The Cube Generalizing Aristotle’s Square in Logic of Determination of Objects (LDO)

Abstract

In this paper we present a generalization of Aristotle’s square to a cube, in the framework of an extended quantification theory defined within the Logic of Determination of Objects (LDO). The Aristotle’s square is an image which makes the link between quantifier operators and negation in the First Order Predicate Language (FOPL). However, the FOPL quantification is not sufficient to capture the “meaning” of all quantified expressions in natural languages. There are some expressions in natural languages which encode a quantification on typical objects. This is the reason why we want to construct a logic of objects with typical and atypical objects. The Logic of Determination of Objects (LDO) (Desclés and Pascu, Logica Univers. 5(1):75–89, 2011) is a new logic defined within the framework of Combinatory Logic (Curry and Feys, Combinatory Logic I, North-Holland, Amsterdam, 1958) with functional types. LDO basically deals with two fundamental classes: a class of concepts ($\mathcal{F}$) and a class of objects ($\mathcal{O}$). LDO captures two kinds of objects: typical objects and atypical objects. They are defined by the means of other primitive notions: the intension of the concept f ($\operatorname{Int}f$), the essence of a concept f ($\operatorname{Ess}f$), the expanse of the concept f ($\operatorname{Exp}f$), the extension of the concept f ($\operatorname{Ext}f$). Typical objects in $\operatorname{Exp}f$ inherit all concepts of $\operatorname{Int}f$; atypical objects in $\operatorname{Exp}f$ inherit only some concepts of Intf. LDO makes use of all the above notions and organizes them into a system which is a logic of objects (applicative typed system in Curry’s sense—Curry and Feys, Combinatory Logic I, North-Holland, Amsterdam, 1958—with some specific operators). In LDO new quantifiers are introduced and studied. They are called star quantifiers: Π^⋆ and Σ^⋆ (Desclés and Guentcheva, in: M. Bőttner, W. Thűmmel (eds.), Variable-Free Semantics, pp. 210–233, Secolo, Osnabrűck, 2000). They have a connection with classical quantifiers. They are considered as the determiners of objects of $\operatorname{Exp}f$. They are different from the usual quantifiers Π and Σ expressed in the illative Curry version (Curry and Feys, Combinatory Logic I, North-Holland, Amsterdam, 1958) of Frege’s quantifiers. They are defined inside the Combinatory Logical formalism (Curry and Feys, Combinatory Logic I, North-Holland, Amsterdam, 1958) starting from Π and Σ, by means of abstract operators of composition called combinators (Curry and Feys, Combinatory Logic I, North-Holland, Amsterdam, 1958). The system of four quantifiers Π, Σ, Π^⋆ and Σ^⋆ captures the extended quantification that means quantification on typical/atypical objects. The cube generalizing the Aristotle’s square visualizes the relations between quantifiers Π, Σ, Π^⋆ and Σ^⋆ and the negation.