Mathematical Logic as based on the Theory of Types


Published in: American Journal of Mathematics, vol.30(1908), pp. 222-262
Transcribed into hypertext by Burtzev B.I., Jul., 07, 2003
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The Meaning and Range of Generalized Propositions.

BH.In this section we have to consider first the meaning of propositions in which the word all occurs, and then the kind of collections which admit of propositions about all their members.
BI.It is convenient to give the name generalized propositions not only to such as contain all, but also to such as contain some (undefined). The proposition φx is sometimes true is equivalent to the denial of not-φx is always true; some A is B is equivalent to the denial of all A is not B; i.e., of no A is B. Whether it is possible to find interpretations which distinguish φx is sometimes true from the denial of not-φx is always true, it is unnecessary to inquire; for our purposes we may define φx is sometimes true as the denial of not-φx is always true. In any case, the two kinds of propositions require the same kind of interpretation, and are subject to the same limitations. In each there is an apparent variable; and it is the presence of an apparent variable which constitutes what I mean by a generalized proposition. (Note that there, can not be a real variable in any proposition; for what contains a real variable is a propositional function, not a proposition.)
BJ.The first question we have to ask in this section is: How are we to interpret the word all in such propositions as all men are mortal? At first sight, it might be thought that there could be no difficulty, that all men is a perfectly clear idea, and that we say of all men that they are mortal. But to this view there are many objections.
BK.(1) If this view were right, it would seem that all men are mortal could not be true if there were no men. Yet, as Mr. Bradley has urged,13) Trespassers will be prosecuted may be perfectly true even if no one trespasses; and hence, as he further argues, we are driven to interpret such propositions as hypotheticals, meaning if anyone trespasses, he will be prosecuted. i.e., if x trespasses, x will be prosecuted, where the range of values which x may have, whatever it is, is certainly not confined to those who really trespass. Similarly all men are mortal will mean if x is a man, x is mortal, where x may have any value within a certain range. What this range is, remains to be determined; but in any case it is wider than men, for the above hypothetical is certainly often true when x is not a man.
BL.(2) All men is a denoting phrase; and it would appear, for reasons which I have set forth elsewhere,14) that denoting phrases never have any meaning in isolation, but only enter as constituents into the verbal expression of propositions which contain no constituent corresponding to the denoting phrases in question. That is to say, a denoting phrase is defined by means of the propositions in whose verbal expression it occurs. Hence it is impossible that these propositions should acquire their meaning through the denoting phrases; we must find an independent interpretation of the propositions containing such phrases, and must not use these phrases in explaining what such propositions mean. Hence we can not regard all men are mortal as a statement about all men.
BM.(3) Even if there were such an object as all men, it is plain that it is not this object to which we attribute mortality when we say all men are mortal. If we were attributing mortality to this object, we should have to say all men is mortal. Thus the supposition that there is such an object as all men will not help us to interpret all men are mortal.
BN.(4) It seems obvious that, if we meet something which may be a man or may be an angel in disguise, it comes within the scope of all men are mortal to assert if this is a man, it is mortal. Thus again, as in the case of the trespassers, it seems plain that we are really saying if anything is a man, it is mortal, and that the question whether this or that is a man does not fall within the scope of our assertion, as it would do if the all really referred to all men.
BO.(5) We thus arrive at the view that what is meant by all men are mortal may be more explicitly stated in some such form as it is always true that if x is a man, x is mortal. Here we have to inquire as to the scope of the word always.
BP.(6) It is obvious that always includes some cases in which x is not a man, as we saw in the case of the disguised angel. If x were limited to the case when x is a man, we could infer that x is a mortal, since if x is a man, x is a mortal. Hence, with the same meaning of always, we should find it is always true that x is mortal. But it is plain that, without altering the meaning of always, this new proposition is false, though the other was true.
BQ.(7) One might hope that always would mean for all values of x. But all values of x, if legitimate, would include as parts all propositions and all functions, and such illegitimate totalities. Hence the values of x must be somehow restricted within some legitimate totality. This seems to lead us to the traditional doctrine of a universe of discourse within which x must be supposed to lie.
BR.(8) Yet it is quite essential that we should have some meaning of always which does not have to be expressed in a restrictive hypothesis as to x. For suppose always means whenever x belongs to the class i. Then all men are mortal becomes whenever x belongs to the class i, if x is a man, x is mortal; i.e., it is always true that if x belongs to the class i, then, if x is a man, x is mortal. But what is our new always to mean? There seems no more reason for restricting x, in this new proposition, to the class i, than there was before for restricting it to the class man. Thus we shall be led on to a new wider universe, and so on ad infinitum, unless we can discover some natural restriction upon the possible values of (i.e., some restriction given with) the function if x is a man, x is mortal, and not needing to be imposed from without.
BS.(9) It seems obvious that, since all men are mortal, there can not be any false proposition which is a value of the function if x is a man, x is mortal. For if this is a proposition at all, the hypothesis x is a man must be a proposition, and so must the conclusion x is mortal. But if the hypothesis is false, the hypothetical is true; and if the hypothesis is true, the hypothetical is true. Hence there can be no false propositions of the form if x is a man, x is mortal.
BT.(10) It follows that, if any values of x are to be excluded, they can only be values for which there is no proposition of the form if x is a man, x is mortal; i.e., for which this phrase is meaningless. Since, as we saw in (7), there must be excluded values of x, it follows that the function if x is a man, x is mortal must have a certain range of significance,15) which falls short of all imaginable values of x, though it exceeds the values which are men. The restriction on x is therefore a restriction to the range of significance of the function if x is a man, x is mortal.
