The Heterological Paradox
Few things have had a more stimulating effect on the development of modern logic than the discovery of the antinomies of set theory round the turn of the century and the subsequent discussion of these and other paradoxes. The idea that there is a »solution» to the problem of the antinomies such that, the solution having been found, the paradoxes would cease to puzzle logicians, may have been natural to entertain in the days when Russell put logic and Zermelo set theory on a new basis. To-day it appears wiser to think that the antinomies will remain a permanent topic of discussion to receive fresh attention over and over again by logicians. Rather than speaking of the efforts of the logicians as of so many proposed »solutions» to a problem, we should think of them as different, but not necessarily mutually exclusive or even mutually competing, ways of treating a subject-matter.
In this paper one of the antinomies, viz. the one known as the Heterological or Grelling Paradox, is selected for treatment along lines which, as far as the Author knows, are somewhat novel. The treatment to which this one Paradox is subjected is meant to be an illustration of a more general method which can be applied to other antinomies as well. Some remarks about the general significance of the method are made in the five Appendices (pp. 21–28) of the present essay.
1. The Grelling or Heterological Paradox is one of the so-called semantic antinomies. It is called »semantic», because it concerns the relation of linguistic expressions to their meanings. The linguistic expressions concerned in this particular antinomy are words, and the meanings are properties.
It may be asked, whether a property should be called the meaning or rather the reference (denotation, »bearer») of the (word which is a) name of this property. I do not believe, however, that the distinction between meaning and reference, important as it is in itself, is relevant to our discussion of the paradox.
The concept of a »word» gives rise to interesting problems. One may distinguish between the word and the word-sign (»icon», »picture»), and
say that a word is a word-sign in use (or associated with a meaning). One also distinguishes between the word(-sign) as a type and as a token (of a type). — I do not think that these distinctions are relevant to our discussion of the paradox. I shall here regard the notion of a word as being unproblematic. (See, however, remarks in Appendix I.)
Words which are names of properties are, for example, »red» or »new» or »animal».
It is clear that words may not only name but also have properties. (I assume that a word can be regarded as a kind of thing. For comments on this assumption see Appendix I.) A property of a given word can be, for example, that it contains so and so many letters or syllables, that it is long or short, that it occurs so and so many times in a certain context, that it is a proper name or a noun or a verb, etc.
Since words have properties and some words are names of properties, it may happen that a word has exactly that property of which it is a name.
Examples are easily found. Thus the word »pentasyllabic» is pentasyllabic (fivesyllabic). The word »short» is a short word, the word »old» is an old word in the English language, the word »used» is used, »noun» is a noun, etc.
A word which has a property of which it is itself a name is called autological. A word which is not autological is called heterological. It is clear that a vast majority of words, even of those which are names of properties, are heterological.
(Be it observed in passing that one and the same word may be a name of several properties; we then call it autological, if it has at least one of the properties which it names.)
2. Before we proceed to constructing the antinomy, we ought to make the two notions »autological» and »heterological» more precise. We propose the following definition:
x is autological, if and only if, x has a property of which x is a name.
The definition contains a variable, »x». What does this variable »stand for», what is its range (range of significance)? The range of the variable might also be called the Universe of Discourse of the definition.
A most general answer to our question is that »x» stands for any thing (i.e. for anything to which some property may be significantly attributed). The definition then says, that a thing is autological, if – – –.
I think that this is a quite legitimate answer to our question, and that it is not impaired by the generality and vagueness of the concept
of a thing, nor by the difficulties which may be caused by the idea of a totality of all things.
One can, however, also narrow the Universe of Discourse of the definition. One can limit it to words, or even to names of properties only. Then the definition says, on which conditions a word is autological, or on which conditions a name of a property is autological.
If the Universe of Discourse of the definition is the totality of things, then in order to decide, whether a given thing is autological or not, we ought to make up our minds on the following three questions:
1. Is x a name of a property? If the answer is no, we decide that x is not autological. If the answer is yes, we proceed to question
2. Of which property (properties) is x a name? When this has been settled, we have still to answer question
3. Has x this property (at least one of these properties)? If and only if the answer to this last question is yes, is x autological.
It may be thought that question 1., whether x is the name of a property, can be settled in the affirmative only by mentioning some property of which x is a name. I shall accept this. Thus an affirmative answer to question 1. automatically answers, at least partially, question 2. as well.
If the Universe of Discourse of the definition is the totality of words, then, when the definition is being applied to a given x, it is already assumed that this x is a word. And if the Universe of Discourse is the totality of names of properties, then, when the definition is being applied to a given x, it is already assumed that question 1. above can be answered in the affirmative for this x. Since, in discussing the paradox, anything which has been assumed in constructing it may afterwards be questioned, it is practical to conceive of the Universe of Discourse in a manner which makes a minimum of assumptions and which does not beg any of the questions 1.–3. above.
We proceed to the concept »heterological». Its definition is as follows:
x is heterological, if and only if, it is not the case that x is autological.
Substituting, in this definition, for »autological» its defining expression, we get:
x is heterological, if and only if, it is not the case that x has a property, of which x is a name.
Since »autological» and »heterological» are interdefinable, we could dispense with one of the terms (and yet have the paradox). We need not introduce the word »heterological» at all, but could simply say »not auto-
logical» in its place. But we could equally well dispense with the word »autological» in favour of the two word phrase »not heterological».
It might be observed in passing that the extension of the term »autological» is not affected by the choice between the alternative conceptions, which we mentioned, of the range of the variable. But the extension of the term »heterological» is affected by the choice.
If the Universe of Discourse of the definition is restricted to names of properties, then, for example, the word »red» is heterological, but the word »Napoleon» is neither autological nor heterological, — since it is not the name of any property at all, but of a man. If the universe is restricted to words, the word »Napoleon» is heterological, but the emperor Napoleon is neither autological nor heterological. If, however, the universe comprises the totality of things, we shall have to say of the emperor too that he is heterological, — since he is not a word, but a man.
