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© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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1. Self-membership.

2. The Russell Paradox.

3. Impermutability.

1. Self-membership.

With regard to the issue of self-membership, more needs to be said. Intuitively, to me at least, the suggestion that something can be both container and contained is hard to swallow.

Now, self-membership signifies that a nominal is a member of an exactly identical nominal. Thus, that all X are X, and therefore members of “X”, does not constitute self-membership; this is merely the definition of membership in a first order class by a non-class.

We saw that, empirically, at least with ordinary examples, “X” (or the class of X) is never itself an X, nor therefore a member of “X”. For example, “dogs” is not a dog, nor therefore a member of “dogs”.

I suggested that this could be generalized into an inductive postulate, if no examples to the contrary were forthcoming. My purpose here is to show that all apparent cases of self-membership are illusory, due only to imprecision of language.

That “X” is an X-class, and so a member of “X-classes”, is not self-membership in a literal sense, but is merely the definition of membership in a second order class by a first order class. For example, “dogs” is a class of dogs, or a member of “classes of dogs”, or member of the class of classes of dogs.

Nor does the formal inference, from all X are X, that all X-classes are X-classes, and so members of “X-classes” (or the class of classes of X), give us an instance of what we strictly mean by self-membership; it is just tautology. For example, all dog-classes are members of “classes of dogs”.

Claiming that an X-class may be X, and therefore a member of “X”, is simply a wider statement than claiming that “X” may be X, and not only seems equally silly and without empirical ground, but would in any case not formally constitute self-membership. For example, claiming “retrievers” is a dog.

As for saying of any X that it is “X”, rather than a member of “X”; or saying that it is some other X-class, and therefore a member of “X-classes” — such statements simply do not seem to be in accord with the intents of the definitions of classes and classes of classes, and in any case are not self-membership.

The question then arises, is “X-classes” itself a member of “X-classes”? The answer is, no, even here there is no self-membership. The impression that “X-classes” might be a member of itself is due to the fact that it concerns X, albeit less directly so than “X” does. For example, dog-classes refers to “retrievers”, “terriers”, and even “dogs”; and thus, though only indirectly, concerns dogs.

However, more formally, “X-classes” does not satisfy the defining condition for being a member of “X-classes”, which would be that ‘all X-classes are X’ — just as: “X” is a member of “X-classes”, is founded on ‘all X are X’. As will now be shown, this means that the above impression cannot be upheld as a formal generality, but only at best as a contingent truth in some cases; as a result, all its force and credibility disappears.

If we say that for any and every X, all X-classes are X, we imply that for all X, “X” (which is one X-class) is X; but we have already adduced empirical cases to the contrary; so the connection cannot be general and formal. Thus, we can only claim that perhaps for some X, all X-classes are X; but with regard to that eventuality, no examples have been adduced.

Since we have no solid grounds (specific examples) for assuming that “X” or “X-classes” is ever a member of itself, and the suggestion is fraught with difficulty; and we only found credible examples where they were not members of themselves — we are justified in presuming, by generalization, that: no class of anything, or class of classes of anything, is ever a member of itself.

I can only think of one possible exception to this postulate, namely: “things” (or “things-classes”). But I suspect that, in this case, rather than saying that the class is a member of itself, we should regard the definition of membership as failing. That is, though this summum genus is a thing, it is not ‘a member of’ anything.

2. The Russell Paradox.

The Russell Paradox is modern example of double paradox, discovered by British logician Bertrand Russell.

He asked whether the class of “all classes which are not members of themselves” is or not a member of itself. If “classes not members of themselves” is not a member of “classes not members of themselves”, then it is indeed a member of “classes not members of themselves”; and if “classes not members of themselves” is a member of “classes not members of themselves”, then it is also a member of “classes which are members of themselves”. Thus, we face a contradiction either way.

In contrast, the class of “all classes which are members of themselves” does not yield a similar difficulty. If “self-member classes” is not a member of “self-member classes”, then it is a member of “classes not members of themselves”; but if “self-member classes” is a member of “self-member classes”, no antinomy follows. Hence, here we have a single paradox coupled with a consistent position, and a definite conclusion can be drawn: “self-member classes” is a member of itself.

