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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 45. ILLICIT PROCESSES IN CLASS LOGIC.
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.
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.
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. |
