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

## 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.^{2}I = I-A, analogously E from O, i.e.^{2}O = O-E. Although the philosophical possibility of A ⊂ I is thus lost in^{2}I, 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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