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Abstract

Part I. General Hypothesis

1. The possibility to which this paper responds is that the Paradoxes of Self Reference have remained unsolved because it has not been sufficiently understood how self reference itself works and to what peculiar fallacies its use is consequently exposed. I propose here a general hypothesis about the workings of self reference and, in its context, seek to resolve both Russell’s and Lukasiewicz’s antinomies. While I do not claim that this hypothesis can explain all uses of self reference, I do claim that if it serves to resolve some of the paradoxes, then this suggests the need to regard self reference as a legitimate kind of discourse whose peculiar utility, in its various uses, requires further study.
My hypothesis is presented in two parts. The initial hypothesis is concerned with identifying the logical phenomenon to which a certain kind of use of self reference gives rise, and is illustrated by preliminary analyses of Russell’s Antinomy (Part II) and of Lukasiewicz’s Paradox (Part III). The amplified hypothesis is concerned with the explanation of this logical phenomenon, and is used to complete the resolutions of the two antinomies (Part IV).
The initial hypothesis is this: The use of self reference is sometimes apt to provide the occasion for the cropping up, in the course of a line of reasoning, of hitherto unknown or simply unusual objects. Russell’s class of classes which are not members of themselves may be regarded as a paradigmatic case of such an object, hitherto unknown. Now such objects are apt to pose the difficulty that certain patterns of inference we have been assuming are inapplicable to them. Thus the fallacy of a paradox of self reference may consist in the valid extrapolation of an accostumed pattern of reasoning to an object that has cropped up. To exhibit the fallacy of such a paradox is to determine why a certain pattern of inference does not hold for a certain object. Also, of course, one must determine whether or not there is a pattern of inference that is applicable to the object and with logically acceptable results.
The amplified hypothesis is concerned with the logical structure of such cropping up of objects and with the nature of the grounds for the use of this logical mechanism in any given case. We start from the consideration that the uses of self reference are characteristically uses of logically contingent forms, so that if some given use of self reference leads to contradiction, this cannot impute self reference unless this use of it is somehow justified. Thus if a variable of a function or propositional form takes a certain value that poses a self referent case, then the justification of this use of self reference is a matter of the grounds for permitting this value of the variable. Now a peculiar difficulty arises in this respect, because the value that poses the self referent case is apt to belong to a subclass of the values of the variable that has not been taken into account in defining the variable. This, then, is the mechanism by whose means an unexpected object can crop up, while the problem posed is to characterize the subclass to which this object belongs. This then enables us to reconsider the whole matter of whether the function can permit a self referent case.

Part II. Preliminary Analysis of Russell’s Antinomy

2. We note that in mathematics the concept of number had to be progressively enriched owing to the cropping up, by the fundamental operations of arithmetic, of negative, irrational and imaginary numbers. We then admit the possibility that an analogous problem may be posed for the conception of logical objects owing to the consequences of fundamental logical principles; and we entertain the hypothesis that Russell’s Antinomy, which obviously involves the cropping up strange logical objects, poses this problem.
Russell’s straightforward and compelling line of reasoning, which is grounded in universally accepted logical principles, rapidly discovers some remarkable logical objects: the reflexive class membership relation, the class of classes that are not members of themselves, and the class of classes that are not members of themselves (this latter to be called w, following Russell). The difficulty then is that, by an accustomed pattern of reasoning the question about whether w is or is not a member of itself leads to a contradiction.
We display the accustomed pattern of inference as follows: We say,

(1) “Murmur” is an onomatopeic word.

Then, since an onomatopeic word is defined as one whose sound is characteristic of the referent of the word, we have,

(2) If “murmur” is onomatopeic, then “murmur” sounds like a murmur.

And clearly, the implication is reciprocal.
Analogously, to reason about the reflexive class membership of w, we assume conditionally that,

(3) w is a member of itself.

Then since a class which is a member of itself is defined as one which possesses its own membership condition property, and since w’s membership condition property is not to be a member of self, it follows that,

(4) If w is a member of itself, then w is not a member of itself.

And again, the implication is clearly reciprocal. So we get the antinomy,

(5) w is a member of itself if, and only if, w is not a member of itself.

