``...it seems that hardly anybody proposes to use different variables for propositions and for truth-values, or different variables for individuals and individual concepts.''--(Carnap 1956, p. 113).
Admitting individual concepts as objects--with concept-valued constants, variables, functions and expressions-- allows ordinary first order theories of necessity, knowledge, belief and wanting without modal operators or quotation marks and without the restrictions on substituting equals for equals that either device makes necessary.
In this paper we will show how various individual concepts and propositions can be expressed. We are not yet ready to present a full collection of axioms. Moreover, our purpose is not to explicate what concepts are in a philosophical sense but rather to develop a language of concepts for representing facts about knowledge, belief, etc. in the memory of a computer.
Frege (1892) discussed the need to distinguish direct and indirect use of words. According to one interpretation of Frege's ideas, the meaning of the phrase ``Mike's telephone number'' in the sentence ``Pat knows Mike's telephone number'' is the concept of Mike's telephone number, whereas its meaning in the sentence ``Pat dialed Mike's telephone number'' is the number itself. Thus if we also have ``Mary's telephone number = Mike's telephone number'', then ``Pat dialed Mary's telephone number'' follows, but ``Pat knows Mary's telephone number does not.
It was further proposed that a phrase has a sense which is a concept and is its meaning in oblique contexts like knowing and wanting, and a denotation which is its meaningin direct contexts like dialing. Denotations are the basis of the semantics of first order logic and model theory and are well understood, but sense has given more trouble, and the modal treatment of oblique contexts avoids the idea. On the other hand, logicians such as Carnap (1947 and 1956), Church (1951) and Montague (1974) see a need for concepts and have proposed formalizations. All these formalizations involve modifying the logic used; ours doesn't modify the logic and is more powerful, because it includes mappings from objects to concepts. Robert Moore's forthcoming dissertation also uses concepts in first order logic.
The problem identified by Frege--of suitably limiting the application of the substitutitivity of equals for equals--arises in artificial intelligence as well as in philosophy and linguistics for any system that must represent information about beliefs, knowledge, desires, or logical necessity--regardless of whether the representation is declarative or procedural (as in PLANNER and other AI formalisms).
Our approach involves treating concepts as one kind of object in an ordinary first order theory. We shall have one term that denotes Mike's telephone number and a different term denoting the concept of Mike's telephone number instead of having a single term whose denotation is the number and whose sense is a concept of it. The relations among concepts and between concepts and other entities are expressed by formulas of first order logic. Ordinary model theory can then be used to study what spaces of concepts satisfy various sets of axioms.
We treat primarily what Carnap calls individual concepts like Mike's telephone number or Pegasus and not general concepts like telephone or unicorn. Extension to general concepts seems feasible, but individual concepts provide enough food for thought for the present.
This is a preliminary paper in that we don't give a comprehensive set of axioms for concepts. Instead we merely translate some English sentences into our formalism to give an idea of the possibilities.