Now, every absurdity which arises in knowledge should be regarded as an opportunity for advancement, a spur to research and discovery of some previously unknown detail. So what is the hidden lesson of this puzzle?

As I will show, the Russell Paradox proceeds essentially from an equivocation; it is more akin to the sophism of the Barber paradox, than to that of the Liar paradox. For whether self-membership is possible or not, is not the issue. Russell believed that some classes, like “classes” include themselves; though I disagree with that, my disagreement is not my basis for dissolving the Russell paradox. For it is not the concept of self-membership which results in a two-way inconsistency. It is the concept of non-self-membership which does so; and everyone agrees that at least some (if not all, as I believe) classes do not include themselves: for instance, “dogs” is not a dog.

What has stumped so many logicians with regard to the Russell paradox, was the assumption that we can form concepts at will, if we but formulate a verbal definition. But this viewpoint is without justification. The words must have a demonstrable meaning; in most cases, they do; but in some cases, they are isolated or pieced together without attention to their intrinsic structural requirements. We cannot, for instance, use the word ‘greater’ without specifying ‘than what?’; many words are attached, and cannot be reshuffled at random. The fact that we commonly, in everyday discourse, use words loosely, to avoid boring constructions, does not give logicians the same license.

3. Impermutability.

The solution to the problem is so easy, it is funny, though I must admit I was quite perplexed for a while. It is simply that: propositions of the form ‘X (or “X”) is (or is not) a member of “Y” (or “Y-classes”)’ cannot be permuted. The process of permutation is applicable to some forms, but not to all forms.

a. In some cases, where we are dealing with relatively simple relations, the relation can be attached to the original predicate, to make up a new predicate, in an ‘S is P’ form of proposition, in which ‘is’ has a strictly classificatory meaning. Thus, ‘X is-not Y’ is permutable to ‘X is nonY’, or ‘X is something which is not Y’; ‘X has (or lacks) Y-ness’ is permutable to ‘X is a Y-ness having (or lacking) thing’; ‘X does (or does not do) Y’ is permutable to ‘X is a Y-doing (or Y-not-doing) thing’. In such cases, no error arises from this artifice.

But in other cases, permutation is not feasible, because it falsifies the logical properties of the relation involved. We saw clear and indubitable examples of this in the study of modalities.

For instance, the form ‘X can be Y’ is not permutable to ‘X is something capable of being Y’, for the reason that we thereby change the subject of the relation ‘can be’ from ‘X’ to ‘something’, and also we change a potential ‘can be’ into an actual ‘is (capable of being)’. As a result of such verbal shenanigans, formal errors arise. Thus, ‘X is Y, and all Y are capable of being Z’ is thought to conclude ‘X is capable of being Z’, whereas in fact the premises are quite compatible with the contradictory ‘X cannot be Z’, since ‘X can become Z’ is a valid alternative conclusion, as we saw earlier.

It can likewise be demonstrated that ‘X can become Y’ is not permutable to ‘X is something which can become Y’, because then the syllogism ‘X is Y, all Y are things which can become Z, therefore X is something which can become Z’ would seem valid, whereas its correct conclusion is ‘X can be or become Z’, as earlier seen. Thus, modality is one kind of relational factor which is not permutable. Even though we commonly say ‘X is capable or incapable of Y’, that ‘is’ does not have the same logical properties as the ‘is’ in a normal ‘S is P’ proposition.

b. The Russell Paradox reveals to us the valuable information that the copula ‘is a member (or not a member) of’ is likewise not open to permutation to ‘is something which is a member (or not a member) of’.

The original ‘is’ is an integral part of the relation, and does not have the same meaning as a solitary ‘is’. The relation ‘is or is not a member of’ is an indivisible whole; you cannot just cut it off where you please. The fact that it consists of a string of words, instead of a single word, is an accident of language; just because you can separate its verbal constituents does not mean that the objective relation itself can similarly be split up.