Now the reasoning from (3) to (5) seems to be justified by our intuitive notion of a class; yet it remains possible that w has peculiar properties such that (3) engenders a different pattern of inference, and my suggestion is that this latter is the case.
The class w has two very strange properties which can be formulated in a preliminary way as follows:
(i) w’s membership condition is always a function of the membership condition of a member class. In its nature, this membership condition is a function in the case of... So if w itself is to enter into the reflexive class membership relation, its membership condition for itself, in the role of the member class, has to be given before its membership condition function can take this membership condition as an argument. Yet w’s membership condition for itself can only be given by the same function.
(ii) Of itself, w’s membership condition involves the notion of the membership condition of a class only in the role of a variable. It therefore turns out that in the self referent case of w’s membership condition for itself, the expression “membership condition” that enters into this membership condition has no other sense than the general sense of this expression.
That (i) and (ii) may have unexpected consequences is suggested in a dramatic way by the following argument: Let v be the class of classes that do not satisfy their own membership conditions. Then v’s membership condition for any class, including itself, is that this class does not satisfy its own membership condition. But v’s own membership condition is, not to satisfy its own membership condition. So for v to satisfy v’s membership condition is for v not-not to satisfy its own (v’s) membership condition!
This argument in intension is strong, and leads not to contradiction but to tautology. On the other hand, Russell’s well known argument in extension (cited in English in the body of the paper) leads to contradiction. How, then, shall we choose between these two arguments? Surely the indicated procedure is to reformulate Russell’s function in a way that permits both intensional and extensional readings, and to reconsider the instantiation problem in this context. We may put,

(α) Wx ≡ ∼Xx; and, (β) ∼ Wx ≡ Xx,

x being the individual class variable and X the corresponding membership condition property variable. In this symbolism, W is w’s membership condition, and since it is defined (in the context of Wx) by the right hand term of α, we assign it the non-committal reading, “w’s membership condition”. Notice, too, that in β we have suppressed the implicit double negation of the right hand term.
Now then, what is the value of X when x is w? As indicated above in (1), w’s membership condition for any class is given by the function ∼Xx, and for the case of w this function gives ∼W as w’s membership condition for itself. Of course, this has always been recognized. But what has not been taken into account is that ∼W is w’s membership condition for itself in its role as the member class term of the reflexive class membership relation. Thus ∼W is the value we put for X! It is, then, the function Xx which, when instantiated for the case of w, gives ∼Ww. But Xx is equivalent by definition not to Wx but rather to ∼Wx. And, indeed, ∼Ww serves without more ado to instantiate the right hand term of β, while in the right hand term of α, the instantiation of Xx as ∼Ww remains to be negated. So we end up with the tautologies,

(α′) Ww ≡ ∼∼Ww; and, (β′) ∼Ww ≡ ∼Ww.

The contradiction has disappeared.
Moreover, this result is interpretable as corroborating the point made above in (ii). When we instantiate Xx as ∼Ww this corresponds to our intuitive notion of what would be w’s membership condition for itself. But this is because we are reading “W” unambiguously as “w’s membership condition”. If we tried to read any more into “W”, we should fall into the fallacy of confounding the negation of “W”, according to its assigned reading, with the intension of “W”. To avoid this fallacy, we have to accept the assigned reading of “W” as its complete reading. And counter-intuitive as this may seem, it is consistent with point (ii). Since in w’s membership condition, the expression “membership condition” has only a general sense, this expression cannot acquire any additional sense in the case of w. It is the negative form of ∼Xx, regarded as the function that gives w’s membership condition for itself in the first step of the instantiation, that permits the instantiation of Xx in α and β to express w’s membership condition for itself.
To be sure, this is all very strange. The import of the instantiation of α and β when x is w turns out to be that Ww and ∼Ww mean respectively that w does and w does not satisfy w’s membership condition, in the general sense of “membership condition”. This is all they mean.
We are thus left with an exceedingly strange pair of propositions. Ww and ∼Ww are neither nonsense nor self-contradictory. Regarded as a pair, they have the form of a pair of contradictories, yet they have no intensional reading which can serve for judging whether they are true or false and thus ground their standing in a truth functional relation to each other. Their meaning has a deprived character. They would seem to be a kind of propositional object we have not met up with before; and if so, they have to be added to the set of strange logical objects which crop up in the development of Russell’s line of reasoning.
3. It must be conceded, however, that the preceding argument does not resolve Russell’s Antinomy. By this argument, the pairs of functions, (Wx, ∼Wx) and (Xx, ∼Xx), are not contradictories for the case of w. This result would invalidate the conception of α and β unless it could be shown that it is a consequence of the character of the functions α and β in so far as they contain as constant components the reflexive class membership relation and the strange membership condition W. Thus in order to go on with this line of attack on the Antinomy, we must perforce adopt some hypothesis about these strange logical objects.
It seems to me that there is a striking analogy between these objects and the imaginary number, so in what follows I entertain the hypothesis that these objects are in some sense imaginary logical objects. The rest of this section explores one aspect of the logical problem that has arisen that suggests this analogy. In Part IV I shall try to show that if this view were adopted, α and  would be interpretable as permitting the self referent case.
If we were to make a distinction between real and imaginary logical objects, we could classify as imaginary the reflexive class membership relation, the class w and other similar classes. Propositions like Ww, whose meaning have a deprived character and which are consequently neither true nor false, would be imaginary. On this interpretation, the notion of the reflexive class membership of w gives rise to imaginary propositions. Such propositions could be thought of as wholly imaginary.
On the other hand, we have also met up with propositions like,

