This is a possibly foolish thought… But I’m a fool, and I can’t help but want to explore it.
I suspect that Russell’s paradox would not arise as a problem for the Medievals, since their philosophy of language implied rules about sentences involving subjects and predicates and rules about having to relate them properly. Let’s take his example:
There is a set of all sets which are not members of themselves.
Here, the set of all sets which are not members of themselves, since it is not a member of itself, should be included in itself. This causes an infinite ‘regress’ if we admit that we can simply include it in the set of all sets which are not members of themselves. What are we to do with such a paradox?
Consider the sentence carefully, and one recognizes that the Subject is the set in: “there is a set of all sets which are not members of themselves.” I submit that perhaps the predicate has to differ from the subject for the sentence to express a proposition. This may seem odd at first blush. Consider, for instance, the sentence “A is A” or “the Dog is the Dog”. Such sentences seem intelligible, coherent and make sense to us superficially, but I think deeper examination will bring us to recognize that such sentences are in any case not propositions; they do not propose anything about the world (at least if we assume that the same nouns are being used in an equivocal sense).
Therefore, in constructing the sentence which seems to be a paradox, I think we (implicitly) ought to mean:
the set of all other sets which are not members of themselves
or, to put it differently:
the set of all sets (other than itself) which are not members of themselves.
The trouble here is that the one set (the Subject) does not seem to be identical to the predicate (all sets which are not members of themselves). However, on deeper analysis one can recognize that what one expresses in the subject is just equivalent to the predicate. Thus, what one was intending to express, was that there is some ‘set’ which contains all (other) ‘sets which are not members of themselves’, and the paradox ensued when one realized that the subject and predicate were identical because the differentiation (‘other’) was not made explicit. Thus, for any proposition, the subject and predicate must differ, or else what is ‘sententially’ expressed is not a proposition. What we should have meant was:
The set of all sets other than which are not members of themselves.
It may seem ironic to accuse the late Mr. Russell of misusing language, but it seems to me that my suggestion will not even be found unacceptable to evolutionary psychologists and cognitive scientists, any less than to the Medieval masters. For, to take an evolutionary view of human language one might think that predication is a cognitive development ‘engineered’ through natural selection to allow us to share/propose information, such that for any subject, we can relate to it a predicate, such that this process conveys new information to others. This, at least, provides an account of the indicative mood. Whatever account of Language we have, it seems that the only way to avoid admitting that such paradoxes are logically indissoluble, is to shore up our ‘Logic’ appropriately (at least as it applies to propositions and predications). Thus, to think that the sentence “there is a set of all sets which are not members of themselves” – if we do not mean by the subject something which differs from the predicate – is a proposition is merely confused. Language, it seems, is still on holiday… Perhaps it will return when we reclaim the legitimate insights of the Medievals, with their philosophy of language, and combine those insights into a synthesis with modern insights.
This post continues the train of thought on which I embarked here.