Permutation is a process we use, when possible, to bypass the difficulties inherent in a special relation; in this case, however, we cannot get around the peculiar demands of the membership relations by this artifice. The Russell paradox locks us into the inferential processes previously outlined; it tells us that there are no other legitimate ones, it forbids conceptual short-cuts.

The impermutability of ‘is (or is not) a member of’ signifies that you cannot form a class of ‘self-member classes’ or a class of ‘non-self-member classes’. These are not terms, they are relations. Thus, the Russell paradox is fully dissolved by denying the conceptual legitimacy of its terms. There is no way for us to form such concepts; they involve an illicit permutation. The connections between the terms are therefore purely verbal and illusory.

The definition of membership is ‘if something is X, then it is a member of “X”‘ or ‘if all X are Y, then “X” is a member of “Y-classes”‘. The Russell paradox makes us aware that the ‘is’ in the condition has to be a normal, solitary ‘is’, it cannot be an ‘is’ isolated from a string of words like ‘is (or is not) a member of’. If this antecedent condition is not met, the consequent rule cannot be applied. In our case, the condition is not met, and so the rule does not apply.

c. Here, then, is how the Russell paradox formally arises, step by step. We will signal permutations by brackets like this: {}.

Let “X” signify any class, of any order:

(i) If “X” is a member of “X”, then “X” is {a member of itself}. Call the enclosed portion Y; then “X” is Y, defines self-membership.

(ii) If “X” is not a member of “X”, then “X” is {not a member of itself}. Call the enclosed portion nonY; then “X” is nonY, defines non-self-membership.

Next, apply the general definitions of membership and non-membership to the concepts of Y and nonY we just formed:

(iii) whatever is not Y, is nonY, and so is a member of “nonY”.

(iv) whatever is Y, is not a member of “nonY”, since only things which are nonY, are members of “nonY”.

Now, the double paradox:

(v) if “nonY” is not a member of “nonY”:

— then, by putting “nonY” in place of “X” in (ii), “nonY” is {not a member of itself}, which means it is nonY;

— then, by (iii), “nonY” is a member of “nonY”, which contradicts the starting premise.

(vi) if “nonY” is a member of “nonY”:

— then, by putting “nonY” in place of “X” in (i), “nonY” is {a member of itself}, which means it is Y;

— then, by (iv), “nonY” is not a member of “nonY”, which contradicts the starting premise.

Of all the processes used in developing these arguments, only one is of uncertain (unestablished) validity: namely, permutation of ‘is a member of itself’ to ‘is {a member of itself}’, or of ‘is not a member of itself’ to ‘is {not a member of itself}’. Since all the other processes are valid, the source of antinomy has to be such permutation. Q.E.D.

d. The existence of impermutable relations suggests that we cannot regard all relations as somehow residing within the things related, as an indwelling component of their identities. We are pushed to regard some relations, like modality or membership, as bonds standing outside the terms, which are not actual parts of their being.

Thus, for example, that ‘this S can be P’ does not have an ontological implication that there is some actual ‘mark’ programmed in the actual identity of this S, which records that it ‘can be P’. For this reason, the verbal clause {can be P} cannot be presumed to be a unit; there is nothing corresponding to it in the actuality of this S, the potential relation does not cast an actual shadow.

Thus, there must be a reality to ‘potential existence’, outside of ‘actual existence’. When we say that ‘this S can be P’, we consider this potentiality to be P as somehow part of the ‘nature’ of this S. But the S we mean, itself stretches in time, past, ‘present’, and future; it also has ‘potential’ existence, and is wider than the actual S.

The same can be argued for can not, or must or cannot. Thus, natural (and likewise temporal) modalities refer to different degrees, or levels, of existence.

Similarly, the impermutability of membership relations, signifies that they stand external to their terms, leaving no mark on them, even when actual.

It seems like a reasonable position, because if every relation of something to everything else, implied some corresponding trait inside that thing, then each thing in the world would have to contain an infinite number of messages, one message for its relations to each other thing. Much simpler, is to regard relations (at least, those which are impermutable) as having a separate existence from their terms, as other contents of the universe.

See also, regarding Russell’s Paradox: Ruminations, chapter 5.7-8, and A Fortiori Logic, appendix 7.5.