Symmetric Properties of the Syllogistic System Inherited from the Square of Opposition

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

Abstract

The logical square \(\Omega\) has a simple symmetric structure that visualises the bivalent relationships of the classical quantifiers A, I, E, O. In philosophy it is perceived as a self-complete possibilistic logic. In linguistics however its modelling capability is insufficient, since intermediate quantifiers like few, half, most, etc cannot be distinguished, which makes the existential quantifier I too generic and the universal quantifier A too specific. Furthermore, the latter is a special case of the former, i.e. A ⊂ I, making the square a logic with inclusive quantifiers. The inclusive quantifiers I and O can produce redundancies in linguistic systems and are too generic to differentiate any intermediate quantifiers. The redundancy can be resolved by excluding A from I, i.e.2I = I-A, analogously E from O, i.e.2O = O-E. Although the philosophical possibility of A ⊂ I is thus lost in2I, the symmetric structure of the exclusive square \(^{2}\Omega\) remains preserved. The impact of the exclusion on the traditional syllogistic system \(\mathbb{S}\) with inclusive existential quantifiers is that most of its symmetric structures are obviously lost in the syllogistic system \(^{2}\mathbb{S}\) with exclusive existential quantifiers too. Symmetry properties of \(\mathbb{S}\) are found in the distribution of the syllogistic cases that are matched by the moods and their intersections. A syllogistic case is a distinct combination of the seven possible spaces of the Venn diagram for three sets, of which there exist 96 possible cases. Every quantifier can be represented with a fixed set of syllogistic cases and so the moods too. Therefore, the 96 cases open a universe of validity for all moods of the syllogistic system \(\mathbb{S}\), as well as all fuzzy-syllogistic systems \(^{\mathrm{n}}\mathbb{S}\), with n-1 intermediate quantifiers. As a by-product of the fuzzy syllogistic system and its properties, we suggest in return that the logical square of opposition can be generalised to a fuzzy-logical graph of opposition, for 2 < n.

Keywords

Fuzzy logic Reasoning Set theory Syllogisms 

Mathematics Subject Classification

03B22 Abstract deductive systems 03B35 Mechanization of proofs and logical operations 03B52 Fuzzy logic 03C55 Set-theoretic model theory 03C80 Logic with extra quantifiers and operators 

Notes

Acknowledgements

Thanks are due to Mikhail Zarechnev for developing applications that generate various data sets of the fuzzy syllogistic systems \(\mathbb{S}\), \(^{2}\mathbb{S}\) and \(^{6}\mathbb{S}\), for analysis purposes.

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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Computer Engineeringİzmir Institute of TechnologyİzmirTurkey

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