It may be suggested that the most natural way of using the term »heterological» is for names of properties which have not got the property they name. This would speak in favour of restricting the Universe of Discourse of our definitions of »autological» and »heterological» to names of properties. This question of naturalness, however, is of no importance to the discussion of the paradox.
3. In the course of our considerations so far, we have introduced two new words into our language, the words »autological» and »heterological».
The two new words, it would seem, are names of properties. The word »autological» names the property which a thing has, if and only if, it a) is a name of a property and b) has a property of which it is a name. The word »heterological» again names the property which a thing has, if and only if, it a) is not a name of any property or b) is a name of some property but has not got any property of which it is a name.
We could also say that the word »autological» names the property of autologicality, and the word »heterological» the property of heterologicality.
To say that »autological» and »heterological» are names of properties is, however, to make a very consequential commitment. In view of what will happen later, we may have to question our willingness to make this commitment.
4. I now proceed to the construction of the antinomy. The paradox originates from an application of the definition either of the concept
»autological» or of the concept »heterological» to the word »heterological». (No paradox, however, springs from an application of either concept to the word »autological».)
We ask: Is the word »heterological» autological or heterological? Let the thesis that the word is heterological be presented for consideration. Is this thesis true or false?
When a definition containing a free variable is being applied to an individual thing in the range of the variable, we substitute for the variable in the definition a name of this thing in the range. What is substituted for the variable is thus a name of a thing, — not the thing itself. This is obvious enough, if the things under consideration are some extra-lingual entities such as, say, men or plants or numbers. It is perhaps less obvious, if the things are linguistic entities, such as names or words or sentences. If the thing, to which the definition is to be applied, is a name, then what has to be substituted for the variable is the name of a name.
Thus when the definition of the concept »heterological» is applied to the word »heterological», we substitute for »x» in the definition a name for this word. The customary way of naming words is by the use of quotes. I shall use this customary way here too. And although the quotationmark method of naming words is not free from problems, I shall not for a moment doubt that its use in the reasoning which takes us to the paradox is perfectly sound and uncontroversial.
The application of the definition of »heterological» to the word »heterological» gives us in the first place:
»heterological» is heterological, if and only if, it is not the case that »heterological» is autological.
Substituting for the phrase »'heterological' is autological» its definiens, we get:
»heterological» is heterological, if and only if, it is not the case that »heterological» has a property, of which »heterological» is a name.
Now in order to decide, whether the thesis that »heterological» is heterological is true or not, we must answer the above questions 1.–3., when for »x» we substitute »'heterological'». The questions are:
1'. Is »heterological» a name of some property?
2'. Of which property (properties) is »heterological» a name?
3'. Has »heterological» any property of which it is a name?
As regards questions 1'. and 2'. we tentatively gave the answer that »heterological» names the property of heterologicality, viz. the property which a thing has on condition that it is not autological.
Accepting this, the truth of the thesis that »heterological» is heterological hinges on question 3'., whether »heterological» has or has not this property which it is thought to name. To say that »heterological» is heterological is tantamount to saying that it is not the case that »heterological» has the property which it names. But to say that »heterological» has the property which it names is tantamount to saying that »heterological» is heterological. The truth-condition of our thesis therefore becomes:
»heterological» is heterological, if and only if, it is not the case that »heterological» is heterological.
This truth-condition is of the general form: p, if and only if, it is not the case that p. (In the symbolism of the propositional calculus: p ↔ ~p.) According to »ordinary» logic this is a contradiction. For, if p is or is not true, then to say that p is true, if and only if p is not true is tantamount to saying that p is both true and not true. I shall not for a moment doubt the contradictory character of the form »p, if and only if, it is not the case that p».
By the Grelling or the Heterological Paradox I shall understand the finding that the truth-condition of the thesis that »heterological» is heterological is a contradiction.
*
In derivations and discussions of the Grelling paradox it is assumed as a matter of course that »heterological» is not the name of any other property beside (possibly) the property of heterologicality.
It is, however, trivially the case that the word »heterological» could be the name of just any property. The word could, for example, also mean »pentasyllabic».
Let us therefore assume, for the sake of argument, that the word »heterological» already occurred in our language prior to its introduction by the definition which we have given to it here. Then it will either be the case that it names some property (other than heterologicality) which it also possesses, or it will be the case that it does not name any such property.
In the first case we shall have to say that the word »heterological» is autological. (The word »heterological» is autological if, for example, it also means »hexasyllabic».) And in this case there will be no paradox at all. For, the truth-condition of the thesis that »heterological» is hetero-
logical cannot now be stated in the form of an »if and only if» proposition. We should have to weaken the bi-conditional to a (simple) conditional and say that
if »heterological» is heterological, then it is not the case that »heterological» is heterological.
This is of the general form »p → ~p» which again is equivalent to ~p alone. »~p» here stands for »it is not the case that 'heterological' is heterological», and this is the same as »'heterological' is autological».
Only in the second case is there a paradox. For then the only ground on which the word »heterological» could be declared autological is that it happened to possess the property of heterologicality. The truth-condition of the thesis that the word »heterological» is heterological could be stated in the form of a bi-conditional, — and thus become a contradiction.
As is shown by these considerations, it is not essential to the Grelling paradox that the words »autological» and »heterological» (or any other words) should be univocal. What is essential is only that the truth-condition of the thesis that »heterological» is heterological can be stated in the form of a bi-conditional. The assumption of univocality warrants this possibility and thus the occurrence of the paradox.
5. We must now stop to consider exactly what we have done, when we produced the antinomy.
When an antinomy is described in the usual, carefree manner, which one may find even in respectable works on logic, it easily looks as though we in an antinomy, quasi, prove a thesis and its contradictory, i.e. prove a contradiction. One says, for example:
The word »heterological» must surely be either autological or heterological (i.e. not autological). Now assume that it is autological, i.e. that it has the property which it names. Then the word »heterological» is heterological. But if it is heterological it has the property which it names, viz. the property of heterologicality. And if it has the property which it names, it is by definition autological. Thus if »heterological» is autological, it is heterological, — and if it is heterological, it is autological.
