hx =: 0, if x=0; , else xy =: 1, if xy; , else Finally we extend those operations to tensors in the same way as the one referred to above. (Notice that this structure is mapped into L by a homomotphism h such that for  Zd every xR hx = 2x, while h=0, and, for any zD such that z = x (= x+), with xR, hz = hx+ (= hx).)   Z/ !3. Applications ) I am going to show in this Section how the above technical system can cope with several problems which have been deemed to constitute serious challenges to [the application of] classical logic, namely those of partial truth, fringes and motion. The first problem is raised by the fact that when a predicate correctly or truthfully applies to a zone, or part, or area, of some object but not to other parts thereof, it can only be said with partial truth that the object satisfies that predicate or has the property it denotes. Thus, e.g., if some large part of a country is barren while another part is fertile, is the country fertile or is it not? Likewise, is it true that it rains in Madrid on 29th. August if on that day it rains there but only for seven hours on end? The second problem arises because of the fact that many predicates can be neither completely assigned to some things nor completely withheld from them. Thus, predicates like `young', `kind' `poor', `red', `healthy', `near', `old', `new', `mountanous', and the like, apply in degrees. But so do other predicates such as `table', `car', and all implementterms, of course. And so are species terms bound to do, if evolution theory is correct as most of us nowadays think: there is bound to have been something or other to which the predicate `monkey' applies (or applied, if you prefer) neither wholly nor yet not at all " some creature which was only to some extent a monkey; or a mammal, or a vertebrate, or8 In any such case is the sentence to the effect that the predicate applies true? Or is it not? The 3d. problem is nothing else but Zeno's paradox of the arrow. Is the travelling body here or is it there, or both, or neither here nor there? A number of accounts have been proposed. One of them is that terminological distinctions allow to dispel all those perplexities. Thus, the predicate `red' could be applied, e.g., only to extensionless points, but in such a way that to say that point x is red would amount to what in our ordinary way of speaking is expressed, more or less, as `x belongs to a uniformly red surface'. I regard that [Aristotelian] account as merely programmatic and in fact probably impossible to carry out. It would cripple the role of logic and sever the links between language and reality. Yet without resorting to some such procedure, classical logic is unlikely to be able to cope with any of those problems.c* o.,,::ԌAn alternative approach is that implemented by the kind of fuzzy set theories proposed by Lofti Zadeh and other researchers working within similar frameworks (ukasiewicz's logics or other logics of a similar kind). The logical systems they favour give up excluded middle (and noncontradiction). Thus what they in the end advise is for us to renounce the very same question Is it or isn't it?  Not that according to them it neither is nor fails to be, no. What they advise is that you cannot say that it is, nor that it is not, not that it neither is nor is not, nor that it both is and is not. You can say nothing of the sort. So, the question Yes or no?  is " according to them " misapplied, misasked. No answer, no question. That is that simple. However, that approach seems to me to be in serious trouble. It amounts to a kind of ineffabilism. Moreover, it requires, for any predicate to be assigned to an object, that the  Z object should completely have the property the predicate denotes, which seems to me to constitute an implausible alethic maximalism. But what about using a logic which would e.g. be like ukasiewicz's infinite valued logic but with all values  designated? It would enforce excluded middle, and so be immune to the objection above. But if we add a strong negation, , to be read `not8 at all' (such that  d  v (p) = 0 iff v (p) / 0 and else v (p) = 1), the system would still fail to enforce strong  Z excluded middle , i.e.: pVp , which also seems to me to be a drawback. Moreover, many applications of the system would be desastrous: let's suppose that a predicate h  applies to an object o to a degree of ; then the logic under consideration would warrant our asserting both p  and not p , with p  meaning that o has the property denoted by h ; now, in all ukasiewicz's systems we have the Scotus rule: p, Np  q. So any such application would yield a deliquescent theory. In order to avoid that result we would have to change the conditional functor of ukasiewiczian logics. (Notice that