The Calculus of Concepts

In this article, the authors propose a new version of a natural deduction system for a typed lambda calculus of concepts as functional abstracts in Frege's fashion. The introductory section provides a general description of the approach. In so doing, the authors briefly consider different theories of concepts and Fregean ideas among them. Next, the authors introduce their conception of neurophenomenology as far as the cognitive procedure of categorization is concerned. They also describe their previous results in the formal (logical) representation of concepts as functions. The second section introduces the calculus of concrete concepts (concepts whose extensions consist of substances) as a natural deduction system. The authors consider concepts as object-to-object functions expressing the result of the cognitive processing of perceptive stimuli. The current version of this calculus is designed to establish standard relations between concepts and, first of all, the inclusion relation, which allows specifying the remaining fundamental and secondary relations. The consequence relation between two concepts means that the extension of the first is included into the extension of the second. The authors formalize both simple and complex types of objects constructed with the help of operations that are analogs of conjunction, disjunction and negation. The section ends with the discussion of the meaning of some rules of inference. In the final part of the article, the authors sum up the findings and outline the prospect of further research in this area. The prospect is, first of all, the construction of adequate semantics for the proposed calculus and extension of the calculus to the case of quantified attributes. This calculus could be further modified in an obvious way to provide a formal representation of abstract concepts. All this allows considering the prospects of this formalism for computer science and the project of artificial intelligence in the field of knowledge representation and processing, as well as for modeling the machine learning process (in particular, instance-based learning).

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