Abstract
All valid Aristotle’s modal syllogisms can be obtained by adding modal operators to 24 valid classical syllogisms. On the basis of the 20 valid modal syllogisms obtained by adding modal operators to valid classical syllogisms AAA-1 and EAE-1, this paper not only shows that the validity of the other 326 Aristotle’s modal syllogisms can be derived by making full use of truth definition and symmetry of Aristotelian quantifiers in generalized quantifier theory, and propositional deformation rules in proof theory, but also shows that there are reducible relations between/among Aristotle’s modal syllogisms. These innovative results are embodied in the 29 theorems proposed in this paper. The research methods used in the paper provide a simple and reasonable mathematical model to study generalized modal syllogisms. It is hoped that these innovative achievements will make contributions to further research on Aristotle’s and generalized modal syllogistic logic, and to promote knowledge representation and knowledge reasoning in computer science, and natural language information processing.

Keywords
generalized quantifier theory; Aristotle’s modal syllogisms; reducible relations; validity

Cite this paper
Xiaojun Zhang, Reducible Relations between/among Aristotle’s Modal Syllogisms , SCIREA Journal of Computer. Volume 5, Issue 1, February 2020 | PP. 1-33.

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