Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic

Moktefi, Amirouche

Schang, Fabien

English

PhilPapers

1Depicting negation in diagrammatic logic: legacy and prospectsFabien Schang, Amirouche Moktefischang.fabien@voila.f  r  amirouche.moktefi@gersulp.u-strasbg.fr AbstractHere are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic.1. From dichotomy to bivalence     In the primary diagrams suggested in the logic of classes, each predication of the form S is P was presented by a circle or closed curve symbolizing the class P and in which an element S occurred; in other words, S is P is a true proposition if and only if the individual value occurs  in the region purported to figure the class of individuals which satisfy the property  P.  A  translation  of  classes  in  terms  of  propositions  or  predications  is  thus possible, provided that the class P is considered as a predicate term attached to a subject term S. As to the set of individuals which don't satisfy the property P, they must occur outside the closed curve that determines the set of P's.     One first epistemological obstacle to non-classical logics is the lack of any universe of discourse in the first suggested diagrams, as shown by the distinction below.        Figure 1       Figure 2                                                                                U                                   × S                                     × S      non-P P          P2     In Figure 1, the set of individuals that are not P (i.e. the non-P's) are not located in a specific space since only the closed curve is taken into account; whereas in Figure 2, all the non-P's correspond to the region located outside the closed curve and are within the universe  of  discourse U.  We  call  by dichotomy the  opposition  between  the  whole individuals located within (the class) P and the whole remaining ones located outside (the class) P. The opposition between P and non-P thus characterizes logical negation as a dichotomy; it is presented in diagrammatic logic as a spatial contrast in vs. out.     As to the semantic notion of bivalence, it extends dichotomy in splitting the universe of discourse into two classes of truth-values. If  S is located in the region of P's, then the proposition S is P is true; if S is not located in the region of P's, then S is P is false. Now since S cannot be both in and outside P, the truth of the proposition  S is P entails the falsehood of its negation S is not P, and the falsehood of the proposition S is P entails the truth of its negation S is not P. Consequently, every proposition does have a truth-value within the universe of discourse: either it is true, or it is false; no proposition can be both in and outside the closed curve, so that if a proposition is true then it is not false and if it is false then it is not true. Another way to state these two properties of bivalence is to say that a diagram is complete and consistent.    It seems difficult to challenge bivalence in logical diagrams, assuming that it implies for any  element  S  in  U  both  its  occurrence  and  non-ubiquity.  Nevertheless,  the  logic  of diagrams gave way to the introduction of non-classical constants in its modern variants, including  non-classical  negation.  How  to  embed  the  latter  without  questioning  both preceding requirements of completeness and consistency?      Our answer is: by internalizing truth-values within the logic of diagrams, and by making a  distinction  between  two  readings  of  negation.  Our  final  claim  is  that  non-classical negation  is  less  revolutionary  than  it  might  appear  first,  insofar  as  it  doesn't  really challenge the general property of dichotomy. 2. From exclusion negation to choice negation      By an internalization, we mean the process that consists in introducing a metalanguage notion within the object-language. While truth-values belong to metalanguage in the logic of diagrams, given that they don't occur in the logical space U, some non-classical logics make use of the following process in order to depict non-classical constants, namely: to see the general proposition of the form S is P as an element in a class of truth-value (a “semantic class”, say); e.g. [S is P: p] is true, with [S is P: p] for S and true for P. We thus obtain  the  subsequent  illustrations  of  non-classical  logic,  where  the  Venn  scheme  is internalized in the semantic level of truth-values.     In Figure 3, a diagram is generally presented as a range of possible relations between two distinct classes and the combinations of which are counted as 2² = 4; by analogy with 3the  classes  of  propositions,  the  classes  of  truth-values  equally  propose  four  possible relations between the class T of true propositions and the class F of false propositions.                                   