Let us put fuzzy sets to one side for the moment and take up the matter of belief you raised, as I believe that connection is very apt.
As it happens, there was a moment in the history of philosophy when a thinker of note was driven almost to the point of despair in his trials to analyze situations of belief by means of cobbled together dyadic relations, and I truly believe that had he persisted just a little while longer he might’ve broken through the mental blockade of hidebound habits to try triadic relations instead. But it would not be, partly because he was discouraged in his efforts by the criticism of another thinker of note whose opinion, I believe, he may have trusted too much.
At any rate, let us enter the fray in medias res at the point where Bertrand Russell asks the following question:
I want to try to get an account of the way that a belief is made up. That is not an easy question at all. You cannot make what I should call a map-in-space of a belief. You can make a map of an atomic fact but not of a belief, for the simple reason that space-relations always are of the atomic sort or complications of the atomic sort. I will try to illustrate what I mean.
The point is in connexion with there being two verbs in the judgment and with the fact that both verbs have got to occur as verbs, because if a thing is a verb it cannot occur otherwise than as a verb.
Suppose I take ‘A believes that B loves C’. ‘Othello believes that Desdemona loves Cassio’. There you have a false belief. You have this odd state of affairs that the verb ‘loves’ occurs in that proposition and seems to occur as relating Desdemona to Cassio whereas in fact it does not do so, but yet it does occur as a verb, it does occur in the sort of way that a verb should do.
I mean that when A believes that B loves C, you have to have a verb in the place where ‘loves’ occurs. You cannot put a substantive in its place. Therefore it is clear that the subordinate verb (i.e. the verb other than believing) is functioning as a verb, and seems to be relating two terms, but as a matter of fact does not when a judgment happens to be false. That is what constitutes the puzzle about the nature of belief.
You will notice that whenever one gets to really close quarters with the theory of error one has the puzzle of how to deal with error without assuming the existence of the non-existent.
I mean that every theory of error sooner or later wrecks itself by assuming the existence of the non-existent. As when I say ‘Desdemona loves Cassio’, it seems as if you have a non-existent love between Desdemona and Cassio, but that is just as wrong as a non-existent unicorn. So you have to explain the whole theory of judgment in some other way.
I come now to this question of a map. Suppose you try such a map as this:
Othello | | believes | v Desdemona -----------> Cassio loves
This question of making a map is not so strange as you might suppose because it is part of the whole theory of symbolism. It is important to realize where and how a symbolism of that sort would be wrong:
Where and how it is wrong is that in the symbol you have this relationship relating these two things and in the fact it doesn’t really relate them. You cannot get in space any occurrence which is logically of the same form as belief.
When I say ‘logically of the same form’ I mean that one can be obtained from the other by replacing the constituents of the one by the new terms.
If I say ‘Desdemona loves Cassio’ that is of the same form as ‘A is to the right of B’.
Those are of the same form, and I say that nothing that occurs in space is of the same form as belief.
I have got on here to a new sort of thing, a new beast for our zoo, not another member of our former species but a new species.
The discovery of this fact is due to Mr. Wittgenstein.
(Russell, POLA, pp. 89–91).
And just by way of picking up the res into the middle of which we’ve jumped:
- Bertrand Russell, “The Philosophy of Logical Atomism”, pp. 35–155 in The Philosophy of Logical Atomism, edited with an introduction by David Pears, Open Court, La Salle, IL, 1985. First published 1918.