(6) The class of classes having more than five members is a member of itself.

It may now strike us that (6) can be regarded as a composite of real and imaginary elements and that, in addition, it has two different but interrelated readings. To see how this works out, let b be the class of classes having more than five members and u the class of classes that are members of themselves. Also, let “i” and “e” be used as subscripts to indicate intensional and extensional readings respectively. Then (6) says both (Pe) b is a member of b, and (Qi) b possesses the property, member of self. Moreover, (Pe) has the corresponding reading, (Pi) b is a class having more than five members; and (Qi) has the corresponding reading, (Qe) b is a member of u. Of these four readings, (Pi) is stated wholly in the domain of real objects and is unequivocally true. But the rest of the members of this set each relates a real logical object, the class b, to an imaginary logical object, whether as a term of an imaginary relation (in Pe), or as possessing an imaginary property (in Qi), or a belonging to an imaginary class (in Qe). Now all four members of this set are interrelated by precisely the pattern of reasoning which we have seen to fail in the cases of the wholly imaginary propositions, Ww and ∼Ww. It is thus very tempting to think that the pattern does not fail in the case of the set implied by (6) because this set contains the real (Pi), in which imaginary objects disappear from the reasoning.

Part III. Preliminary Analysis of the Lukasiewicz Paradox.

4. It is pointed out that the form (T), which Tarski attributes to Lesniewski and which this paradox exploits (“X is true if, and only if, p”, p being a sentence of the language to which “true” belongs, and X being the name of p), was used by Aristotle himself in Categories (14b11 ff.). My assumption is that this form is valid within ordinary language and that a careful analysis of the paradox will vindicate this view.
The Lukasiewicz Paradox is stated in the later form given to it by Tarski in his paper, “The Semantical Conception of Truth”, Sec. 7. The self referent sentential construction ‘s’ is on page 125, lines 2 and 3. Prop. (I) is the equivalence of the form (T); Prop. (II) is the identity, claimed to be an empirical fact, which is presented as the ground of the substitution; and Prop. (III) is the antinomy.
My first point then is that Prop. (II) is not empirical. If the substitution is valid and Prop. (II) is its ground, then Prop. (II) must have the import that ‘s’ and the sentence ‘s’ is about are the same sentence. If (II) did not have this import, then the substitution would rest on a verbal sophistry. The required import is assured by reading the expression, “The sentence on page 125, lines 2 and 3 of this paper”, in the sense of a definite description. Prop. (II) then means that there is one and only one sentence in the said place, and that sentence is ‘s’. But on this interpretation, Prop. (II) is a problematical theoretical premise. Since ‘s’ is used to instantiate the truth locution form of the form (T), it is unquestionably being treated as a statement. However, ‘s’ was constructed by putting in a certain place a sentence of the type that can be used to make different statement. It is, then, a question of linguistic theory whether this verbal sentence, as I call it, is also there on page 125, lines 2 and 3. Moreover, it may not simply be assumed that there is only possible statement-making use of that verbal sentence in that place. This is a matter requiring analysis. It follows, therefore, that if Prop. (II) has the import needed to ground the substitution, it is a theoretical judgment about what sentential objects are available to be denoted by the subject of ‘s’.
This result puts the paradox in a new light. There is no hard empirical fact blocking the path to the theoretical treatment of the singular self reference sentence. Moreover, we now glimpse the possibility of interpreting such a sentence as being about the verbal sentence used to state it; and the vicious circle interpretation becomes just another claimant. Are there, then, theoretical grounds for rejecting this latter? The attempt to answer this question is postponed until after the semantical interpretation of the paradox is considered.