But this is a superficial and gravely misleading way of presenting the paradox. It takes for granted the very things which a true presentation of the antinomy will force us to question.
Under no circumstances must we let ourselves be induced into saying that we have proved a contradiction. For a contradiction is something
which by its very nature cannot be proved. The meaning of »proof», one might also say, excludes this.
If, instead of saying that we have proved a contradiction we say that we have derived one, this would be a legitimate way of speaking. But then the meaning of »derivation» stands in need of clarification. I here propose to clarify it in the following manner:
When we say that in the Grelling antinomy a contradiction is being derived what we mean is that we have proved that from certain premisses a contradiction logically follows.
Thus in producing the antinomy, something has been proved. But that which has been proved is not a contradiction, but some true proposition. This true proposition has the form of a conditional, and can be stated as follows:
If »heterological» names a property which a thing has, if and only if, it is not autological, then »heterological» is heterological, if and only if, it is not heterological.
The conditional proposition, which we have proved, is of the general form »If p, then (q if and only if not q)». Or, in the symbolism of the propositional calculus: p → (q ↔ ~q).
The consequent of the proved conditional proposition is, as was already noted, a contradiction. And a contradiction is a logically false proposition. »q, if and only if, not q» is false for any value of »q». Thus also for the particular value »q» = »'heterological' is heterological». The proposition that »heterological» is heterological, if and only if, »heterological» is not heterological is false for general logical reasons, the soundness of which no »paradox» must make us doubt.
Since »q, if and only if not q» is false for any value of q, its negation (contradictory) »not: q, if and only if, not q» will be true for any value of q and thus also for the particular value »'heterological' is heterological».
But from the truth of a conditional proposition and the falsehood of its consequent, we may conclude to the falsehood of its antecedent. From »If p, then (q if and only if not q)» and »not: q, if and only if, not q» logically follows »not p». This is true for any values of »p» and »q» and thus also for the particular values which they have as parts of the conditional proposition which we said we had proved. Thus we are entitled to say that we have disproved the proposition that »heterological» names a property which a thing has, if and only if, it is not autological. And this again means that we have proved the proposition that
it is not the case that »heterological» names a property which a thing has, if and only if, it is not autological.
This is what is ultimately established or proved by the reasoning process which took us to the Grelling antinomy. Let us now inspect this »result» more closely.
6. The proposition, which we reached, says that the word »heterological» does not name a certain property. But how can this be a provable (logical) truth? The connexion between a word and its meaning is established by arbitrary convention. No grounds of logic could prevent the word »heterological» from being the name of any property.
The answer to this puzzle, as far as I can see, is as follows:
If there is a property which a thing has, if and only if, it is not autological, then no »proof» can exclude the word »heterological» from being its name. Therefore the only reason, why it can be logically true to say of the word »heterological» that it does not name a property of a certain description, is that there is no property of this description. (And by saying that there is no such property I am not saying that the property is »empty», non-existing in the sense that no thing actually has this property. What I am denying is that an entity of the description in question is a property.)
Thus we can reformulate or reinterpret the conclusion which we reached in the Grelling paradox in this way:
That a thing is not autological, does not constitute a property of this thing. (Not to be autological, i.e. to be heterological, is no 'property'.)
7. It may, in view of the somewhat strange nature of our conclusion, be asked: Is this conclusion really »inescapable»? To raise doubts about the inescapability of the conclusion, would be to doubt the soundness of the means by which this conclusion was reached. Let us therefore, before we proceed, briefly recall which these means were. They can be schematically divided into two groups. To the first belong a few very elementary principles of logic, such as, e.g., the laws of contradiction and of contraposition. To the second belong the definitions which we gave of the concepts »autological» and »heterological».
I see myself no way of doubting the soundness of the principles of logic which we have been using. I cannot even see what it would mean to doubt, say, whether some proposition may not »after all» be both true and not true, or neither true nor not true. Some logicians profess to dispute the validity of the Law of Excluded Middle, — but what they dispute is, as far as I can see, not any principle of a kind which we have been using here.
Thus if our means of proving that »heterological» does not name a property are subjected to doubt, then the doubt must be directed against the »extra-logical» part of the means, viz. the definitions of »autological» and »heterological». Such doubts are conceivable and have in fact often been raised in discussing the paradox (and corresponding doubts in discussing other paradoxes).
Of these doubts I shall here only say that I do not find them profitable. The definition of »autological», which we gave, seems to me to be as good (solid, sound) a definition as one could ever wish for. And the definition of »heterological» just consisted in the creation of one word (»heterological») for the two word phrase »not autological». I find this absolutely uncontroversial too. The concept-formation involved in the paradox I therefore accept as being in order. (Cf. below p. 19.)
After this review of the means of proof, I see no possibility of escape from our conclusion through doubt about these means, i.e. the principles of logic used and the concept formation involved. If there is any possibility of »escape» at all, it must be because in deriving the contradiction we have tacitly relied on some other premiss (assumption) beside the one that »heterological» is the name of a property. If some such premiss (assumption) can be found, it would be an alternative candidate for rejection in the modo tollente argument which was our way out of the paradox.
For the time being, however, let us ignore this possibility. The question will not be raised again until we have concluded our discussion of the rejected premiss that to be heterological, i.e. not to be autological, is not to have a property (the property of heterologicality).
8. We shall now consider an objection against the conclusion, to which our presentation of the Grelling paradox has taken us, which is not an objection against the proof of the conclusion but rather an objection against its truth. This objection can be stated as follows:
I agree to saying that the proposition »'pentasyllabic' is autological» is true and also to saying that »'monosyllabic' is heterological» is true. Both propositions are simple and, for all we can see, quite uncontroversial examples of true propositions of the (grammatical) subject-predicate form. And do we not say that in true propositions of the subject-predicate form a property is being truly predicated of or attributed to a thing? Would it not be sheer »verbalism» to say that in some true propositions of the subject-predicate form that which is being truly predicated of the
subject is not a property (of the subject)? Yet this is what we do, if we deny that the word »heterological» names a property.
