How to Derive the Other 37 Valid Modal Syllogisms from the Syllogism ◇A£I◇I-1

Syllogistic reasoning plays an important role in natural language information processing. In order to provide a consistent interpretation for Aristotelian modal syllogistic, this paper firstly proves the validity of the syllogism ◇A£I◇I-1, and then takes it as the basic axiom to derive the other 37 valid modal syllogisms on the basis of some reasoning rules in classical propositional logic, the transformation between any one of Aristotelian quantifiers and its three negative quantifiers, the symmetry of the Aristotelian quantifier some and no, and some relevant definitions and facts. In other words, there are reducibility between the modal syllogism ◇A£I◇I-1 and the other 37 valid modal syllogisms. There are infinitely many modal syllogism instances in natural language corresponding to every valid modal syllogism, thus this study has important practical significance and theoretical value for natural language information processing in computer science

Rejection in Łukasiewicz's and Słupecki's Sense

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic

How to Deduce the Other 91 Valid Aristotelian Modal Syllogisms from the Syllogism IAI-3

This paper firstly formalizes Aristotelian modal syllogisms by taking advantage of the trisection structure of (modal) categorical propositions. And then making full use of the truth value definition of (modal) categorical propositions, the transformable relations between an Aristotelian quantifier and its three negative quantifiers, the reasoning rules of classical propositional logic, and the symmetry of the two Aristotelian quantifiers (i.e. some and no), this paper shows that at least 91 valid Aristotelian modal syllogisms can be deduced from IAI-3 on the basis of possible world semantics and set theory. The reason why these valid Aristotelian modal syllogisms can be reduced is that any Aristotelian quantifier can be defined by the other three Aristotelian quantifiers, and the necessary modality ( ) and possible modality ( ) can also be defined mutually. This research method is universal. This innovative study not only provides a unified mathematical research paradigm for the study of generalized modal syllogistic and other kinds of syllogistic, but also makes contributions to knowledge representation and knowledge reasoning in computer science

Inductive Logic

Inductive Logic is a ‘thematic compilation’ by Avi Sion. It collects in one volume many (though not all) of the essays, that he has written on this subject over a period of some 23 years, which all demonstrate the possibility and conditions of validity of human knowledge, the utility and reliability of human cognitive means when properly used, contrary to the skeptical assumptions that are nowadays fashionable