5. My contention is that, owing to its logical structure, the Lukasiewicz Paradox can neither place the form (T) in doubt within its linguistic context, nor can it give ground for considering that ‘s’ is contradictory within this context because it is a truth locution.
Let us accept for the sake of the argument that the antinomy is validly derived. Clearly, the reasoning proceeds by using ‘s’ to instantiate the form (T). If, then, there is a latent contradiction in Prop. (I) which is elicited by the substitution, this contradiction can be traced either to the form (T) or to ‘s’, but not to both. Because: Prop. (I) is a counter-example to the validity of the form (T) only if ‘s’ is indubitably consistent; or, if the form (T) is valid, the source of the contradiction can be inferred to be ‘s’. Thus all that can be inferred about the source of the contradiction is that S (“ ‘s’ is consistent”) and T (“The form (T) is valid”) are contraries. We have, not-S or not-T. Then to prove that ‘s’ is inconsistent we must say, T, and therefore, not-S. Nor is there any reason not to say this since it would be absurd to pit the consistency of ‘s’, interpreted as a vicious circle by (II), against the self evident validity of the form (T) which, to my knowledge, no one ultimately denies. But if we assert T in order to infer the inconsistency of ‘s’, we have conceded the consistency of the truth locution form which is a component of the form (T), so we cannot then turn about and say that ‘s’ is inconsistent because it is a truth locution. Thus even if the paradox were validly derived (which is not the case), its semantical interpretation would be groundless. We must perforce look to the self reference of ‘s’, on its vicious circle interpretation, for the source of the trouble.
6. It is now argued that a sentence which is a vicious circle can indeed be rejected on the ground that it is not well formed. By this interpretation, the self referent sentence purports to be identically the same statement as the statement it says something about.* Now surely, the vicious circularity of such a sentence consists just in that it has no distinguishable subject and predicate, and so is utterly impervious to grammatical analysis. To analyze such a sentence into subject and predicate is to destroy it. In this respect, therefore, it is not in the form “S is P”. Yet at the same time, the conception of such a sentence in that its subject denotes itself. Thus the conception of the viciously circular sentence involves the contradiction that this sentence is and is not in the subject-predicate form. It would surely be surprising, then, if one could reason validly from such a sentence.
The purported line of reasoning of the Lukasiewicz Paradox can now be reinterpreted as follows: The right hand term of (I) is the viciously circular ‘s’. The substitution is then an analytical operation on ‘s’ which destroys it. ‘s’ vanishes, and the right hand term of Prop. (III) appears. Notice that if (III) is an antinomy, then both terms of the “if and only if” must be in the subject-predicate form, and both must be about precisely the same sentence. But the left hand term of (I), “ ‘s’ is true”, which subsists unchanged in (III), is non-self referent and is about the circular ‘s’. So if the right hand term of (III) is its contradictory, where did it come from? This is the strictly logical question posed by the paradox; and we can now answer it. The right hand term of (III) resulted from a misuse of Leibniz’s Law to exploit one aspect of the ambiguous form of ‘s’ and thus dissolve ‘s’.
It follows that the reasoning of the paradox is a pseudo-argument. ‘s’ is to be rejected, not because it leads to a contradiction, but rather because it leads nowhere. It is not a well formed sentence from which one can reason validly.
However, there is a residual problem: Are we to regard the circular ‘s’ of the paradox as a natural use of self reference that constitutes a counter-example to the reliability of self reference as such? Or, may we regard ‘s’ a misuse of self reference? In Part IV it is argued that there are theoretical grounds for the latter alternative.