To this I shall have to answer that I am not denying that the word »heterological» names a property, if by »property» we understand the meaning of a word which can (with this meaning) stand for the predicate in a true proposition of the (grammatical) subject-predicate form. To do this would be inconsistent with admitting, say, that »'monosyllabic' is heterological» is a true proposition; we would then have to say some such thing as that any predication of heterologicality of a word is »meaningless», since there is no such property to be predicated of anything. But this is certainly not what I wish to say.
If I do not (indeed cannot) deny that the word »heterological» names a property, if by »property» we mean anything which can function as a predicate in a true proposition of the subject-predicate form, then I must be denying that »heterological» names a property in some other sense of the word »property». What could this other sense be? Which are the criteria, on the basis of which something is to be called a »property»? Which criteria, if any, could be more natural and sound than that the entity in question can function as a predicate in a true subject-predicate proposition?
I shall say of a word which names the predicate in a true proposition of the subject-predicate form that it »sounds» like the name of a property or that it has the »property-ring». And possessing the »property ring» could be taken as a defining criterion of naming a property. But it need not be thus taken. Let us therefore look round to see which other criteria for something's being a property might be given.
There is a branch of logic, commonly known as the predicate calculus. It can also be called a »general theory» of properties and relations. It provides us with a set of rules or network of norms, to which entities ought to conform, if they are going to qualify as properties. One of these rules is the Law of Contradiction. It says that no property can both belong and not belong to one and the same thing.
The predicate calculus can be said to constitute an »implicit definition» of the notion of a property. Its rules give us a standard which may be used for saying of a certain entity that it is not a property, on the ground that it does not satisfy the requirements imposed by the rules. An example of such an entity is the concept »heterological» which, if it were a property, would both belong and not belong to one and the same thing, viz. the word »heterological».
(Be it observed that from our refusal to admit that the concept »heterological» is a property it does not follow that we must refuse the status of property to the concept »autological» as well. The assumption that the word »autological» names a property does not lead to a contradiction. (See above p. 7 and below p. 16.))
If the sole criterion of something being a property is that this »something» can be truly predicated of some thing, then we would have to acknowledge that not every property obeys the laws of the predicate calculus in general and the law of contradiction in particular. It could then happen that we had two propositions »x is P» and »x is not P» which are both true. But this would be no antinomy or paradox or contradiction. For, the predications »P» and »not P» would not be predications in the sense of the predicate calculus, i.e. they would not both affirm and deny that one and the same thing has a certain property. This might also be expressed by saying that »not» here has a different meaning from that which it has in the (»classical») propositional and predicate calculi, since the truth of the denial does not exclude the truth of the denied. (Cf. Appendix III.)
We can now once more review the »result» which we said emerged from our construction of the antinomy. This result was that the word »heterological», as defined by us, did not name a property. But in so saying, which were our criteria or standards for saying of an entity that it is, or is not, a property? This we had not made clear. And therefore it was also not clear, in which sense of the word »property» — according to which standards for judging the propertyhood of an entity — the word »heterological» was declared not to name a property. Thus we were led to considering the criteria of propertyhood. And then we found that the sense of »property», in which heterologicality is no property, is a sense which makes conformity to the law of contradiction a necessary condition of propertyhood.
One may think that this clarification of the meaning of the result, which we reached in discussing the paradox, must annihilate the value of the result. For, in the end, our only objection to admitting that the word »heterological» is the name of a property, is that this admission takes us to a contradiction.
But although our discussion thus ends in a triviality, I hope to have succeeded in showing that the road to this end is an interesting road. It is interesting, because it forces us to challenge assumptions which at first sight seem absolutely innocent and obvious, such as that any word which
can stand for the predicate in a factually true and grammatically flawless statement of the subject-predicate form must name a property. We are forced to admit that not anything which has the »ring» of a property, is one according to some other standards too which normally go with this ring. It is in such insights into the nature of the assumptions which we make and the commitments we thereby take upon ourselves that our »solution» of the paradox consists.
9. It is sometimes thought that the error which is implicit in the heterological paradox is the idea that a division of words into autological and heterological ones could be all-embracing. It is suggested that this idea must lead to difficulties, when the principle of division is applied to the words »autological» and »heterological» themselves.
If we take the view which I have tried to advocate in this paper, then there is no logical difficulty about the idea that all words may be divided into two mutually exclusive and jointly exhaustive classes, viz. the autological words and the heterological words, and that this division applies also to the words »autological» and »heterological» themselves. For, let us see what follows as regards these words from what we have said about the concepts which they name:
For the word »heterological» the conclusion is that it is heterological. It is heterological, not because it has not got a property of which it is itself a name, but because it does not name any property at all. Thus »'heterological' is heterological» expresses a true proposition. But from this truth we cannot conclude to the truth of the proposition that »heterological» is autological. The apparent analogy with, say, »'short' is short» must not be allowed to mislead us. This latter proposition is true, because »short» is the name of a property and this property belongs to the word »short» itself. (Assuming, of course, certain standards of shortness for words.) And for this reason we conclude from »'short' is short» to »'short' is autological». But »'heterological' is heterological» is true for an entirely different reason, viz. that »heterological» does not name any property. And hence we cannot pass from it to »'heterological' is autological».
If we took the view that the word »autological» too does not name any property, the conclusion would be that the word »autological» is heterological. (And this, of course, is no paradox, even if we replace the word »heterological» here by its defining phrase »not autological». »'autological' is not autological» is no contradiction.) A reason for taking this view of the word »autological», viz. that it does not name a property, could be that
the paradox forces us to conclude that the absence of autologicality, i.e. heterologicality, is not a property, and it may be thought that, for symmetry's sake, the presence of autologicality cannot then be a property either. But there is nothing in the paradox which forces this conclusion upon us, and I can myself see no good ground for accepting it. It seems to me perfectly logical to think that something is a property, but that the absence or lack of that »something» does not eo ipso constitute a property too, which may then be called the »negation-property» of the first. (This, of course, is contrary to what some logicians would say and to what I have explicitly said myself in past publications.)
