Paradoxes and Their Resolutions

A Thematic Compilation by Avi Sion

7. The Russell paradox (early)


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.

4.    The Barber paradox

The Barber paradox[1] may be stated as: ‘If a barber shaves everyone in his town who does not shave himself, does he or does he not shave himself? If he does, he does not; if he does not, he does’.

This double paradox arises through confusion of the expressions ‘does not shave himself’ and ‘is shaved by someone other than himself’.

We can divide the people in any town into three broad groups: (a) people who do not shave themselves, but are shaved by others; (b) people who do not shave themselves, and are not shaved by others; (c) people who shave themselves, and are not shaved by others. The given premise is that our barber shaves all the people who fall in group (a). It is tacitly suggested, but not formally implied, that no one is in group (b), so that no one grows a beard or is not in need of shaving. But, in any case, the premise in fact tells us nothing about group (c).

Next, let us subdivide each of the preceding groups into two subgroups: (i) people who shave others, and (ii) people who do not shave others. It is clear that each of the six resulting combinations is logically acceptable, since who shaves me has no bearing on whom I can shave. Obviously, only group (i) concerns barbers, and our premise may be taken to mean that our barber is the only barber in town.

Now, we can deal with the question posed. Our barber cannot fall in group (a)(i), because he is not shaved by others. He might fall in group (b)(i), if he were allowed to grow a beard or he was hairless; but let us suppose not, for the sake of argument. This still allows him to fall in group (c)(i), meaning that he shaves himself (rather than being shaved by others), though he shaves others too.

Thus, there is no double paradox. The double paradox only arose because we wrongly assumed that ‘he shaves all those who do not shave themselves’ excludes ‘he shaves some (such as himself) who do shave themselves’. But ‘X shaves Y’ does not formally contradict ‘X shaves nonY’; there is no basis for assuming that the copula ‘to shave’ is obvertible, so that ‘X shaves Y’ implies ‘X does not shave nonY’.

If the premise was restated as ‘he shaves all those and only those who do not shave themselves’ (so as to exclude ‘he shaves himself’), we would still have an out by saying ‘he does not shave at all’. If the premise was further expanded and restricted by insisting that ‘he somehow shaves or is shaved’, it would simply be self-contradictory (in the way of a single paradox).

Further embellishments could be made to the above, such as considering people who shave in other towns, or making distinctions between always, sometimes/sometimes-not, and never. But I think the point is made. The lesson learned from the barber ‘paradox’ is that without clear categorizations, equivocations can emerge (such as that between ‘shaves’ and ‘is shaved’), which give the illusion of double paradox.

5.    The Master Catalogue paradox

A class may be viewed as an imaginary envelope, which flexibly wraps around all the class’ purported members, however dispersed in place and time, to the exclusion of all other things. The question arises, can the figurative envelope of the class “classes” wrap itself too, or not?

Reviewing the Russell paradox[2], we must conclude that not all ‘word-objects’ are ‘things’ – measures of things are not themselves to be counted as things. Since classification is an expression of our measurement of things, it cannot itself be counted as a thing. To do so gives rise to a paradox, we should avoid it.

In other words, the problem involved is that the iterative form (“class of classes”) is not identical with the simple form (“class”), except very superficially and verbally – so the former cannot logically be subsumed under the latter. There is a sufficiently significant modification of the subject-predicate relation involved, caused by the iteration of the same term, to exclude the reflex of subsumption. The paradox arising if we do not restrain this impulse is precisely what teaches us to exercise such restraint.

The word ‘things’, note, has many meanings. Sometimes, we intend by it all possible objects of thought. Sometimes, we mean to exclude words from it[3]. Sometimes, we mean to exclude classes; or more narrowly, as just pointed out, classes of classes; ditto, with regard to concepts or to concepts of concepts. Sometimes, the word ‘things’ includes only material objects, whatever their category. Sometimes, we mean by it ‘entities[4]’ (material, mental or spiritual bodies, or delimited substances, individual cases of which are generally subjects of propositions) in contrast to their ‘properties’ (the predicates of place, time, quality, action, quantity, relation, and so forth). Sometimes, in everyday discourse, we refer to ‘things’ in contrast to ‘persons’ – i.e. ‘things’ here means inanimate or non-volitional entities. And there are yet more senses of the word.

Thus, whenever logicians refer to ‘things’, they ought to try and first make clear just what is to be included under that heading.

Incidentally, even worse than ‘self-membership’ as a concept to swallow, is the notion of “classes that seem contradictory to what they include” – the latter seems inconceivable at the outset, at least in verbal appearance! Thus, for instances: “no relationship” is a relationship of sorts; “non-classes” is in a sense a class. There has to be some fallacy involved in such terms, which needs to be clarified. Perhaps the problem is a hyperbole or misnomer?

The answer to this question would be that we are here again dealing with classes of classes, and these need not be outwardly consistent with their member classes. Thus, the class of non-relationships still involves a relationship. The class of non-classes is nonetheless a class. The class of empty or null classes does have members. The class of meaningless or self-contradictory classes is itself neither meaningless nor self-contradictory. And so forth.