Part IV. The Workings of Self Reference in Russell’s and Lukasiewicz’s Problems

7. It may be asked, On just what grounds can a paradox be thought to impute self reference as such? The uses of self reference are characteristically uses of logically contingent forms. Now the deduction of a contradiction from a premise in a contingent form ordinarily implies only that the content which has been put into that form is inappropriate. The contradiction becomes paradoxical only if there is nonetheless a good reason for putting that content in that form. Then we have a conflict of criteria and a paradox, some reason supporting the acceptance and the contradiction supporting the acceptance and the contradiction supporting the rejection of a given use of contingent discourse.
In both Russell’s Antinomy and the Lukasiewicz Paradox, the justification of the respective uses of self reference is a matter of the reason for permitting a certain value of a variable term that poses a self referent case of a function or propositional form. In Russell’s problem, we have a paradigmatic example of a justified use of self reference. Russell’s unanswerable preliminary argument justifies the function which rests on accepting that w is a class, so it is very hard to see how one could accept the function and at the same time rule out w as a permitted value of the class variable. But in the Lukasiewicz Paradox, we are dealing with an arbitrarily formulated verbal sentence —hereafter called ‘sv’— in the form of a truth locution. Now surely, any statement-making use of ‘sv’, whether self referent or non-self referent, is permissible if, and only if, it is a correct use of language. Thus the very criterion by which it could be hoped to justify the ‘s’ of the paradox is just that criterion by which this circular ‘s’ has already been rejected. And since there is no argument of any kind which establishes a reason for constructing ‘s’, there simply is no conflict of criteria in this case. The prima facie possibility of constructing the defective ‘s’ of the paradox is merely an instance of the prima facie possibility of misusing language. This construction misuses both self reference and “S is P”. Burt if this is so, how could it strike us as a natural use of self reference?
8. The problem of the concept of self reference affords a very pretty example of Bacon’s Idols of the Market Place. The name, “self reference”, that has been given to a set of indicated linguistic phenomena does indeed respond in a preliminary way to the appearance of these phenomena. But it does not follow that the properties of self reference or its natural uses can be analytically deduced on the ground of the antecedently given meaning of the expression, “self reference”. To form a valid notion of the expressive potential of this linguistic resource, we have to go to the self referent phenomena themselves and analyze them.
We have seen that the use of self reference may lead to the cropping up of unexpected objects. To consider the logical structure of this phenomenon, I now recur to the amplified hypothesis already outlined in Part I.**
We start from the accepted notion that the permitted values of a variable constitute a class. Now a value that poses a self referent case of a function may exploit the systematic ambiguity of the variable term, in that this value may belong to a subclass of the values of the variable that has not been taken into account in defining the variable, or perhaps has never even been conceived. If, then, it is not noticed that the value that poses the self referent case belongs to such a subclass, it may happen that a pattern of inference which is being assumed to hold for all the values of the variable, does not in fact hold for the value that poses the self referent case. We thus get the fallacy that the variable is conceived too narrowly to permit the self referent case at the same time that this case is permitted. The problem, then, is whether or not the very conception of the function is compatible with enlarging the scope of the variable to permit the self referent case. But of course such a reinterpretation of the function becomes urgent only if its self referent use must be regarded as justified.
Russell’s Antinomy provides a clear example of the fallacy in question and of the possibility of reinterpreting the function to permit the self referent case. But as for the ‘sv’ of the Lukasiewicz Paradox, we find that the very conception of this propositional form seems to be incompatible with its valid self referent use. In what remains of this summary, I shall merely indicate how this matter seems to me to work out in the respective problems.
With respect to the α and β of Russell’s problem, the pairs of functions (Wx, ∼Wx) and (Xx, ∼Xx) are, of course, contradictories; yet this is compatible with there being a subclass of classes for which the reflexive class membership relation does not hold. Contradictory functions are such on the assumption that their use is within the domain of true or false discourse. Then if the value w of x takes the use of these functions out of this domain, with the result that Ww and ∼Ww are neither true nor false (as I have suggested in Part II), this does not refute that (Ww, ∼Wx) and (Xx, ∼Xx) are contradictory pairs when used within the domain of true or false discourse. However, to develop this solution requires that we revise our conception not only of the class variable, to include the subclass of classes to which w belongs, but also of the functions α and β. Assuming (as suggested at the end of Part II) that the strange objects which have cropped up are imaginary, we could say that these functions contain constant imaginary components —most notably, “W” itself— and thus admit only of complex or wholly imaginary use. On this view, when x is a real class, their use is complex and within the domain of true or false discourse; but when x is imaginary, their use leaves this domain. If this way of handling the problem were adopted, then, among the remaining difficulties, there would be a need for a special symbol (or perhaps more than one) to designate the imaginary components of expressions.