If, however, we take the view that »autological» names a property of a certain kind of words — and this, I think, we are perfectly free to do — then we cannot, on the basis of our definitions, decide whether the word »autological» is autological or heterological. There is absolutely no basis for ascertaining, whether the word »autological» possesses or does not possess the property of which it is admittedly a name. The proposition »'autological' is autological» and its negation, the proposition »'autological' is heterological» are both undecidable.
The emergence of such undecidable propositions from the discussion of the paradox seems to me interesting. But I shall not here stop to consider the significance of this fact.
10. It might be suggested that the conclusions which we reached in our discussion of the Grelling paradox are dependent upon the particular mode or way in which we choose to present the paradox. If we had chosen a different mode of presentation, we might have reached different — and maybe more plausible — conclusions.
It is, for example, possible to present the paradox in a way which does not speak of properties at all. Instead of saying that x is autological, if x has a property, which it names, we could say that x is autological, if x falls under a concept, which it names, or simply if x is something which it names. If we had presented the paradox in either of these last two ways, the conclusion ultimately reached would have been that there is no such concept as heterologicality, or that »heterological» does not name anything.
I shall not say of either of these conclusions that it is more or less plausible than the conclusion that heterologicality is not a property. It is the observation that some such conclusion must be drawn which I find important. An alternative way of presenting the paradox would therefore
possess an independent interest only if it could altogether avoid conclusions of this character.
At this stage it might be suggested that the nature of the conclusions, which we have so far been considering, depends upon the »informal» character of our presentation of the paradox, which makes use of such loose words as »property» or »thing». In a »formalized» presentation of the paradox we can altogether dispense with this jargon.
A neat way of »formalizing» the paradox would be the following:
Let us set up a formal calculus.
The symbols are truth-connectives, an unlimited number of what I shall call T-symbols »a», »b», … , and an unlimited number of what I shall call P-symbols »A», »B», ….
By an atomic expression we mean a complex formed by a T-symbol standing immediately to the right of a P-symbol.
By a molecular expression we mean a complex formed of one or several atomic expressions by means of truth-connectives. (We need not here reproduce the familiar recursive definition.)
By an expression we mean an atomic or a molecular expression.
The axioms are a set of axioms of the propositional calculus (with atomic expressions of our calculus instead of propositional variables).
The theorems are any expression which may be obtained from an axiom or theorem by
i. substituting for a T-symbol in the axiom or theorem another T-symbol throughout, or for a P-symbol another P-symbol throughout; or
ii. detachment (modus ponens).
(We might also have quantifiers in our calculus, but this complication is not necessary for our purposes here.)
We next enrich the calculus by a new symbol »' '» (quotation marks). The introduction of the new symbol necessitates a modification of our previous definition of an atomic expression. By an atomic expression we from now on understand a complex formed of a P-symbol immediately followed by a T-symbol or by a P-symbol enclosed within quotes. No other modification in the rules of our calculus is needed.
We could call the complex symbols »'A'», »'B'», … , which are formed of P-symbols and quotes, N-symbols. The N-symbols resemble P-symbols in that they are complexes containing a P-symbol as one of their parts. On the other hand N-symbols resemble T-symbols in the way they are combined with P-symbols to form atomic expressions.
Finally we introduce one more symbol into our calculus, the Greek letter »ξ». The introduction of it necessitates a further modification of the definition of an atomic expression. By an atomic expression we from now on understand, in addition to the atomic expressions already defined, also a complex which is formed of the letter »ξ» immediately followed by a P-symbol in quotes.
But this is not the only modification in the rules, which the introduction of »ξ» calls for. We also add a new clause to our previous definition of a theorem. This is a definition of the symbol »ξ» through the identity »ξ'X' = ~X'X'», — »X» here standing for an arbitrary P-symbol. The definition permits us to derive from any theorem of the calculus a new theorem by substituting — not necessarily throughout — for parts of the form »~X'X'», which occur in the theorem, parts of the form »ξ'X'», or vice versa.
A theorem of our calculus is the expression »~A'A' ↔ ~A'A'». Replacing one occurrence of »~A'A'» in the theorem by »ξ'A'», we obtain the new theorem »~A'A' ↔ ξ'A'».
If in the last theorem we substitute »ξ» for »A» throughout, we get »~ξ'ξ' ↔ ξ'ξ'» which is a contradiction. We could call the contradiction The Heterological Paradox.
Since we do not wish to have a contradiction in our calculus, we ought to take steps to remove it. In order to see, how the contradiction could be removed, we examine, how the contradiction came about.
The last step in producing the contradiction was a substitution. Was this substitution permitted? By the rules, as we had formulated them, it was not permitted. For the rules only said that for a P-symbol in a proved formula, another P-symbol may be substituted throughout. But we had not said that »ξ» could be handled as a P-symbol with regard to substitutability.
That the contradiction occurs could now be taken as a ground for saying that »ξ» must not be regarded as a symbol of the same kind (type, category) as the P-symbols, although it resembles P-symbols by virtue of
the way in which it is combined with N-symbols to form expressions.
This way of avoiding the contradiction would correspond to the »way out», adopted in our previous informal discussion of the Heterological Paradox. Although the word »heterological» resembles property-words as far as its use to form senten[c]es is concerned, it cannot be treated as a property-word in the sense that, if something is true for the meaning of any property-word, then it is also true for the meaning of »hetero-
logical». We can of course decide to call »ξ» too a P-symbol (by virtue of its resemblance to the other P-symbols), — but then we must distinguish between two senses of being a P-symbol: the sense in which anything is a P-symbol which may occur to the left of a T- or a N-symbol in a well-formed expression, and the sense in which anything is a P-symbol which may be substituted for the symbol standing to the left of a T- or a N-symbol in a theorem.
It might be suggested that another »way out» would be to declare, not the substitution which we made, but the definition of »ξ» which we gave, »illegitimate».