in ukasiewicz logic proper " with only value 1 designated " neither of these schemata is theorematic: p(qr).pUqr  [importation], pNpNp  [abduction]; with all values  designated, only the former becomes theorematic.) But what would we choose as a correct conditional? For reasons I'm not going to canvass here, I think the only satisfactory [mere] conditional, D, would be defined in a classical way: pDq  as abbreviating pVq . But then in order for modus ponens to hold, all values are to be designated except 0. Now, if we have all real numbers in the interval ]0,1] designated, then within the kind of ukasiewiczoid logic we are entertaining the following result can ensue (let's call it a strong 3oversinconsitency): let's suppose that there is a set of terrains such that for any dry member of that set there is some other member which is less dry, the sequence of them tending towards zero (total lack of dryness), none of those terrains being wholly nondry, though. Then of each of those terrains it would be truthfully assertable that it was dry but  Z# yet it would be utterly false (hence by no means truthfully assertable) that all them are dry. Separate truth of each and every would not entail conjoint truth of all. That surely is mistaken. Hence the need for hyperreals.  ZE' Now, scalar semantics, even with hyperreals, are not sufficient either. We need tensors. For otherwise we couldn't have functors such as `in all respects' (or `it is truthfully assertable') which are pluridimensional. For in some cases " e.g. cases of partial truth " besides an object having (let's say: all in all) a property to some extent, it is true that in some* o.,,:: respects it has it to a much higher degree than in other respects. Hence the need for a kind of logic like the one presented above, in Sect. 2. I am now going to separately examine each of the three domains of applicability of that logical approach which I have listed above.   Z  3.1 Application to the treatment of partial truth ) I am now going to explore a number of alternative ways of dealing with one of the  Zq issues I want to tackle namely that of partial truth , in the sense of that which, being true of a part of some object, is said to be partly, or partially, true of that object. I also include under this head cases of some predicate being satisfied by an object [only] in some ways , i.e. of a sentence's being true [only] when expanded with some circumstantial complement , be it temporal, or locative, or introduced by a prepositional phrase like with regard to , or whatever. Our present approach could be expected to handle that issue in the following way. Let's e.g. take as our truthvalues the alethic tensors, as defined hereinabove. An alethic tensor,  Zo t, will be said to be encompassed by another, t', iff there is a mapping - such that for any  df projectionfunction p there is a projectionfunction p' = - p , with p  p' (projection functions are wellordered, each of them picking out the ith component of a tensor whatever, for some  dX positive integer i), - being monotonic (i.e. for any p 1, p 2, - p 1  - p 2 iff p 1  p 2) and for every  dY  p p t = - p t'. We might lay down that, when an area z' was a part of another area z, then the truthvalue of p obtains at z'  would be encompassed by the one of p obtains at z . That account  Z I'll call the encompassing approach . Winsome though it seems to be, I'm not satisfied with it. For it would countenance our drawing from at z' p  the conclusion J(at z p)  (i.e. that it's at least in a way, or relatively, true that, at z, p), and, although very often that seems right (e.g., when the southern part of some country is very dry, it seems safe to say that it's relatively true that that country is very dry), I'm far from being convinced that such is always the case. Recall that the areas we are thinking of are taken to be any relativizing respects whether spatial, temporal or whatever. Now it's far from obvious that, since in 1792 France is undergoing a revolution, it's relatively true that in the 18th century France is undergoing a revolution, if, at least, `in' means the same as `through' or `over'. However, my purported counterexample may well turn out to rest on a confusion since `through' or `over' may embody some universal quantifier in their underlying structures: `through that year' may well mean `at each time interval included in that year'. So, it may after all be right to say that it's  Z" relatively true that in ( not `through') the 18th century France is undergoing a revolution.  