berke@CS.UCLA.EDU (09/09/87)
(Note: I am using the form Title (Author) in the subject line of my messages. If you follow up on this, please change the name in parentheses to your own.) Personally, I find M&M's indescribably delicious. I must admit this as a concept that cannot be expressed in words, or face some contradiction in terms. J.S. Mill, in "A System of logic, Ratiocinative and Inductive," (1840, 1865, Sec. 5, p.23) gives another example: "A child knows who are its brothers and sisters, long before it has any definite conception of the nature of the facts which are involved in the signification of those words." Another example is a crying baby that can only be comforted by its mother. It must be admitted that the baby knows its mother though it has not learned to speak. If I speak of concepts as units of knowledge, I must allow that the baby has some concept of its mother. Perhaps this is not the best way to speak, of "units of knowledge," but so far, it is the only formal (well-defined) way we have of speaking. It may be observed that I used words to speak about inexpressible concepts. This may appear to be confused or contradictory. Confusion may be avoided by distinguishing between using words to express a concept and using words to name a concept. Naming (denoting) a concept is similar to naming (denoting) a real object such as the terminal you are sitting at now. Expressing a concept (putting it into words) is different than naming (denoting) it. Church has shown the necessity for abstract objects if one adopts a Fregean theory of meaning. The abstract objects, whether called 'concepts' or 'intensionalities', are required by the assumption that names name (denote) things. When we use a word, we usually (purport to) denote an object and express a concept. There are problems with this. The main one is called the 'paradox of the name relation by Church', the 'antinomy of the name relation' by Carnap. It was discovered by Frege when he asked "How can A=B, if true, differ in content from A=A?" It is commonly thought that Russell's theory of descriptions solves this paradox, but it does not. Russell's theory of meaning requires intensionalities as does Frege's. Husserl and Korzybski are among the many who have proposed other explanations of language. If you compare their writings, you may notice that Husserl's and Korzybski's are all words, while Frege's, Russell's, and Church's have math. Frege invented a formal, symbolic, language in which he could prove and disprove things and not just debate them in words. Russell too insisted on the formal expression of logic, and wholly adopted Frege's language. Russell linearized it - Frege's was two-dimensional. Russell added ramified type theory to avoid certain paradoxes. Frege thought up the idea of a variable as a place-holder, and invented quantification to explain our way of speaking, e.g., "Let x be a number greater than 5," or "consider all the blue things in your office." Previously there was a lot of debate about "arbitrary objects" and such. The syntactic approach to language invented by Frege quickly swept aside all other views of the matter. There is very little else that everyone has agreed on ever since Russell explained Frege's system to the world and used it in Principia Mathematica. Though there has been some recent work trying to return to a less syntactic view of language, it has not been formalized, so much of it remains just opinion. No other system has been formalized to get around the problems with Frege's logic. Husserl and Korzybski among them. There are, of course, many writings including these which help us think clearly about words, they are just not useful in the same sense as Frege's. This is not an insult to the other work, just testament to Frege's view of language. It enabled Goedel to do his work, and Turing to invent computers. I call that useful. As explained above, once you admit concepts as abstract entities, the problem is not the existence of inexpressible concepts. Not all objects can be named (denoted) in a given language. E.g., the real numbers are more numerous than names for them. So we admit objects without individual names. We also admit names without individual denotations, e.g., 'Pegasus', 'The present king of France'. Concepts, abstract entities, are just a third kind of entity required by the supposition of the first two: names and things they name. Just as we can have names without objects and objects without names, we can have concepts that cannot be expressed in words, and concepts that do not apply to any objects. The real problem comes in (1) formalizing the relationships among names, concepts, and objects which Church has tried to do; or (2) providing some alternative, non-Fregean formalization, which neither Husserl or Korzybski provided; or (3) showing that these relationships are not formalizable, that is, not expressible in words, which Wittgenstein hinted at in his poetic way, but did not prove. The third option carries with it the onus of explaining how the illusions of words and objects (and thus concepts) arise, since they do arise, or we wouldn't "be talking about" "them." I am preparing an article putting forth a theory of a "semantic gradient" to explain how physical stimuli acquire what we tend to call meaning. If you are interested in the above arguments of Mill, Frege, and Church, (and how, ignoring them, we have vastly over simplified knowledge representation, concepts, etc.) I can send you a copy of a draft of my article "Naming and Knowledge: Implications of Church's Arguments about Knowledge Representation," which I have submitted for publication. It covers this topic of names, objects, and concepts as thouroughly as I can. I would like to thank the many people who responded to my notes "An Unsearchable Problem of Elementary Human Behavior," and "How to measure learning Ability?" As soon as I get a moment, I'll post a summary or mere agglutination of the articulate replies I've received. Respectfully, Peter Berke