To say that the definition is illegitimate merely because of its somewhat unusual form seems to me to be, not only arbitrary, but plainly false. Since we can use it to define »ξ» without contradiction. Actually, the definition as introduced by us into our calculus was flawless. (It was only the unwarranted assumption that the symbol defined was a P-symbol which led to the contradiction.) The definition teaches us the use of »ξ» for purposes of forming expressions and proving theorems in the calculus. It is true that it does not serve any purpose which a definition may serve. It has not the same powers of eliminating the defining terms from contexts, where they occur, as has an »ordinary», explicit definition. This is a peculiarity, not a defect, of it. Definitions with this peculiarity are sometimes called non-pascalian.
But even to say that a non-pascalian definition is illegitimate for the purpose of defining P-symbols seems to me to be false. For if we defined a symbol »
Ø'» by means of the definition »
Ø''X' = dfX'X'» and said that the symbol so defined was to be treated as a P-symbol, we should not get any contradiction in our calculus. (This would correspond to a definition of »autological».) The non-pascalian definition would now be of the kind called impredicative.
Thus a definition cannot be declared illegitimate because it is non-pascalian or because it is impredicative. I can see no ground for declaring the definition of »ξ» illegitimate at all. It is true that it cannot be used to define a P-symbol, because this would lead to contradiction. But this only amounts to saying that the symbol »ξ», which it is used to define, is not a P-symbol. The only thing which may be called »illegitimate» here is the assumption, if we happen to make it, that »ξ» is a P-symbol.
*
Actually, in setting up the calculus, it is not necessary to have two distinct sets of symbols, the T- and the P-symbols. We can have only one set. Let us call them V-symbols and denote them by lover case letters »a», »b», … . An atomic expression would now be defined as a succession of two V-symbols. For example: »ab» would be an atomic expression, and so would »ba» be. The rule of substitution would now be that for any V-symbol, which occurs in an axiom or theorem of the calculus, another V-symbol may he substituted throughout. For example: from »ab → (ab v cd)[»], which is a theorem, we derive by substitution »aa → (aa v aa)[»], which is therefore a theorem too.
We could, however, even within this simplified calculus mark a distinction which would correspond to our previous distinction between T- and P-symbols. For, we could say that, in a given expression »xy», the V-symbol to the left occurs as P-symbol, and the V-symbol to the right as T-symbol. (One and the same V-symbol may then occur both as P-symbol and as T-symbol in the same atomic formula, or as P-symbol in one and as T-symbol in another atomic formula.)
Of »ξ», when added to the calculus, we should then have to say either that it is not one of the V-symbols, or that it is a V-symbol but not subject to the rule of substitution when it occurs as P-symbol.
*
I hope that these considerations suffice to show that it was not the informal character of our presentation which was responsible for the peculiar nature of our proposed »solution» of the paradox, and that also a formalized presentation would have taken us to essentially the same end.
This being so, it may be asked, why we should have given an informal presentation at all of the paradox. The answer is that it is only in the informal presentation that the derivation of the paradox can be stated as an argument from certain premisses to a certain conclusion (the contradiction), and the »way out» as an argument modo tollente from the falsehood of the conclusion to the falsehood of some of the premisses. In the formalized presentation the paradox originates thanks to a manipulation of symbols which is not permitted by the rules of the calculus. This not-permitted move in the game may be tempting, because of its strong resemblance to certain permitted moves. If the move was not expressly forbidden by the rules, then making it shows that it must
be forbidden because of leading to contradiction. Therefore, if making the move can be described as acting on the assumption (premiss) that it could be permitted, then the paradox shows that this assumption (premiss) was false. But this argument from the assumption to the contradiction and back from the rejection of the contradiction to the rejection of the assumption, cannot be formalized in the calculus.
APPENDIX I
A characteristic of the treatment of the Heterological Paradox, which we have given in the paper, is that it tries to show that in the derivation of the antinomy use is being made of an entity, viz. the property of heterologicality, which the paradox proves to be non-existing. The derivation of the antinomy may thus be regarded as a kind of reductio ad absurdum argument.
The observation that in the derivation of some of the best known antinomies — the Russell Paradox, the Heterological Paradox, and the Liar — use is made of a provably non-existing entity, is not new. The first to make the observation was, as far as I know, Bochvar.1
The formula of the predicate calculus, which may be said to show that there can be no such property as that of heterologicality, is, ultimately, the formula Px v ~Px or its equivalent according to the laws of the propositional calculus ~(Px & ~Px). The formulae mentioned say that, given an arbitrary property P and an arbitrary thing x, then it is always the case that x either is P or is not P, and never the case that x both is P and is not P. To express that these formulae hold quite generally for any thing we may »quantify» them in the variable x, thus obtaining the new formulae (x)(Px v ~Px) and (x)~(Px & ~Px), of which the second can also be written in the form ~(Ex)(Px & ~Px). And in order to express that these last formulae hold quite generally for any property as well, we may quantify them in the variable P too, thus obtaining the new formulae (P)(x)(Px v ~Px) and (P)~(Ex)(Px & ~Px), of which the second can also be written in the form ~(EP)(Ex)(Px & ~Px).
These formulae can be used as a ground for saying that there is no such property as heterologicality, only if we take for granted that the fault in the paradox is with the property and not with the thing with which we are manipulating in constructing the antinomy. This thing is the word »heterological». An alternative way out of the paradox would therefore be to say that there is no such thing as this word.
This second way out may seem more »artificial» than the one adopted here. If we deny that the entity named by »'heterological'» is a thing but regard words as a kind of thing, then we would have to dispute that the named entity is a word. The conclusion would be that there is no such word as »heterological»2. An alternative conclusion would be that not all words are things (since the word »heterological» is not one).
I do not think that either of these conclusions can be brushed aside as being absurd. To do this would be to take a much too simpleminded view of the notions of »thing» and »word».
What I have done in my discussion of the paradox is therefore to show what follows for the concept of heterologicality, if we take the thing- and word-character of »heterological» for granted. If, instead, I had taken for granted the property-character of heterologicality, I might have used the paradox for drawing conclusions about this property's name, viz. that it is not a thing or not a word.
Which road we wish to go, which moral we wish to draw from the paradox is, I believe, very much a matter of indifference. We can let the paradox, i.e. the contradiction which we reach as the result of a certain argument, »recoil» modo tollente either on the notion of a property, as we preferred to do in the paper, or on the notion of a thing or on the notion of a word.