BU.(11) We thus reach the conclusion that all men are mortal means if x is a man, x is mortal, always, where always means for all values of the function if x is a man, x is mortal. This is an internal limitation upon x, given by the nature of the function; and it is a limitation which does not require explicit statement, since it is impossible for a function to be true more generally than for all its values. Moreover, if the range of significance of the function is i, the function if x is an i, then if x is a man, x is mortal has the same range of significance, since it can not be significant unless its constituent if x is a man, x is mortal is significant. But here the range of significance is again implicit, as it was in if x is a man, x is mortal; thus we can not make ranges of significance explicit, since the attempt to do so only gives rise to a new proposition in which the same range of significance is implicit.
BV.Thus generally: (x).φx is to mean φx always. This may be interpreted, though with less exactitude, as φx is always true, or, more explicitly: All propositions of the form φx are true, or All values of the function φx are true.16) Thus the fundamental all is all values of a propositional function, and every other all is derivative from this. And every propositional function has a certain range of significance, within which lie the arguments for which the function has values. Within this range of arguments, the function is true or false; outside this range, it is nonsense.
BW.The above argumentation may be summed up as follows:
BX.The difficulty which besets attempts to restrict the variable is, that restrictions naturally express themselves as hypotheses that the variable is of such or such a kind, and that, when so expressed, the resulting hypothetical is free from the intended restriction. For example, let us attempt to restrict the variable to men, and assert that, subject to this restriction, x is mortal is always true. Then what is always true is that if x is a man, x is mortal; and this hypothetical is true even when x is not a man. Thus a variable can never be restricted within a certain range if the propositional function in which the variable occurs remains significant when the variable is outside that range. But if the function ceases to be significant when the variable goes outside a certain range, then the variable is ipso facto confined to that range, without the need of any explicit statement to that effect. This principle is to be borne in mind in the development of logical types, to which we shall shortly proceed.
BY.We can now begin to see how it comes that all so-and's is sometimes a legitimate phrase and sometimes not. Suppose we say all terms which have the property φ have the property ψ. That means, according to the above interpretation, φx always implies ψx. Provided the range of significance of φx is the same as that of ψx, this statement is significant; thus, given any definite function φx, there are propositions about all the terms satisfying φx. But it sometimes happens (as we shall see more fully later on) that what appears verbally as one function is really many analogous functions with different ranges of significance. This applies, for example, to p is true, which, we shall find, is not really one function of p, but is different functions according to the kind of proposition that p is. In such a case, the phrase expressing the ambiguous function may, owing to the ambiguity, be significant throughout a set of values of the argument exceeding the range of significance of any one function. In such a case, all is not legitimate. Thus if we try to say all true propositions have the property φ, i.e., p is true always implies φp, the possible arguments to p is true necessarily exceed the possible arguments to φ, and therefore the attempted general statement is impossible. For this reason, genuine general statements about all true propositions can not be made. It may happen, however, that the supposed function φ is really ambiguous like p is true; and if it happens to have an ambiguity precisely of the same kind as that of p is true we may be able always to give an interpretation to the proposition p is true implies φp. This will occur, e.g., if φp is not-p is false Thus we get an appearance, in such cases, of a general proposition concerning all propositions; but this appearance is due to a systematic ambiguity about such words as true and false. (This systematic ambiguity results from the hierarchy of propositions which will be explained later on). We may, in all such cases, make our statement about any proposition, since the meaning of the ambiguous words will adapt itself to any proposition. But if we turn our proposition into an apparent variable, and say something about all, we must suppose the ambiguous words fixed to this or that possible meaning, though it may be quite irrelevant which of their possible meanings they are to have. This is how it happens both that all has limitations which exclude all propositions, and that there nevertheless seem to be true statements about all propositions. Both these points will become plainer when the theory of types has been explained.
BZ.It has often been suggested17) that what is required in order that it may be legitimate to speak of all of a collection is that the collection should be finite. Thus all men are mortal will be legitimate because men form a finite class. But that is not really the reason why we can speak of all men. What is essential, as appears from the above discussion, is not finitude, but what may be called logical homogeneity. This property is to belong to any collection whose terms are all contained within the range of significance of some one function. It would always be obvious at a glance whether a collection possessed this property or not, if it were not for the concealed ambiguity in common logical terms such as true and false, which gives an appearance of being a single function to what is really a conglomeration of many functions with different ranges of significance.
CA.The conclusions of this section are as follows: Every proposition containing all asserts that some propositional function is always true; and this means that all values of the said function are true, not that the function is true for all arguments, since there are arguments for which any given function is meaningless, i. e., has no value. Hence we can speak of all of a collection when and only when the collection forms part or the whole of the range of significance of some propositional function, the range of significance being defined as the collection of those arguments for which the function in question is significant, i.e., has a value.



13) Logic, Part I, Chapter II  back to text13

14) On Denoting, Mind, October, 1905.  back to text14

15) A function is said to be significant for the argument x if it has a value for this argument. Thus we may say shortly φx is significant, meaning the function φ has a value for the argument x. The range of significance of a function consists of all the arguments for which the function is true, together with all the arguments for which it is false.  back to text15

16) A linguistically convenient expression for this idea is: φx is true for all possible values of x, a possible value being understood to be one for which φx is significant.  back to text16

17) E. g., by M. Poincare, Revue de Metaphysique et de Morale, Mai, 1906.  back to text17

I.    The Contradictions.
II.   All and Any
III.  The Meaning and Range of Generalized Propositions.
IV.   The Hierarchy of Types.
V.    The Axiom of Reducibility.
VI.   Primitive Ideas and Propositions of Symbolic Logic.
VII.  Elementary Theory of Classes and Relations.
VIII. Descriptive Functions.
IX.   Cardinal Numbers.
X.    Ordinal Numbers.
I.    .
II.   “, ”     “, ”
III.  ...

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