A system of relational syllogistic incorporating full Boolean reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are R-related to all B. Such primitives formalize sentences from natural language like `All students read some textbooks'. Here A and B denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.Comment: Available at http://link.springer.com/article/10.1007/s10849-012-9165-

No syllogisms for the numerical syllogistic

Abstract. The numerical syllogistic is the extension of the traditional syllogistic with numerical quantifiers of the forms at least C and at most C. It is known that, for the traditional syllogistic, a finite collection of rules, similar in spirit to the classical syllogisms, constitutes a sound and complete proof-system. The question arises as to whether such a proof system exists for the numerical syllogistic. This paper answers that question in the negative: no finite collection of syllogism-like rules, broadly conceived, is sound and complete for the numerical syllogistic.