There is, however, a further way out which we have not so far mentioned at all. This way consists in pronouncing neither upon the property-character of heterologicality nor on the thing-character of »heterological» but in saying simply that »heterological» and heterologicality do not match one another as thing and property, according to the standards of the predicate calculus. This would be the most »neutral» conclusion of all. It might be termed a »Janus face conclusion», from which one could then proceed in one of two directions:
The road in the one direction terminates in the conclusion that heterologicality is no property, since, if it were, it would both belong
and not belong to one and the same thing, viz. the word »heterological». This was the direction we choose and the end we reached in our discussion in the paper. But we need not proceed as far as this end. We could also say that heterologicality is a property (»behaves like a property») up to a singular point. This point is, when we predicate the property of its own name.
The road in the second direction terminates in the conclusion that »heterological» is no thing, since, if it were, it would both possess and lack one and the same property, viz. the property of heterologicality. But we need not proceed as far as to this conclusion. We might also say that the word »heterological» is a thing (»behaves like a thing») up to a singular point, namely to the point when we attribute to the word the property which it names.3
APPENDIX II
It is an interesting observation that we get an antinomy only by trying to answer the question, whether »heterological» is autological or heterological, — and not by raising the same question about »autological». Similarly, the Russell Paradox originates, when we ask whether the extension of the property »extension which is not a member of itself» is a member of itself or not, — whereas the question, whether the extension of the property »extension which is a member of itself» is a member of itself or not, does not lead to paradox. The Liar again originates from the question, whether the proposition »this proposition is not true» is true or not, — but no paradox springs from the question whether the proposition »this proposition is true» is true or not. Remembering that a thing is heterological, if it has not got any property, which the thing itself names, we may take these observations as an indication that the concept of negation holds a crucial position in the construction of these three antinomies.4
The »negative» nature of heterologicality must not be misunderstood. It does not lie in the fact that we defined »heterological» as meaning »not autological», thus introducing »autological» first and »heterological» second. We could equally well have started with a definition of »heterological» (not mentioning »autological») and subsequently defined »autological» as meaning »not heterological». (See above p. 5 f.) By calling »heterological» negative I do not mean that the word has been introduced as an abbreviation for a phrase (»not autological») containing the word »not» or some other word signifying negation. The negativity of »heterological» is a peculiarity of the concept and not of the word, and consists in the fact that a thing is said to be heterological, not on the ground that it has such and such characteristic features, but on the ground that it has not those features, the presence of which in a thing (word) makes this thing autological.
Similar remarks could be made about the negative nature of the concept »extension which is not a member of itself» and the concept »not true».
The »way out» of the Grelling and Russell antinomies, which is here suggested, is the conclusion that the negative property involved is not a property in the sense of the predicate calculus. The corresponding »way out» of the Liar is the conclusion that the negative proposition involved is not a proposition in the sense of the propositional calculus. The negative concepts, which give raise to antinomies, are thus of a different logical category or level or type from those concepts, of which they are the negatives. This can also be expressed by saying that, if »P» names a property, it does not always follow that »not P» names a property in the same sense of »property»; and if »p» expresses a proposition, it does not always follow that »not p» expresses a proposition in the same sense of »proposition». When the conclusion does not follow, I shall call the entity named by the phrase containing the negation-word essentially negative.
The antinomies of Grelling and Russell and the Liar can be said to establish or demonstrate the »essential negativity» of certain concepts. But this is not to say that the notion of essential negativity could be used to »explain» or »solve» the antinomies. This would be the case only if the notion of essential negativity had independent criteria which would make it possible to show — independently of the antinomy — that e.g. heterologicality is not a property in the same sense as those properties, the absence of which in their names makes us call the names heterological. This condition, for all I can see, is not fulfilled. It is not the essential
negativity of the entities, which shows that they must not be treated as entities of the same category as those, of which they are the negatives. It is the fact that treating them thus leads to antinomies, which shows that the entities are what I have here called »essentially negative».
APPENDIX III
The way of treating the Heterological Paradox which I have adopted n this paper seems to me related to Wittgenstein's comments on the paradox in Remarks on the Foundations of Mathematics V-21 (p. 178) and to some other of his remarks on paradoxes and contradictions in general.
»Why shouldn't it be said», Wittgenstein asks, »that such a contradiction as: 'heterological' ε heterological ≡ ~('heterological' ε heterological), shews a logical property of the concept 'heterological'.»
This, in my view, is exactly what the antinomy, when rightly understood, does. It reveals to us a logical feature of the concept »heterological». This feature is that the concept, although it may function as a predicate in true propositions of the subject-predicate form, yet does not, unlike most concepts with this function, in all cases obey the law of contradiction. The exception is the paradox. Therefore we do not notice the exception, until we become aware of the paradox.
Wittgenstein then goes on to saying that »'h' ε h ≡ ~('h' ε h) might be called 'a true contradiction'». That the contradiction is true means, he says, that »it is proved; derived from the rules for the word 'h'».
To say that the contradiction is »true» would seem to me to be very misleading, — and therefore also to say that it is »proved». For the lesson which the contradiction teaches us, I have tried to argue, is a lesson modo tollente, i.e. because we insist that a contradiction is not true and therefore cannot be proved, we are forced to question the validity of an assumption which we made in the derivation of the contradiction. (To say, as Wittgenstein does, that the contradiction is derived from the rules for the word »heterological» is, of course, quite in order.) That this assumption is not valid for the concept »heterological» is the truth which the paradox reveals.
Wittgenstein also says that the truth of the contradiction means: this really is a contradiction, and so you cannot use the word »'h'» as an argument in »ξ ε h». To this we can agree. For it amounts to saying that
26
the truth is not of the contradiction, but of the proposition that, on such and such assumptions, we get a contradiction.
