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Fuzzy Syllogisms, Numerical Square, Triangle of Contraries, Inter-bivalence

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Around and Beyond the Square of Opposition

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

A new Predicate Calculus, called “Distinctive” D, is presented in the form of the Hexagon of Opposition, in which the Particular Yba (= only some b are a) is the contradictory of the Universal Uba (= all or no b are a). Y is preferred, as a primitive, to I or O, because it is more “natural” than the others. Typical inferences of the systems are the obversions: Yba=Yba′ (= only some b are not a), Uba=Uba′ (= all or no b are not a).

D-Systems include traditional Syllogisms. Polygonal and Numerical developments are being developed, including intermediate quantifiers (“the majority of”, …). Polygons are finally absorbed into the Numerical D-Square NDS. Isomorphisms are discovered between bivalent D-systems and some Non-Standard Logics by depriving the subject-class of the quantifier and transferring its (pre)numerical attribute to the truth value of the judgement. These ‘(poly-)inter-bivalent’ logics admit intermediate values between true and false. In that way we get Fuzzy Logic, and present the Fuzzy Square.

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Notes

  1. 1.

    For the purposes of this study we will utilise the terms “class” and “set” as synonyms.

  2. 2.

    Examples: case 4: UD=quadrilaterals, b=rhombuses, a=rectangles; case 5: UD=polygons, b=polygons with fewer than five sides, a=polygons with more than three sides.

  3. 3.

    This number result from the combination of the three quantifiers with the three predicates in the four figures.

  4. 4.

    Blanché [3] and Sesmat [19] already presented oppositive hexagons, but we have found no evidence of a calculus ever having been developed. As from 1910, Vasiliev [25] developed a ‘triangular’ syllogistic based on the “only some” quantifier, but different from ours in that it is similar to paraconsistent systems. See Béziau [1], Suchon [24] and Moretti [13].

  5. 5.

    In the last three lines there are expressions not included in the initial hexagon, but rather generated from another 3 hexagons constructed on the pairs b′a′, ab, a′b′ (the remaining 4 possible hexagons [or triangles] are equivalent to the others by obversion).

  6. 6.

    If the first premise is converted, it is possible to reach EYO in the fourth Figure from EYO in the third Figure; this mood may result in YAI, still in the fourth Figure, by means of indirect reduction. From here, conversion of the conclusion and the exchanging of premises results in AYI in the first Figure, which by obversion of conclusion and the first premise generates EYO, which, by conversion of the first premise, generates EYO in the second Figure. AYO in the second Figure is generated from AYI in the first Figure as follows: contraposition of the first premise and obversion of second premise and conclusion.

  7. 7.

    An example of QD may be: (Only a majority part of the animals is ill) * (no ill animal is prized) → (no, or only a minority part of animals is prized) (>ai ∗ Eip)→(E<ap).

  8. 8.

    In general, the validity of the 8 deductive formulas of modern Singular Syllogisms are confirmed, while the U-proposition schemes will be added. See [2, chapter 5, paragraph 45].

  9. 9.

    Conclusions are indicated by means of bi-categorical propositions, simple distinctive particulars, and traditional particulars, conjunctions of bi-categoricals being excluded for the purpose of conciseness.

  10. 10.

    A long tradition, from the scholastic “excepta” to J.H. Lambert to A. De Morgan to the numerical propositions of Murphree [14], had made use of the numerical exceptive, which can be compared to the set operation of asymmetric difference: s∖p (or s−p)=the s’s that are not p (= the complement of p in s).

  11. 11.

    If s has an even number of elements, a numerator will be placed exactly on the median axis and will be its own reciprocal.

  12. 12.

    For example:

  13. 13.

    Many thinking systems adopt it, e.g: Imaginary logic of N. Vasiliev [25], Non-Monotonic Logics, Psychoanalysis (contrariety conscious-unconscious, bi-logic of I. Matte-Blanco) and, in general, the non-reductionist descriptions of reality, from biology to historical and sociological processes.

  14. 14.

    Naturally, the measure of truth is relative to the totality of the subject-class (usable in various ways: absolute commensurable, correlative-statistical, relative-probabilistic, fractionary-percentages, etc.). Here, as in the following systems, all the deductive rules and laws for numerical calculus can be found.

  15. 15.

    The 50 % uncertainty that the refrigerator contains an apple is everything but the certainty that there is half an apple (Kosko [9]).

  16. 16.

    Zadeh [26] proposed the term “Fuzzy syllogism” as a reasoning pattern based on vague quantifiers, like “most”. Since then the term assumed different meanings in scientific literature. See for example the work of Dubois and Prade [6], Naddeo et al. [16] or, more recently, Kumova and Çakir [10]. The latter use the term for representing the truth degree of the traditional 256 moods. Their “fuzzy syllogistic system” is an original approach, whose objective is to model automated reasoning based on inductive knowledge. On the contrary, we use the same term to highlight that new quantifications (and new deductive moods) derivable from classical syllogistics are interpretable as fuzzy reasoning, without probability or statistical bases.

  17. 17.

    For example, in case 3, all possible predicates are: those that are both true and false, those that are true but not false, those that are neither true nor false, the gap being between those that are false but not true. A model of this logic may fit into the idea that all falsehood hides a truth, as in the Freudian ‘lapsus’. A model for case 2 is given by a philosophy that considers all truth relative (that is, false at the same time) but also admits the existence of pure (not-true) false predicates. As regard case 1, a model may be a sophistic position where all propositions are either contradictory or meaningless.

  18. 18.

    However, the analyses of the distinctive propositions reveal a relation with four-five arguments (and even more in compound calculi) in which the first two (subsets) share the third (subject) and relate to a fourth and a fifth argument (the predicate and its negation). In the historical development of classical logic, the Exceptive Proposition [“Every b, except c, is a” or A(b−c)a] manifests a tri-argument structure.

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  1. v. della Libertà 5, 47042, Cesenatico, FC, Italy

    Ferdinando Cavaliere

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  1. , Institute of Philosophy, Federal University of Rio de Janeiro, Largo São Francisco de Paula 1, Rio de Janeiro, RJ 20051-0, Brazil

    Jean-Yves Béziau

  2. , Institute of Philosophy, University of Bern, Länggassstrasse 49a, Bern, 3000, Switzerland

    Dale Jacquette

Appendix: Deductive Conclusion in D6

Appendix: Deductive Conclusion in D6

See Table 6. The headings of the columns and lines function respectively as I and II premises. Each proposition inside the cells represents the ‘stricter’ or ‘stronger’ conclusion of the syllogism (subordinate moods are implied).

Table 6 Distinctive simple deductions
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Cavaliere, F. (2012). Fuzzy Syllogisms, Numerical Square, Triangle of Contraries, Inter-bivalence. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_17

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