Relational Syllogistics

Details Relational Syllogistics Relational Syllogistics

2011-02-22

arxiv.org/abs/1102.4496v4

Computer Science

Logic in Computer Science

Scientific paper

We present a quantifier-free Hilbert-style axiomatization, based on classical propositional logic, of a system of relational syllogistic formalizing the following binary relations between classes (of objects): a\leq b \Leftrightarrow \forall x(x\in a \Rightarrow x\inb) and (a,b)[{\alpha}] \Leftrightarrow (Q_1x \in a) (Q_2y \in b) ((x,y) \in {\alpha}), where a and b denote arbitrary classes, Q_1,Q_2 \in {\forall,\exists}, and {\alpha} denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting classes, determining the set of class terms, and variables denoting binary relations between objects, determining the set 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 {\alpha}^[-1]. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.

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