Bertrand Russell illustrates his paradox with reference to:

  1. a catalogue of all books that mention themselves, and
  2. a catalogue of all books that do not mention themselves.

Case (a) presents no problem: the catalogue can list itself without contradicting its own definition; whereas, if it does not list itself, it betrays that definition[5]. Case (b), on the other hand, is a problem: if it does not list itself, in accord with its own definition, it thereby becomes eligible for inclusion in itself[6]; but, if it does indeed list itself, it contradicts its own definition. The latter is the double paradox under discussion.

Now, my first objection would be as follows. The catalogue’s title (and even, perhaps, a brief description of its contents, an abstract) could perhaps be listed within the book itself– but such a book would not and cannot include a reproduction of the whole book inside itself (not to mention all the other books it lists or reproduces), for the simple reason that the task would be infinite (a book within a book within a book… etc., or the same in the plural).

The book is therefore not itself a member of itself; strictly speaking, only words about the book are mentionable in it. The terms inclusion or membership, as used here, then, have a very limited meaning. Thus, the plausibility of Russell’s example is very superficial, spurious; he is being fallacious, sophistical, suggesting something impossible.

Moreover, every book “includes itself” in the sense that it consists of whatever contents it has and no more. But if a book is conceived as including a number of other books, defined by some statement (e.g. all English books), the book cannot include itself in the sense that this content is only part of itself. This would not only signify infinite regression (a book with other books plus itself in it, the latter in turn with other books plus itself in it, and so forth), and infinite size, but it would constitute a contradiction within the definition. The book cannot both be all its content and only part of its content.

In this perspective, defining the book as ‘the catalogue of all books that do not include themselves’, the Russell paradox is akin to the liar paradox, since the projected book is an entity that has no finite dimension; it can never be pinned down.

A second objection would be the following. Even if we take Russell’s construct as a mere list of books, defined as ‘the catalogue of all books that do not mention themselves’, the definition is absurd, since it cannot logically be realized. We simply cannot write a book listing all books that do not mention themselves (Conrad’s Lord Jim, Hugo’s Notre Dame, etc.), in view of the stated dilemma, that whether we list or not list the book itself in it we are in a contradiction. Therefore, this concept is of necessity a null-class and meaningless.

Logic has not been stumped by the paradox, but has precisely just been taught that the proposed concept is unsound and unusable; it must therefore simply be dropped or at least changed somewhat. There is nothing dramatic in the paradox; it represents one of the functions of Logic. We might try to propose a modified concept, as follows. Perhaps we should instead refer to a library.

(a) Consider a catalogue of all books in a certain library, which is to be placed in that same library. If the book lists itself, it presents no problem. If the book does not mention itself as being in the library, it is simply incomplete and should be expanded; or its title is incorrect and should be modified (“all books but this one”); or it should be left out of the library.

(b) Now, with regard to a catalogue of all books not in our library: such a book cannot both mention itself and be put in the library. If we want to keep it in our library, we must erase its mention of itself. If we want it to mention itself, we must leave it out of the library. These are practical alternatives, which present no problem.

In this perspective, as we seek a practical expression for it, the Russell paradox becomes more akin to the Barber paradox.

6.    Grelling’s paradox

To develop his paradox[7], Kurt Grelling[8] labels a word ‘homological’, if it has the quality it refers to (e.g. the word “short” is short, or the word “polysyllabic” is polysyllabic), or ‘heterological’, if it lacks the quality it refers to (e.g. “long” lacks length, or again “monosyllabic” is not monosyllabic). He then asks whether these two words, themselves, are to be categorized this way or that, arguing:

  • If “heterological” is homological, then it is heterological (contradictory predicates).
  • If “heterological” is heterological, then it is homological (contradictory predicates).

But it is a misapprehension of the meanings of these words to even try to apply them to themselves. In their case, the references are too abstract to have visible or audible concomitants. Neither term is applicable to either of them.

Note first that the apparent contradictions in predication either way apply to the word “heterological” only. For, using similar reasoning with regard to the word “homological”, although it might seem more consistent to say that “homological” is homological than to say that it is heterological, the sequence of predicates would seem consistent both ways, i.e.:

  • If “homological” is homological, then it is homological (consistent predicates).
  • If “homological” is heterological, then it is heterological (consistent predicates).

This could be taken to suggest that the term homological is somehow better constructed, while the term heterological has a structural fault. But this is not the real issue here.

The real issue is distinguishing between the physical words “homological” and “heterological” and their respective intended meanings, viz. homological and heterological. When we intend a word as such, we traditionally place it in inverted commas; and when we intend its assigned meaning we use it simply. In the above propositions, through which a paradox apparently arises, the subjects are words as such (in inverted commas) and the predicates are the meanings of such words.

In this perspective, there is no basis for the claim that “heterological” is heterological implies “heterological” is homological, or vice versa. The inference is very superficial, because it confuses the word as such (intended as the subject) with the meaning of the word (intended as the predicate). That is, the inverted commas in the subject are not used sincerely, but we secretly intend the underlying meaning as our subject.

