1. On Denoting and its history Harm Boukema

2. Everyone agrees that “the golden mountain does not exist” is a true proposition. But it has, apparently, a subject, “the golden mountain”, and if this subject did not designate some object, the proposition would seem to be meaningless. Meinong inferred that there is a golden mountain, which is golden and a mountain, but does not exist. He even thought that the existent golden mountain is existent, but does not exist. This did not satisfy me, and the desire to avoid Meinong’s unduly populous realm of being led me to the theory of descriptions. What was of importance in this theory was the discovery that, in analysing a significant sentence, one must not assume that each separate word or phrase has significance on its own account. “The golden mountain” can be part of a significant sentence, but is not significant in isolation. It soon appeared that class- symbols could be treated like descriptions, i.e., as non-significant parts of significant sentences. This made it possible to see, in a general way, how a solution of the contradictions might be possible. My Mental Development (1944)

3. In the belief that propositions must, in the last analysis, have a subject and predicate, Leibniz does not differ either from his predecessors or from his successors. Any philosophy which uses either substance or the Absolute will be found, on inspection, to depend on this belief. Kant’s belief in an unknowable thing-in-itself was largely due to the same theory. It cannot be denied, therefore, that the doctrine is important. Philosophers have differed, not so much in respect of belief in its truth, as in respect of their consistency in carrying it out. In this latter respect, Leibniz deserves credit. The Philosophy of Leibniz (1900): Section 10

5. Next after subject-predicate propositions come two types of propositions which appear equally simple. These are the propositions in which a relation is asserted between two terms, and those in in which two terms are said to be two. The Principles of Mathematics (1903): §94

6. Indeed it may be said that the logical purpose which is served by the theory of denoting is, to enable propositions of finite complexity to deal with infinite classes of terms: this object is effected by all, any, and every, and if it were not effected, every general proposition about an infinite class would have to be infinitely complex. Now, for my part, I see no possible way of deciding whether propositions of infinite complexity are possible or not; but this at least is clear, that all the propositions known to us (and, it would seem, all propositions that we can know) are of finite complexity. It is only by obtaining such propositions about infinite classes that we are enabled to deal with infinity; and it is a remarkable and fortunate fact that this method is successful. Thus the question whether or not there are infinite unities must be left unresolved; the only thing we can say, on this subject, is that no such unities occur in any department of human knowledge, and therefore none such are relevant to the foundation of mathematics. The Principles of Mathematics (1903): §141

7. The use of inverted commas may be explained as follows. When a concept has meaning and denotation, if we wish to say anything about the meaning, we must put it in an entity-position; but if we it itself in an entity-position, we shall be really speaking about the denotation, not the meaning, for that is always the case when a denoting complex is put in an entity-position. Thus in order to speak about the meaning, we must substitute for the meaning something which denotes the meaning. Hence the meanings of denoting complexes can only be approached by means of complexes which denote those meanings. This is what complexes in inverted commas are. If we say ‘“any man” is denoting a complex’, “any man” stands for ‘the meaning of the complex “any man”’, which is a denoting concept. But this is circular; fir we use “any man” in explaining “any man”. And the circle is unavoidable. For if we say “the meaning of any man”, that will stand for the meaning of the denotation of any man, which is not what we want. On Fundamentals (1905): § 35

8. It might be supposed that the whole matter could be simplified by introducing a relation of denoting: instead of all the complications about “C” and C, we might try to put “x denotes y”. But we want to be able to speak of what x denotes, and unfortunately “what x denotes” is a denoting complex. We might avoid this as follows: Let C be an unambiguously denoting complex (we may now drop the inverted commas); then we have (  y): C denotes y: C denotes z.  z.z=y Then what is commonly expressed by  ‘C will be replaced by (  y): C denotes y: C denotes z.  z.z=y:  ‘y On Fundamentals (1905): § 40

9. The most convenient view might seem to be to take everything and anything as primitive ideas, putting (x).  ‘x.=.  ‘(everything) (x).  ‘x.=.  ‘(anything). But it seems that on this view everything and anything are denoting concepts involving all the difficulties considered in 35-39, on account of which we adopted the theory of 40. We shall have to distinguish between “everything” and everything, i.e. we shall have: “everything” is not everything, but only one thing. Also we shall find that if we attempt to say anything about the meaning of “everything”, we must do so by means of a denoting concept which denotes that meaning, and which must not contain that meaning occurring as entity, since when it occurs as entity it stands for its denotation, which is not what we want. These objections, to all appearance, are as fatal here as they were in regard to the. Thus it is better to find some other theory. On Fundamentals (1905): §44

10. The interesting and curious point is that, by driving denoting back and back as we have been doing, we get it all reduced to the one notion of any, from which I started at first. This one notion seems to be presupposed always, and to involve in itself all the difficulties on account of which we have rejected other denoting concepts. Thus we are left with the task of concocting de novo a tenable theory of any, in which denoting is not used. The interesting point which we have a elicited above is that any is a genuinely more fundamental than other denoting concepts; they can be explained byit, but not it by them. And any itself is not fundamental in general, but only in the shape of anything. On fundamentals (1905): §47

11. The above gives a reduction of all propositions in which denoting phrases occur to forms in which no such phrases occur. Why it is imperative to effect such a reduction, the subsequent discussion will endeavour to show. The evidence for the above theory is derived from the difficulties which seem unavoidable if we regard denoting phrases as standing for genuine constituents of the propositions in whose verbal expressions they occur. Of the possible theories which admit such constituents the simplest is that of Meinong. On Denoting (1905): p. 428

12. The interpretation of such phrases is a matter of considerable difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. On denoting (1905): p. 479

13. Of the many other consequences of the view I have been advocating, I will say nothing. I will only beg the reader not to make up his mind against the view – as he might be tempted to do, on account of its apparently excessive complication – until he has attempted to construct a theory of his own on the subject of denotation. This attempt, I believe, will convince him that, whatever the true theory may be, it cannot have such a simplicity as one might have expected beforehand. On Denoting (1905): p. 493

14. The relation of the meaning to the denotation involves certain rather curious difficulties, which seem in themselves sufficient to prove that the theory which leads to such difficulties must be wrong. On Denoting (1905): p. 485

15. Thus all phrases (other than propositions) containing the word the (in the singular) are incomplete symbols: they have a meaning in the use, but not in isolation. For “the author of Waverley” cannot mean the same as “Scott,” or “Scott is the author of Waverley” would mean the same as “Scott is Scott,” which it plainly does not; nor can “the author of Waverly” mean anything other than “Scott,” or “Scott is the author of Waverley” would be false. Hence “the author of Waverley” means nothing. Principia Mathematica (1910) p. 67