A system of relational syllogistic incorporating full Boolean reasoning

TL;DR

This paper introduces a relational syllogistic system based on propositional logic that formalizes natural language sentences involving set relations and Boolean operations, providing completeness and complexity results.

Contribution

It develops a novel logical framework for relational syllogistics with full Boolean reasoning, including a completeness theorem and complexity analysis.

Findings

Proves completeness of the logical system.

Establishes the computational complexity of satisfiability.

Formalizes natural language sentences involving relations and Boolean operations.

Abstract

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.