A Generalized Syllogistic Inference System based on Inclusion and Exclusion Relations

Entailment relations are of central importance in the enterprise of natural language semantics. In modern logic, entailment relations are characterized from two viewpoints: the modeltheoretic and proof-theoretic ones. By contrast, most approaches to formalizing entailment relations in natural languages have been solely based on model-theoretic conceptions. Thus, the notion of validity is only characterized in model-theoretic terms, and only few attempts have been made at the relevant proof-theoretic notions such as provability and proof as applied to natural language inferences. The main aim of our study is to offer a simple inference system for syllogistic fragment of natural language, thereby making a connection between proof theory and natural language semantics. Indeed, there are several prior attempts in this direction.1 One is the study of syllogisms from modern logical viewpoints, which was started by à Lukasiewicz [7] and subsequently given their natural deduction formulations by Corcoran [2] and Smiley [23], among others. In connection with this, the program of natural logics, including the so-called calculus of monotonicity proposed in van Benthem [24] and Sanchez [20], has been developed by some linguists and logicians.2 More recently, there are important developments in the logic and AI literature which aim to extend the traditional syllogistic system to cover more expressive fragments of natural language (cf. Nishihara, Morita, and Iwata [17], Nishihara and Morita [16], Moss [13, 14, 15], and Pratt-Hartmann and Moss [18]). The central aim of these studies is to characterize natural language inferences at the forms as close as possible to surface forms. Our approach agrees with these studies in that a proof system plays a role in characterizing entailment relations in natural languages. However, an essential difference is that we decompose syllogistic inferences and categorical sentences constituting them in terms of more primitive relations (i.e., inclusion and exclusion relations), whereas most approaches in modern logical studies of syllogisms and natural logic take surface forms as primitive and do