Figure 3      Figure 4                                                                                 (non-P∩non-Q)  = –P–Q                             (non-T∩non-F) = –T–F                                                                               (P∩Q) =                                                         (T∩F) =                                     PQ                                                                 TF   (P∩non-Q) = P–Q         (non-P∩Q) = –PQ      (T∩non-F) = T–F          (non-T∩F) = –TF      Any proposition can thus be considered as: true and non-false, false and non-true, true and false, non-true and non-false. Unlike diagrams in classical logic, Figure 4 displays a non-bivalent space in which a proposition can be neither true nor false or both true and false. In this sense, the space U' of non-classical logics is not complete but paracomplete; it  is  not  consistent,  either,  but paraconsistant.  A sample of  paracomplete three-valued logics are Kleene's intuitionistic logic in [3] or Łukasiewicz's logic of indetermination in [4], in which some propositions are neither true nor false indécidable; a case of paraconsistent logic  is  Priest's  dialetheist  logic  in  [6],  in  which  a  proposition  can  be  paradoxal  and, therefore, both true and false.     However, paraconsistency doesn't mean that a proposition may be both in and outside one and the same class: if it is both true and false, then it is located within the class TF that intersects T and F. Paracompleteness doesn't mean any more that a proposition may be absent from the universe of discourse, given that propositions that are neither true nor false  are located in  U'.  Therefore,  the introduction of  non-classical  diagrams does not entail the revision of dichotomy as such: if a proposition is not true, for instance, then it is located in the non-true (in –T) and, thus, outside the true (i.e. outside T); although it may be both true and false,  such a case does not  constitute  any more an infringement  to dichotomy than for an element S to be both in the class P and the class Q. In other words, the  change  caused  by  non-classical  logics  may  generate  some  plausible  confusion between  the  concepts  of  bivalence and  dichotomy:  the  former  states  that  any  true proposition  is  non-false  and  any  false  proposition  is  non-true,  hence  an  obvious connection  between  such  a  formulation  and  dichotomy. However,  dichotomy  merely 4requires the distinction between an arbitrary class and any other one that is not itself: P and non-P, true and non-true, false and non-false, and so on.      In order to bring out the distinction between mere difference and incompatibility, it can be said that while the class T of  strictly true propositions and the class F of  strictly false propositions are dichotomous in classical  logic and even in a number of  non-classical logics, T's and F's are no more dichotomous in non-classical logics since TF is a new class in U' including both true and false propositions. Consequently, this means that a number of non-classical  negations  are  not  dichotomous whenever  they  don't  always require  true propositions to be outside the false region and false propositions to be outside the true region.      Borrowing a terminology from Terence Parsons in [4], the difference is made here between negation as a  choice operation and negation as an  exclusion operation. While exclusion negation generally consists in projecting any proposition from a class (whether semantic  or  not)  into  another  class,  choice  negation  turns  a  proposition  into  another proposition  but  doesn't  necessarily  projects  it  from  a  semantic  class  to  another  one. Łukasiewicz's or Priest's logics are an illustration of it, since the (choice) negation of a proposition  that  is  neither  strictly  true  nor  strictly  false  (say,  indeterminate:  –T–F =  I) results in an unchanged truth-value when the negation is said to be normal: v(p) = v(~p) = I, so that p and non-p do have the same truth-value. Such a non-classical negation does not exhibit any more the property of exclusion that characterizes dichotomy. Let us note that, while not every logical negation acts as an exclusion operation, some non-classical negations still  do. These are said to be  non-normal,  e.g. Post'c cyclic negation: for an ordered series of {1, ...,i, ...  n} truth-values,  v(~P) = i+1 whenever  v(P) = i, so the truth-values  of  P  and  not-P  are  never  the  same.  How  to  characterize  logical  negation intensionally, if not every negation is to be depicted as a exclusion operation?     As Brady put it in [1], we can consider an intensional property of logical negation that is more  general  than  dichotomy  and  does  justice  to  both  classical  and  non-classical negations, that is: negation as a  mirror-image concept with an axis of symmetry. If one depicts a range of three truth-values with T and F as opposite sides and I in the middle, then the negation of I results in I just as the mirror-image of any point located in an axis of symmetry results in this point itself.     