How did we draw out the consequents from the antecedents? Could we see at a glance that the first thesis implies the second? Let us look at the hypothetical propositions in question more closely:

If in the antecedent we place the emphasis on the property referred to by the word “heterological”, viz. some presumed quality called heterologicality, we would formulate the paradoxes as follows:

  • If the word “heterological” has the property it refers to (i.e. it is heterological), then it apparently lacks the property it refers to (i.e. is homological).
  • If the word “heterological” lacks the property it refers to (i.e. it is homological), then it apparently has the property it refers to (i.e. is heterological).

If on the other hand, in the antecedent we place the emphasis on the word “heterological” having or lacking the property it refers to, we would instead formulate the paradoxes as follows:

  • If the word “heterological” has the property it refers to (i.e. it is homological), then it apparently lacks the property it refers to (i.e. is heterological).
  • If the word “heterological” lacks the property it refers to (i.e. it is heterological), then it apparently has the property it refers to (i.e. is homological).

In any of these cases, the consequent is constructed by comparing the subject “heterological” to the antecedent predicate heterological or homological; if they are the same word, we ‘infer’ homological as our consequent predicate, while if they verbally differ, we ‘infer’ heterological. But in truth, in making these comparisons between antecedent subject and predicate, we have not spotted any quality in the word “heterological” as such, but have tacitly referred to its underlying meaning, and faced that off against the hypothesized predicate.

In other words, the statement that “heterological” is homological (or for that matter that “homological” is heterological) is not as self-contradictory as it appears at first glance; it could conceivably be consistent. In truth, it is indeterminate and therefore meaningless.

More precisely, to resolve the paradox we have to remember how our terms were induced in the first place. We can tell that “short” is short merely by seeing or hearing the word “short” (supposing that any one syllable, however written or pronounced, counts as short). But in the case of a term like heterological, you cannot tell whether the word has or lacks the property it refers to, because that property is not a concrete (visible or audible) quality of the word, but something abstract that we apply to visible or audible components of words. If the quality sought is not visible or audible, it is unknowable and there is no way for us to tell which predicate applies.

That is, our initial definitions of those terms, which mention “a word having/lacking a certain quality it refers to”, are not clear and precise, because they do not specify as they should that the qualities intended are phenomenal, i.e. perceptible aspects of the word. If the word labels something not included in its physical aspects, the terms homological and heterological simply do not apply. To apply them is to play verbal tricks. Thus, neither of these predicates is applicable to either of these words as such.

It might be objected that words do have non-phenomenal attributes. For example, we often consider a word useful or useless. In such case, we might ask: is the word “useful” useful or not? Yes, I’d reply to that. Therefore, “useful” is homological. Likewise, “useless” is useful, therefore “useless” is heterological. In this perspective, one may doubt the exactitude of what we have just proposed, that homological and heterological are terms that presuppose concrete (rather than abstract) predicates.

But to this objection, one could counter that the utility of a word is ultimately something concrete: a word is useful if it makes a perceptible practical difference in the development of knowledge. In that case, our definition could be modified slightly, specifying that the terms homological or heterological are only applicable when we can first directly or indirectly anchor them to some concrete property.

In sum, these terms must refer to something other than themselves before they can at all be used. The fallacy involved is similar to that in the liar paradox, where the term “this” is used with reference to itself, whereas it only acquires meaning when it has something else to refer to. Such terms are relational, and so cannot refer to other relations in a circular manner or ad infinitum: they need to eventually be anchored to some non-relational term.

Notice, by the way, that if we changed the word “short” to say “shortissimo”, with reference to the same meaning, the word would change status and become heterological, since “shortissimo” is not shortissimo. On the other hand, whatever other word we substitute for the word “heterological”, Grelling’s paradox in relation to it remains apparent. This test shows that in the latter case it is not purely the word that we are thinking of, but rather its underlying meaning. With regard to the word “useful”, we could also say that it is useful by virtue of its content, or at most by virtue of its being a word (a unit of language), and not because of its specific shape or sound.


From Future Logic 32, 42 & 45, and Ruminations 5.


[1]              This paradox was offered by B. Russell (in his 1918-19 work, The Philosophy of Logical Atomism) as an illustration of the Russell paradox; but he did not claim it as his own, saying that it was “suggested” to him by someone else. However, Russell considered that “In this form the contradiction is not very difficult to solve,” because one can simply deny the subject (i.e. say that such a barber does not exist).

[2]              See Future Logic, chapters 43-45, on class logic.

[3]              Though of course, this distinction may be paradoxical, since the word ‘word’ refers to words.

[4]              The word ‘entity’, of course, is sometimes meant more generally, with reference to any existent.

[5]              That is, the catalogue is not eligible for inclusion in itself – but that does not affect its exhaustiveness.

[6]              So that, if it is not forthwith included in itself, it can no longer be claimed complete.

[7]              This paradox was inspired by Russell’s paradox.

[8]              Germany, 1886-1942. The paradox is also called the Grelling-Nelson paradox, because it was presented in a 1908 paper written jointly with Leonard Nelson (Germany, 1882-1927).

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