Ze$ Accordingly, let's try another tack, the averaging approach . I am going to assume  ZV% that is some function  carrying alethic tensors into alethic tensors and such that for any tensor  dM& t there and projection function p there are projection functions p 1, p 2 such that p 1t  p t   dN'  p 2t. Then that function  (for Greek ` ):') is an averaging function of sorts. Let's also write `' as the symbol denoting that function (which means that, for a valuation v, and a formula p, v(p) = vp, the former  being the functor, the latter the unary operation on alethic tensors); `' may be read as `on the whole', `on an average', `on balance', `all in all', `allC* o.,,:: things considered', or the like. A postulate will be laid down to the effect that the following schemata are to hold: HpHp; (pUq).pUq; pVq(pVq); (pq).pq Let's now introduce a measure of the distance between two alethic numbers, : uv will be the difference or distance between u and v. The averaging approach will be articulated by laying down that, the greater the overlapping between area z and area z' (with z' being a part of z), the smaller the distance between at z' p  and (at z p) ; at least other things being equal. Outfitted with those semantic tools, we now may account for sundry intuitively appealing correlations. In addition to the degree of overlappingness between the areas, other factors may turn out to bear on the result, too; among them: the importance  of the subarea; how much population  it contains: the distance between the truth values of `All in all, in that country people live well' and `In region R people live well' will be smaller than that between the truthvalue of the former sentence and the one of `In region R' people live well' if e.g. two thirds of the country's population live in R and less than a third in R'. Likewise, supponsing Jacques Coeur had more to do with his Bourges countrymen than with foreign captives, the difference between the truthvalues of `All in all Jacques Coeur was a good man' and `Jacques Coeur was good towards people in Bourges' will be smaller than that between the value of the former sentence and the one of `Jacques Coeur was good towards foreign capitves'. An open question remains, though: what about the distance between the sentences resulting from those considered above when the functor `all in all' is dropped? I want my present approach to remain neutral on that issue, while it affords means for coping with statements of partial truth through the averaging functor. Should our qualms over the meaning of `through' above turn out to be groundless " owing to `through' containing an implicit quantifier, as already hinted at, and so differing in meaning from `at' or `in' " then we could wonder if after all `' is a redundant functor, the meanings of p  and all in all p  then being one and the same. Even so, the averaging approach would not boil down to the encompassing one, since the latter alone entails, e.g., for Jacques Coeur's being a good man to be in some respects true up to a certain degree if Jacques Coeur was up to that degree good towards his Bourges countrymen. Nevertheless, I do not think that the averaging approach can be satisfactory with such a qualification (taking  to be redundant), the trouble being that, as it stands, it applies to all and every sentence, and to all states of affairs those sentences would stand for. But of course even if for a long period a person is completely happy, it may well be [utterly] false that that person is completely happy period " i.e. [if `all in all' is redundant] that she is all in all completely happy. So what is probably required is to narrow down the field of sentences (or of states of affairs) for which `' is redundant (for which  is an identical transformation). But even without such a quelification the averaging account seems to me to face another difficulty. Suppose Helen is for some time most happy and for some time most unhappy. Then the averaging approach would entail that she is all in all most happy and yet also all in all most unhappy. That of course would not entail that she is all in all both most happy and most unhappy (since we would not have as theorematic the schema pUq(pUq) ), but anyway it would be an unwanted result even as it sounds. Thus again, the field}+ o.,,:: of sentences to which the averaging account is going to to apply has to be narrowed down.  