Lexical Meanings Analysed by Means of Typed Applicative Representations

Applicative languages (Church’s λ-calculus and Curry’s combinatory Logic) and functional types are useful logical tools for studying and representing the meanings of verbal predicates and other linguistic operators (prepositions, preverbs ...) of natural languages by means of combinations of abstract and cultural primitives. The situations are semantic expressions associated to sentences; they are written by means of applicative expressions (ae) generated from semantic abstract primitives: (i) cognitive basic types (individual, massive, distributive class, abstract places, activity, situations...); (ii) operators transforming assigned types (as topological operators : take the interior, exterior, boundary, closure of an abstract place); (iii) kinematic, dynamic, cause relators: MOVT and CHANG expressing movement or change the state of an entity; FAIRE, CONTR (to control) and TELEO (to intend a teleonomic situation) introducing a link between a kinematic situation and an entity (agent, intermediary instrument...); CAUSE establishing a link between two different situations (a cause and an effect). These abstract primitives are interpreted inside of the cognitive fields of perception and action. They are sources of numerous grammaticalizations in languages. Verbal predicates involve an actualization over topological intervals of instants; thus, it is necessary to introduce complex operators for transforming a situation into an aspectual situation (state, event, process ...). This article presents systematically these abstract primitives with some examples of meanings represented inside the applicative framework. The applicative expressions of situations (semantic schemes) defined to a semantic level can be integrated into lexical predicates of another level, by using combinators of combinatory logic; this integration process in Cognitive and Applicative Grammar (GAC) has already been presented (in precedent FLAIRS). Applicative representations and functional types Applicative expressions, designated by ‘ae’, are generated from a set of elementary operators and absolute operands by means of the operation of application of an operator ‘X’ doing an act over an operand ‘Y’; the result, designated by ‘XY’, is an ‘ae’ that can be again an operator or an operand Copyright © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. according to the context. The objects are always absolute operands. There are different types of objects and operators. The Church’s functional types are generated from basic types by the following rules: (i) The basic types are functional types; (ii) If ‘α’ et ‘β’ are functional types, than ‘Fαβ’ is a functional type (Church 1941; Curry 1958). A functional type ‘α’ assigned to an ‘ae’ ‘X’ is noted [X: α]. The types of operators are always with the form ‘Fαβ’. Let an operator ‘X’ and an operand ‘Y’ of ‘X’, the applicative rule with types is: [X: Fαβ] , [Y:α] => [XY :β] A product of types is canonically associated to a functional type: any operator ‘fn’ with a product of types is transformed into an unary operator ‘Curry(fn)1’, according to the Curryfication principle : [ fn : F(α1 x α2 ... x αn )β ] ≈ [ Curry(fn)1 : Fα1F α2 ...Fαn β ] Remark : We use two notations; one is prefixed : an operator ‘f’ is always positioned before his operand: ‘fXY =def ((fX)Y) ; the second is infixed : a binary relator ‘f2’ is positioned between two successive operands according to the notational equivalence : [X f2 Y] =def f2YX. The Church’s lambda calculus and Curry’s Combinatory Logic (Hindley and Seldin, 1986) are two examples of applicative languages. In Combinatory Applicative Categorial Grammars (CACG) (Desclés and Biskri 1995), Universal Applicative Grammar (GAU) (Shaumyan 1987) and Applicative and Cognitive Grammar (GAC) (Desclés, 1990), all linguistics units (grammatical and lexical units) are operators or absolute operands with assigned functional types; in these formal frameworks, the meanings of verbal predicates and other linguistic operators are also analysed by ‘ae’ generated from semantic primitives: basic cognitive types; topological operators and locating relators; static, kinematic, dynamic, causal relators; aspectual operators, enuntiative and commitment operators (Desclés 2009). For instance, the meanings of prepositions are analysed by means of topological operators and by change relating different static situations. Thus, in the sentence the book is Lexical Meanings Analysed by