Thus, not every logical negation is an exclusion property when applied to propositions with  a  non-classical  value;  nevertheless,  the  distinction  between  classical  and  non-classical remains as a dichotomy in itself between {T,F} and not-{T,F}.      The overall situation may be summarized as follows: - exclusion negation '–' is an intensional property: it rejects a proposition –(p) outside the semantic class of p, but without determining the class in which –(p) should be located; and conversely, choice negation '~' is extensional since it determines a specific semantic class 5for  ~(p)  without  rejecting  it  necessarily  outside  the  semantic  class  of  p.  Priest's paraconsistent negation, for example, locates the negation of a strictly true proposition in the strictly false region F, the negation of  a strictly false proposition in the strictly true region T, and the negation of a paradoxical proposition (i.e. both true and false) in the paradoxical region TF. - the assimilation of logical negation to dichotomy is due to the behavior of choice negation within classical, bivalent logic: given that only two strictly separate semantic classes occur in it (i.e. T and F), classical negation behaves exactly like an exclusion negation and the distinction  between  choice  '–'  and  exclusio  '~'  is  thus  made  impossible  extensionaliy speaking, contrary to the case with most of non-classical logics and especially the three-valued ones.- being inconsistent does not mean the same as being  contradictory if, by contradictory, we mean the possibility to be both X and non-X whatever the syntactic category of X may be (whether a class of propositions or a class of truth-values). Paraconsistent logic does not include both true and non-true propositions or false and not-false propositions: T–T and  F–F  cannot  occur  in  U'.  Such  a  non-classical  (choice)  negation  doesn't  infringe dichotomy at all: it seems to do so only for whoever goes on to think of it in a classical universe of discourse U without including the more complex universe U'. But to accept TF in U is an impossible thing to do, so that the confusion between paraconsistency and true contradictions is nothing more than a confusion between the logical spaces U and U' as different frameworks for thinking about truth-values.Conclusion: non-classical negations are harmless for dichotomy as such     In sum, the introduction of non-classical logics into diagrams entails neither a significant revolution within the logical space, nor the obligation to modify our representation of such a space so as to allow the ubiquity of propositions both in and outside one and the same class. The process of internalization merely helps to make propositions relatively true and false and to go beyond the strictly separate relation between T and F within a bivalent framework; now if such a process if refused, then it appears as literally impossible to think of  negation  from  a  non-classical  perspective,  given  the  spatial  relation  in-out  as  a undebatable  statement  of  dichotomy.  In  a  nutshell:  either  logical  negations  are dichotomous or not, as choice operations; but they don't infringe to dichotomy in a logical space at all, because such an infringement would assume a substantial change in the very geometrical  properties of the logical space. Nothing similar occurs in paraconsistent or paracomplete logics in U', anyway.     One task remains to be accomplished in non-classical diagrammatic logic once the internalization  is  accepted,  namely:  assuming  that  not  every  non-classical  logic  is  a exclusion operation and, thus, an operation of complementation, how to represent the non-6complementary  laws  of  projection  between  p and  not-p for  paraconsistent  and paracomplete  negations  in  U',  and  which  of  these  projections  should  replace  the dichotomous  relation  in-out  that  marks  classical  negation?  Such  a  task  will  be accomplished in a later paper about the specific case of da Costa & Béziau's overclassical logic (see [3]), a case of paraconsistent negation that rejects the laws of non-contradiction and excluded middle while maintaining De Morgan's laws and double negation.References[1] R. T. Brady: “On the formalization of the Law of non-Contradictionˮ, in The Law of Non-Contradiction (New Philosophical Essays), G. Priest & J.C. Beall & B. Armour-Garb. (ed.), Clarendon Press, Oxford (2004), pp. 41-8[2] N.C.A. da Costa & J.-Y. Béziau: “Overclassical Logicˮ, Logique et Analyse 157(1997), 45-66[3] S. Kleene: Introduction to Metamathematics, Amsterdam, Groningen, New York (1952)[4] J. Łukasiewicz: “On three-valued logicˮ, Ruch Filozoficzny 5(1920), 170-1[5] T. Parsons:  “Assertion, Denial, and the Liar Paradox”,  Journal of Philosophical Logic 13(1984); 137-152[6] G. Priest: “The Logic of Paradoxˮ, Journal of Philosophical Logic 8(1979), 219-41

Depicting Negation in Diagrammatic Logic: Legacy and Prospects

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Depicting Negation in Diagrammatic Logic: Legacy and Prospects

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