Intensions as Such: An Outline

The formal expression of pro positional attitudes, especially when nested (iterated), is an important problem for A I . An interesting first-order extensional logical system for such expression has been proposed by Creary. In this system concepts (and concepts of concepts, etc.) are made explicit. The system includes "concept functions", which are special functions which act on and deliver concepts. We point out a difficulty with these functions. A alternative system is proposed, in which there is a conceptforming function corresponding roughly to complexconcept formers (especially the phrase "the proposition that") implicit in English sentences. The resulting system has a more primitive and natural notional base than Creary's has. We avoid problems with quantification inside propositions which are the objects of propositional attitudes by recasting quantified expressions into variable-free form by means of certain functions ("combinators"). I I N T R O D U C T I O N In the past few years AI research has included some attacks on the important problem of representing beliefs, concepts, intensions, referential opacity, and the like [see e.g. (Brachman, 1979), (Hofstadter et al, 1980), (Konolige, 1982), (Maida and Shapiro, 1982), (McCarthy, 1979), (Moore, 1977), (Shapiro, 1979), (Weyrauch, 1977, 1980)]. We propose an alternative to Creary's system as described in (Creary, 1979) (which we shall call "system C" ) ; that system is an extension of a system of (McCarthy, 1979) and is loosely based on the work of Frege (Geach and Black, 1952). Like system C, our own system keeps to first-order extensional logic, makes concepts explicit, has an infinite hierarchy of orders of concepts (concepts of concepts, concepts of concepts of concepts, etc.)*, and takes propositions to be a sort of concept. With Creary, we regard concepts as abstractions from mental entities. We claim, however, that a basic feature in system C concept functions is not well conceived. An example of a concept function is a function which takes a concept of a man and delivers a concept of the man's wife; another concept function might deliver the proposition that the man's wife is French. Concept functions in (Creary, 1979) are based on similar functions in McCarthy's system (McCarthy, 1979), and are akin to the "characterizing functions" in (Church, 1951, 1973, 1974). We shall claim that the use of concept functions does not allow satisfactory formalisation of nested propositional attitudes (beliefs about beliefs, beliefs about beliefs about beliefs, etc.). Our system avoids concept functions by introducing a special function which takes some concepts and delivers a complex concept similar to a definite description. Such definite-descriptional concepts include propositions, which are regarded as concepts whose extensions are truth-values (cf. the view of propositions in (Church, 1951, 1973, 1974)). We shall use the term "denotation" exclusively for the relationship between expressions in English or in a logical formalism and entities in an interpretation of the system of expressions. This relationship is to be carefully distinguished from the relationship which holds between a concept and its extension (which is often also called denotation). We shall say that a concept extends to its extension. Thus the phrase "the concept of Mary" denotes a concept which extends to the person Mary. We do not attempt to specify how to deduce extra beliefs that a cognitive agent holds on the basis of beliefs already ascribed to it. Such deductions are left to arise from particular axioms that some user of our formalism may choose to include in a theory. Similarly, we do not legislate about the connections between knowledge and belief — such matters are again left to the whim of the user. Our work differs from that of (Konolige 1979) in that we avoid the casting of beliefs as expressions in some language so that statements about agents' beliefs are formalised in a metalanguage which talks about that language. The work differs from that of (Konolige 1979) and that of (Maida and Shapiro, 1982) by the inclusion of the abovementioned concept-constructing function and concepts of it. It differs also from (Maida and Shapiro, 1982) in not insisting that items in the formalism cannot denote "extensions". The work differs from that of (Moore, 1977) in avoiding a possible-world approach to propositional attitudes. This paper describes work done in the Department of Computer Science at Indiana University. The research was supported by the National Science Foundation under Grant MCS-8102291. Although there have been objections to such hierarchies, e.g. Carnap [2|.