Meanwhile, it should not be overlooked that to regard W as an imaginary class membership condition involves exploiting the systematic ambiguity of the membership condition property variableX. We found in Part II that “W” is the mere general idea of the membership condition of w. Now we see that, regarded as a value of X, W is a very strange kind of property. Just as there is no square root of —1, there is no univocally specified property W. There is nothing but the idea of it; and in order that X may take W as a value, the conception of the logical property variable has to be pushed, so to speak, to a limit of sense.
On the ground of these results it is concluded that Russell’s Antinomy is resolvable on the basis of accepting his preliminary argument, if it is conceded that he discovered a new class of logical objects. Moreover, this type of solution, if it is tenable, is surely desirable. The great strength of this paradox consists precisely in that its impeccable and intuitively valid preliminary argument, leading to the conception of the function, discovers a problem whose solution clearly requires a development of logical theory. Thus no solution that recurs to a theoretical development which in its turn invalidates the preliminary argument, can wholly dispel the logical disquiet to which this paradox has given rise.
9. We now turn our attention to the workings of self reference in the case of the singular self referent sentence. We note first that by any method of constructing such a statement, the sentence that the statement is about is actually present, in the role of an indicated object, and in this role has the essential ambiguity of any object that can be pointed at. It being a linguistic object, a sentence, what we can make of this object is first of all a matter of what kind of sentence we can take it to be. For example, the moment we focus on the self referent sentence as a statement about a sentence, we see that if its predicate is applicable to a verbal sentence, then such a self referent sentence is interpretable as being about the verbal sentence used to state it. Moreover, this interpretation does not involve regression, because, logically, the verbal sentence is a propositional form whose subject term is a variable and as such has no referent. (See Note 16.) Another possibility is that if its predicate is applicable both to statements and verbal sentences (e.g., “well formed”, significant, etc.), a self referent sentence is interpretable as being about a statement which is in turn about a verbal sentence. Still another possibility is that a self referent sentence may be stipulated to be a non-regressive statement about a sentence that involves an infinite regression. This interpretation is grounded in the consideration that the main statement made by the self referent sentence, in so far as it is a statement, could in principle be asserted in different words. Examples of self referent sentences that can be interpreted significantly in these ways are given in the body of the paper.
The riddle posed by the self referent sentential construction is simply to determine in how many different ways this kind of construction can exploit the systematic ambiguity of the term “sentence”. Depending upon its predicate, the self referent sentence can be used to express something significant about one or another kind of sentence. So we have here again the exploitation by self reference of the systematic ambiguity of a variable term.
It remains to consider whether the ‘sv’ of the Lukasiewicz Paradox, which has the form of a truth locution, admits of self referent use.
Now it has been shown that the circular ‘s’ of the paradox is not a well formed sentence. This use of ‘sv’ commits the fallacy of permitting a value of its subject term by which the subject-predicate form of ‘sv’ is not conserved; and the result is that no pattern of inference for reasoning from a sentence in the subject-predicate form is applicable to this ‘s’.
Next. No matter how one tries to reinterpret the statement ‘s’, it seems to turn out at best to be a category mistake. Its predicate being one of the complimentary pair (“true”, “not-true”), it is impossible to enlarge the scope of the variable subject term of ‘sv’ to permit as values those classes of sentences which are neither true nor false, whether they be verbal sentences, or category mistakes, or infinite regressions, or vicious circles, or whatever. Nor does there seem to be any way of interpreting ‘s’ to be about a true or false statement. Thus is may safely be concluded that, as far as one can see, there is no significant, self referent use of ‘sv’; and it follows that there is no such thing as a singular, self referent truth locution.
Does this, then, imply that the use of the predicate “true” fails within natural language? Indeed not. To be sure, if there were a reason that compelled us to use ‘sv’ self referently, then there would be a conflict of criteria and a paradox. But in the absence of such a reason, we may simply conclude that the self referent use of ‘sv’ involves a use of “true” that is a category mistake. Nor is this at all odd, since it is notorious that the use of “true” is peculiarly exposed to this fallacy.
For the rest, it is surely evident that in so far as its expressive value is concerned, the singular self referent sentence is a mere curiosity. Nonetheless, this logicians brainchild has a different kind of utility. By confronting a statement with the verbal sentence used to make it, this construction provides a highly suggestive context for the further study of the distinction between the statement and the verbal sentence. Thus we see once again how a self reference can be useful to logic itself.

Notas a pie de página

* I wish to emphasize that in this paper I am concerned only with the singular self referent sentence, and not with the general proposition which may have a reference to itself as a case of the generalization. It does not seem to me that the latter kind of self reference involves vicious circularity.
**It is noted that this hypothesis might perhaps be developed in a more fundamental way in the terms of a logical group theory in which the group is associated with a universe of discourse; but in view of its great difficulties, this focus on the problem is not adopted here. —See Note 15 of the paper.