Aristotle's Syllogisms And Its Extensions

Although it had been more than two thousand years,there was no outstanding breakthrough for the theory of Aristotle's syllogisms before the emerging of the Mathematical Logic.Chapter 1 draws the outline of the content of the Aristotle's syllogisms,including its connotation,representation,classification,and the transformation between different syllogistic patterns.Since logicians took account of some logic problems with mathematical ideas,it also has been developed a new content for syllogisms.In the second chapter,starting point is the quantification of natural languages and the logical meanings of quantifiers are explained by Model theory.Also,formal definitions of quantifiers are given.Then,we use mathematical language to depict properties of quantifiers,such as monotonicity,symmetry and so on.There is a close relationship between these properties and the reasoning of syllogisms.Using some properties of a quantifier,combined with operations between the quantifier and its negation,it can be proved that all Aristotle's syllogisms are valid.Not only that,but in fact,all quantifiers which have particular properties are applicable to those reasoning patterns.In the last part,from the perspective of the relation,we borrow some ideas from Ivan Hartmann,cutting a syllogistic fragments,R system,from natural languages.By using p,l,r,t,c as initial symbols,added universal and existential quantifier,wellformulas are formed.Also,based on Model theory,we define semantics of the formulas and inference rules of the system.Finally we discuss the completeness of the system.