Z A prima facie likely candidate would be the set of atomic sentences, but, to be sure, that notion has by now rightly become suspect. Inspite of those difficulties and perplexities, it seems to me that the averaging approach, endowed with careful qualifications of that sort, is the most promising one. However,  Z- a safeguard clause of other things being equal may turn out to constitute the wisest qualification, even if " as it is always the case with such a clause " it also weakens the thesis to the point of somehow blunting it. Since our present exploration does not aim at finding fully  Z satisfactory results but rather at pondering the pros and cons of different approaches, we had better stop our discussion of this issue here by provisionally recommending the averaging approach with a number of qualifications and provisos. One important point to be highlighted, though, is that, for all we have said hitherto,  may either be a uniformizing operator, or fail to be so. A uniformizing operator on alethic  Z tensors is a function, -, such that, for any tensor t and projection functions p1, p2, p1(-t) =  Z p2(-t). It could be felt to be natural that, while a sentence like Helen is happy  can be truer in some respects, less true in others, a sentence like All in all Helen is happy  would have to be as true in any given respect as in any other. I'm far from sure that such is the case, though. One of the advantages of our choosing alethic tensors " rather than scalar unities like the alethic numbers " as truth values is that by doing so we allow for a great many alternative ways of coping with the just discussed problems. The encompassing approach would by impossible, were it not for the acceptance of tensors. And even the averaging approach would become onesided and scarcely fruitful within a scalar truthvalues framework, no interplay being then possible between the operator `all in all' on the one hand and, on the other, `it is truthfully assertable that' or `it is at least relatively true that'. Let me bring this subsection to a close by warning the reader. Th present array of approaches may turn out to allow for ways of looking upon historical facts that, natural though they seem to be to the present writer, may well deserve rejection in accordance with other ways of thinking. Thus, for instance, even though both Sulla and Marius ordered many killings, the former being [all in all] murderous may be much more true than the latter being so.  Z Therefore, it is not only a question of knowing if one of them had a property (like that of murderousness) but also how much he had it, or if he had it more, or less, than [certain] other personages. I can imagine some people frowning at such results, but I refrain from elaborating on those points here.   Z#  v 3.2 Application to the treatment of fringes (or fuzziness) ) Most predicates we avail ourselves of are fuzzy, or vague. Many philosophers have contended that fuzziness, or vagueness, is not " or even cannot be " a matter of how the world is, but only a feature of our concepts , of our linguistic usage, speech patterns or whatever; or at most something pertaining to semantics, but nowise to ontology. The view that fuzziness is a real trait of reality itself has been spurned and contemptiously styled `the ugly view'.) o.,,::Ԍ Z Fuzziness, or vagueness, has often been characterized as failure of excluded middle. ( ~J ԍOne of those who so characterize vagueness is Kenton F. Machina in his Truth, Belief,, and Vagueness , Journal of Philosophical  ~JI Logic 5/1 (feb. 1976), pp. 4777. See, for an interesting discussion, Susan Haack, Deviant Logic, Cambridge U.P., 1974, chap. 6, pp. 10925. Akin to the characterization of fuzziness as failure of excluded middle are those which regard vagueness as truth ~J Ԛvaluelessness (e.g. Bertil Rolf, A Theory of Vagueness , Journal of Philosophical Logic 9/3 (aug. 1980), pp. 31526). There are lots of variations on those kinds of approaches, whereas unfortunately I know of no other approach more or less along the lines of the one I am now offering. I want to challenge such a characterization. Let's take the example of dryness. Desertic lands are dry; steppes are dry; many regions which are neither deserts nor steppes are dry, too. But of course there are degrees of dryness. Now pick out any region which lies in the fringe of the predicate `dry', i.e. which is neither completely dry nor completely wet; call it `Alboran'. Does excluded middle fail  for the sentence `Alboran is dry'? What does such a failure mean? Perhaps that the sentence `Either Alboran is dry or it is not dry' does not hold. But that sentence does hold. What we would usually say is that Alboran is neither dry not wet, or that it neither is nor fails to be dry. Now, `Alboran neither is dry nor fails to be dry' is equivalent to `It is not the case that Alboran is dry and it is not the case (either) that Alboran is not dry'. Which, in virtue of involutivity of simple negation, means the same as `Alboran is not dry and (yet) it is dry', i.e. (in virtue of commutativity of conjunction): `Alboran is dry and it isn't. And " as even people unprepared to countenance true contradictions acknowledge " such sentences are very often used to represent situations consisting in a thing's lying in the fringe of a predicate. But some people are concerned