But, as we have seen (p. 14), there is a sense in which we can say that »'h' ε h ≡ ~('h' ε h)» is true, — but not a true contradiction. To call »'h' ε h ≡ ~('h' ε h)» true entails saying that the pair of predications »P» and »not P» are not always mutually exclusive, and this again amounts to saying that there is a use of »not» for predication which is different from that use of »not» which conforms to the Law of Contradiction. »'h' ε h ≡ ~('h' ε h)», when true, is therefore not a contradiction (unless we wish to give a new sense to the term »contradiction»).
APPENDIX IV
The analogy between an antinomy and division by 0 in arithmetic seems to me a good analogy and worth further exploration.5
Let us assume that we formulated a rule to the effect that for any real numbers m, k, and l, if mk equals ml, then k equals l. Then someone uses the rule to »prove» that, say, since 0 times 5 equals 0 times 7, therefore 5 equals 7.
What shall we then say? Since we insist upon the falsity of the conclusion, one thing to do would be to let it »recoil» modo tollente on some of the premisses used in the proof. One such premiss (perhaps better »presupposition») is that 0 is a (real) number. This we might now reject and say that the »paradox» shows that 0 is not to be regarded as one of the (real) numbers.
Historically, the idea that 0 is one of the numbers was a hard one to acquire. At an earlier stage in the history of mathematics, the above conclusion from the »paradox» might therefore have seemed plausible. Not so to-day. The reasons which we have for regarding 0 as one of the (real) numbers are strong enough to withstand the »paradox».
The obvious thing to say after the discovery of the »paradox» would therefore be, not that 0 is not a number, but that we have discovered an exception to the rule of division, which we gave. It ought to be reformulated. The rule, when reformulated, would say that for any numbers m, k, and l, if m is different from 0 and mk equals ml, then k equals l. This rule holds good for all numbers (including 0), but it enables us to
derive the result that k equals l only for such values of m which are different from 0.
Let us now compare this with our proposed »way out» of the Grelling Paradox.
The conclusion modo tollente from the contradiction was that heterologicality is not a property. This corresponds to the suggested conclusion from 5 = 7 that 0 is not a number. If the conclusion in question from the Grelling Paradox seems plausible, it must be because we do not feel that the grounds for calling heterologicality a property outweigh this ground (viz. the contradiction) against calling it thus.
We could, I think, imagine circumstances under which we would insist upon calling heterologicality a property. (These circumstances would have to be somewhat similar to those which make us insist upon calling 0 a number.) Then the conclusion which we drew from the paradox would appear highly »artificial». But we could now draw a different conclusion, corresponding to the one which we actually draw from the paradox that 5 equals 7. We could say that, for any word (or thing) x, if x is not a name of the property of heterologicality itself, then x is heterological, if and only if, it is not the case that x has got a property, of which x is a name. This holds good for all words (including the word »heterological»), but it enables us to derive a truth-condition for the thesis that x is heterological only for such values of x which do not name heterologicality.
APPENDIX V
Is an antinomy a »danger» to logical thinking? Was reasoning with the concepts, which are involved in the antinomy, not »safe» or »in order» before we had discovered the antinomy? (These questions are very unprecise.)
Questions of this kind are raised and discussed over and over again by Wittgenstein in the Remarks on the Foundations of Mathematics. They are raised, not for the antinomies only, but for all contradictions which may occur in a context of logical or mathematical proof.
Wittgenstein says (II-82) that his aim is »to alter the attitude to contradiction and to consistency proof». He seems, in places, to say that antinomies are harmless (I-Appendix I-12) and that contradictions might be accepted (III-55–60), and he speaks of »the superstitious fear and awe of mathematicians in face of contradiction» (I-Appendix I-17). It seems
to me that Wittgenstein never succeeded in saying quite clearly what he wanted to say on the subject of antinomies, contradictions, and consistency-proofs.
I have tried to advocate a view in this paper which, if correct, would show in what sense antinomies may be regarded as »harmless». What I mean by this can, for the Heterological Paradox, be stated as follows:
The only way, in which the false premiss or illegitimate move in the calculus which is responsible for the heterological paradox shows its falsehood or illegitimacy, is in the production of the antinomy. To assume that the word »heterological» names a property (or that »ξ» in our formalism is a P-symbol) and to rely on this assumption in reasoning works without contradiction up to a singular point. This point is when we assume that the property of heterologicality in relation to its name, the word »heterological», is subject to the same rule which holds for all other properties in relation to their names, viz. that the name is heterological, if and only if, it does not possess the property. To assume this of heterologicality is, by definition, self-contradictory.
If I am right, the antinomies of logic do not require any »general theory» for their solution, — be it a doctrine of distinction of logical types, a Vicious Circle Principle or some other general restriction on the definition of concepts. The antinomies do not indicate any disease or insufficiency in the »laws of thought» as we know them at present. The antinomies are not the result of false reasoning. They are the result of right reasoning from a false premiss. And their common characteristic seems to be that it is only the result, viz. the paradox, which makes us aware of the falsehood. Without the discovery of the paradox, the falsehood would therefore have remained for ever unknown, — just as people might never have realized that fractions cannot be divided by zero, unless they had actually tried to do it and reached contradictory results.
Printed January 1960
Notes:
1 See the review in The Journal of Symbolic Logic 11, 1946, p. 129. See also Quine, Mathematical Logic (1940), p. 128 and, for a more detailed exploration of the idea, Valpola, »Elementare Untersuchungen der Antinomien von Russell, Grelling-Nelson und Eubulides», Theoria 19, 1953, p. 183–188.
2Cf. Valpola, op.cit. p. 186.
3Wittgenstein, in Remarks on the Foundations of Mathematics (III–59), compares an antinomy (he is then speaking of the Russell Paradox) to »something that towers above the propositions and looks in both directions like a Janus head». I do not understand quite clearly the meaning of the simile. It does not seem to me to be related to the idea, propounded in this paper, of the Janus face character of the conclusion which, in the most non-committing way, takes us out of the paradox.
4Cf. Valpola, op.cit., p. 187. The author, however, seems to overlook the existence of »negation-free» antinomies, such as e.g. Curry's Paradox.
5The analogy is referred to in several places in Wittgenstein's Remarks on the Foundations of Mathematics.