Graded Structures of Opposition in Fuzzy Natural Logic
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Logica Universalis
Graded Structures of Opposition in Fuzzy Natural Logic Petra Murinov´a Abstract. The main objective of this paper is devoted to two main parts. First, the paper introduces logical interpretations of classical structures of opposition that are constructed as extensions of the square of opposition. Blanch´e’s hexagon as well as two cubes of opposition proposed by Morreti and pairs Keynes–Johnson will be introduced. The second part of this paper is dedicated to a graded extension of the Aristotle’s square and Peterson’s square of opposition with intermediate quantifiers. These quantifiers are linguistic expressions such as “most”, “many”, “a few”, and “almost all”, and they correspond to what are often called “fuzzy quantifiers” in the literature. The graded Peterson’s cube of opposition, which describes properties between two graded squares, will be discussed at the end of this paper. Mathematics Subject Classification. Primary 68T30; Secondary 03A05, 03B05, 68T37. Keywords. Graded square of opposition, Graded hexagon of opposition, Graded cube of opposition, Fuzzy natural logic, Generalized intermediate quantifiers, Generalized syllogisms.
1. Introduction The Aristotle’s square of opposition has been a source of inspiration for logicians and philosophers for centuries (cf. [1]). It has been analyzed using propositional logic [2,3] as well as classical first-order logic [4–7]. The hexagon of opposition as an extension of the square was studied again by Blanch´e’s independently of some other authors (see [8]). Recall that the hexagon consists of three squares of opposition. At the beginning of the new century, the mentioned structures were studied from philosophical point of view in [9–11]. It is ˇ The work was supported by the MSMT project NPU II project LQ1602 “IT4Innovations excellence in science”.
P. Murinov´ a
Log. Univers.
surprising that the square of opposition was applied not only in philosophical and mathematical logic but also in linguistics and psychology. A cube of opposition including six squares of opposition was proposed by Moretti [12,13]. Another cube of opposition proposed by Keynes and Johnson was studied by Dubois and Prade with respect to the possibility theory. The differences between the two mentioned cubes of opposition were discussed by Dubois et al. [14]. Moyse and Bouchon-Meunier [15] introduced the square of opposition to the concept of linguistic summarization. A generalization of the square of opposition to many-valued logic was introduced by Dubois and Prade [16–18]. The authors proposed a graded Aristotle’s square of opposition and a cube of opposition including its graded version that associates the traditional square of opposition with a dual square of opposition. Ciucci et al. [19,20] introduced an application of the gradual cube in the possibility theory. In [14], the group of authors, namely, Dubois, Prade and Rico discussed a gradual extension of Moretti’s cube in relation to a gradual extension of J-K cube. Another extension of the Aristotle’s square, whic
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