over our purported unwillingness to accept counterexamples to the principle of noncontradiction while we would be apparently less disinclined  Zw to put up with counterexamples to excluded middle.6 w@( ~Jp ԍSee A.C.H. Wright, Verificationism and the Principle of NonContradiction , History and Philosophy of Logic 5 (1984), pp. 195217. In n.31 (on p. 213) Wright says that vague concepts are not counterexamples against his claim that it is absurd to think that reality is inconsistent; and he adds: `Speakers may have discretion as to whether, on occasion, something is to be dubbed `red' or `orange' but an individual speaker cannot consistently say something is red and orange all over. And the truth of `It is raining and it is not raining' merely reflects the vagueness of the concept `drizzle'.' Of course you cannot consistently say something which is contradictory if `consistently' here means `complying with negational consistency', as different from absolute or Post consistency (nondeliquescence, nontriviality). As for the truth of `It is raining and it isn't, well my opinion is that the existence of such truths is reflected by our use of the fuzzy predicate `to rain'; as for the fuzzy predicate `to drizzle', it denotes a fuzzy subproperty of the fuzzy property of raining. Concepts I meither know nor need to posit. In the same way as sometimes it both rains and does not rain, some things are both red and orange all over, namely such as are of a reddish orange hue.6 Such a difference seems to me to amount to nothing else than a stylistic preference. For simple negation noncontradiction and excluded middle are strictly equivalent. Counterexamples to the one are to the same extent counterexamples to the other. Yet there are stylistic, pragmatic constraints about how to put a message, whether or not other ways of putting it are semantically as acceptable. Furthermore, the meaning of `counterexample' in those sentences ought to be clarified. What emerges is not a case which compels us to jettison either noncontradiction or excluded middle, but quite another thing: those are cases wherein negations of instances of those principles are shown, or at least believed, to be (up to a point) true " which does not completely rule out the truth of the thus negated instances of the principles. Alboran neither is nor fails to be dry; in virtue of DeMorgan, that amounts to its being true that the following is not the case: Alboran is dry or it is not. Still, the latter may also be true; why not? Then, by means of adjunction, we can draw the conclusion that Alboran's being either dry or not9 o.,,:: dry is and is not the case. You may have other, independent, grounds for rejecting excluded middle; e.g. you may be a constructivist. But our customary ways of using fuzzy predicates does not compel you to do so, far from it. Unless of course you are bent on avoiding contradictions at any cost. But why? Probably because of Scotus' rule (p, notp  q) and fear of your body of beliefs foundering " becoming deliquescent. No worry! Paraconsistent approaches are devised to ward that danger off. A paraconsistent account of the use of fuzzy predicates is quite straightforward, if it is carried out within the framework of a logical system like the one put forward in Section 2 above. Neither noncontradiction nor excluded middle need to be waived; in fact both are kept as valid principles, with all instances of either recognized as logical truths. Still, negations of a number of those instances can also be countenanced or stated with no loss of logical coherence (nondeliquescence). All such situations ensue upon some thing's neither wholly failing to exemplify some property nor entirely exemplifying it either. Hence fuzziness has nothing to do with our being unable to say whether a thing exemplifies a property or not. We are unable to say whether King John's head was on 01.01.1201 covered by an odd number of hairs, but that makes `odd' none the fuzzier. On the other hand, we know what we are to say concerning some people which are neither wealthy nor entirely destitute: we say that they neither are nor fail to be poor " and hence that they are and are not poor; but some of them are less poor than others. Those fringes make up the field where comparatives are in order. But then, why to remain adamantly loath to considering fuzzy properties real " as real anyway as nonfuzzy properties may be? You may be a nominalist, of course. Then, you'll accept neither real fuzzy properties nor real nonfuzzy ones. Well and good. Or you may be a realist, of whatever variety. Then, in order to rule out fuzzy properties from the world you need some convincing argument. Such arguments as I have thus far come across fail to carry conviction. For instance, that speech or linguistic mechanisms can sufficiently account for fuzziness phaenomena. Well, that is not generally demonstrated in any enlightening way, and, withal, such pragmatic accounts as have been devised (since that is what they are " or at most pragmaticsemantic) lack the smooth, simple, clear straightforwardness ensuant on alethicsemantic accounts like the one I am here putting forward. Another argument: that the world as such  (?) can be neither fuzzy nor nonfuzzy. Why not? The discussion about whether it is fuzzy or not (that is to say about whether there are fuzzy properties and situations or not) makes clear sense, doesn't it? Another way of putting the same argument: that we lack any verification procedure in order to ascertain whether the world is fuzzy; so8 Well, verification procedures we may fail to have, but not so criteria on ways of shaping our worldview according to some epistemological principles; furthermore, that contention seems  Z" to me a little dogmatic, since many people would quite sincerely say that they see that the man yonder is neither bald nor not bald, and so forth. Thus, our track having thus been blazed and cleared, we can now sketch out our approach to fuzziness. First, a nominalistic, satisfactional account of predicates can be articulated Tarskiways; only, our truthvalues will be the alethic tensors introduced above. Rather  Z' than saying that a sequence of things satisfies a predicate tout court, we'll be saying that the degree of its satisfying it is the alethic tensor into which the predicate carries the sequence  Z) under consideration. We also may relativize the satisfaction relation to some given respect , a respect being represented either by some projection function or, better still, by an infinite  d+ sequence of projection functions, thus saying that d = the degree to which a sequence of things,+ o.,,::  d  s , satisfies a predicate, -, in a respect r iff, if < p i1, 8, p in, 8> is the sequence of projection  Z functions which represents r and  q  is the formula resulting from writing m variables to the  Z right of the mplace predicate -, then, upon the m first components of s being respectively  d assigned to those m variables by a valuation v , < p i1( v q), 8, p in( v q), 8> = d / <0,0, 8> In an alternative way, we may prefer a realistic semantic account of predication, e.g. by positing both properties and facts. Let's for simplicity sake conceive of atomic facts as wholes made up by an madic property along with m individuals. Then we may claim that, since the characteristic function, /, of the madic predicate - has a range included in  d& the set of alethic tensors, the fact that -x1,8,xm will have a degree of existence d if d =  Z' /-(x1,8,xm). We also may, for the sake of ontological economy, identify a property with its characteristic function. Thus, a sentence will be true insomuch as the fact it puports to signify or represent exists; when the sentence's truthvalue = 0 (i.e. <0,0,8>) that fact does not exist at all in any respect, so the sentence represents nothing at all " and thus it is absolutely false, completely false, that is, in every respect. You may still prefer to do without degrees of existence, even if you are patient or kind enough to envisage degrees of truth. Then perhaps you'd better say that all facts are existent (existent to the same extent); only, besides existing, and unrelated to the existence they have, they possess a degree of trueness (or of obtaining ). Or you may contend that what makes a sentence more, or less, true is not a feature of the fact it represents, but the degree of its representing it. This track seems to me to be fraught with unsurmountable difficulties, which I refrain from elaborating upon here. I think what I have hitherto said clearly shows that there are viable, workable ways of shaping semantic accounts of fuzziness (or predicate fringes) by means of the logical system set forth above, thus keeping clear of all snags into which certain accounts unavoidably come. For if a thing x is less red than y while being more red than z, what, according to my approach, is happening is not x's redness stretching on an area  in such a way that redness and nonredness are so tangledly mingled that at each subarea there still are both redness and nonredness, without thereby merging or blending. No, my approach posits precisely a blending of sorts, since what, acccording to it, happens is for x to exemplify redness in a smaller degree than y but in a greater degree than z. If my present approach is correct, fuzziness has nothing to do with uncertainty or indeterminacy (unless by `indeterminacy' we understand just for something to be determined as neither absolutely obtaining nor absolutely failing to obtain) or with truthvaluelessness, or failure of excluded middle (which would indeed constitute  Z indeterminacy) or anything like that. Fuzziness is ensuant upon degrees , such diverse degress in which several things exemplify fuzzy properties. The present approach thus prompts us to set up theories of fuzzy modal, deontic, temporal and doxastic extensions, in order to handle degrees of possibility, obligatoriness, simultaneousness, belief and so on: a bump crop in the fields of ontology, epistemology, ethics, philosophy of mind, philosophy of science, philosophy of language, with novel solutions to sundry puzzles which arise when just the  ZW& two classical alternatives of absolutely yes and absolutely no are envisaged, or, at the very most, enriched with supervaluational devices, indeterminacy cases, oscillations or inextricableH' o.,,::ԫ Z intermingling of those two alternatives and so on. ( ~J ԍWhat I am calling `an inextricableintermingling approach' is G.H. von Wright's proposal in a couple of papers, namely Time,  ~JI Change, and Contradiction , and Truth and Logic , both reprinted in Philosophical Papers by the same author, resp. vol. II (Blackwell, 1983), pp. 11531, and vol. III (Blackwell, 1984), pp. 2641. I shall discuss von Wright's approach in a paper entitled Von Wright on Truth and Contradiction . As against all those approaches, the present one has it that there is real continuousness, shades or transitions consisting in nothing else but in things exemplifying properties in such a way that they up to a point exemplify them and yet also to some extent or other fail to exemplify them, in different degrees which pass into one another through intermediate degrees " those degrees forming a set on which a partial order is defined as pointed out above: the order  which corresponds to the functor `8'. Notice finally how hard it seems to be for supervaluational, indeterminacy or inextricableintermingling approaches to fuzziness to account for comparative constructions. If colour fringes lie in some kind of inextricable intermingling of redness and nonredness, or the like, then how to explain that, even within the fringe, some objects are less red than others? Is there more redness in their constitution? Not so since the inextricability in question demands for those elements of redness, or whatever, to be infinitely many in all those cases " and surely the cardinality will be the same in all such cases of the same kind, be it denumerable or not. More obviously still comparative construcions fail to be clearly accountable for by means of supervaluational or indeterminacy approaches. This is why little is left " besides an alethicsemantic account of degrees like the present one " except pragmatic approaches, about which, I feel bound to confess, I have misgivings " lest genuine alethic issues may appear to dissolve into ways of using words on different occasions or utterance contexts ,  Ze with nothing in reality corresponding to such variations.Xe( ~J ԍI have discussed approaches to the semantics of comparative constructions articulated within classical logic in my Ph.D. diss.  ~J (quoted above, in n.4) and in my paper Contribuci;n a la l;gica de los comparativos  ap. Lenguajes naturales y lenguajes formales  ~J^ II (ed. by c. Martin Vide). Barcelona: University of Barcelona, 1987, pp. 33549.   Z3 3.3. Motion and change ) Motion is just a particular case or kind of transition fringes. Motion or more generally change is therefore nothing else but such fuzziness as is displayed in a continuous temporal way. There are degrees of redness; they either do or at least could form a continuous transition, e.g. from red to blue, even should nothing undergo a colour change at all. What pertains to the process (or change) of a blue thing's reddening, or becoming red, is that those shades are successively exemplified by one same thing in a continuous manner and during some timeinterval. Let me sketch how the present approach can account for motion as a sample case of change. Let's assume that a body b does length wise and continuously move over an interval  Z i along a stretch s, running through it from one extreme to the other. Let s1 , s2,8, be sub Z| stretches of b's run (i.e. parts of s), at least as long as b, while i1,i2,8, are subintervals of  dm! i. Then it seems safe to lay down that the following points hold if v is a correct valuation  dh" (if, that is to say, for any sentence  q , v (q) is the truth value  q  really has). (1) Any  dc# projectionfunction p is such that (N being " remember! " the negation operator on alethicc#o.,,::  d tensors) Nu  pv (b is in s1 at i1)  u, u being some fixed alethic number " the same for all  Z the inequalities concerned, even when s1 and i1 vary " such that a