The Hierarchy of Functions and Propositions. Bertrand Russel's QP Tip #33.

Bernard Krigstein

         

The Hierarchy of

Functions and Propositions.

Bertrand Russel's QP Tip #33.

             

RED – Comic Book Shaman ‘clarifications’     

Orange and Pink – Comic Book Shaman emphasis        

Brown – denoted nouns

(person place thing (idea or function)

Purple - superlative

         

By a "propositional function" we mean something which contains a variable x [(xx)], and expresses a proposition as soon as a (another; distinct from) value is assigned to x [(xy)]. That is to say, it differs from a proposition solely by the fact that it is ambiguous [(it contains a variable of which the value is unassigned a person’s worth)]. It agrees with the ordinary functions of mathematics in the fact of containing an unassigned variable: where it differs is in the fact that the values of the function are propositions. Thus e.g. "x is a man" or "sin x = 1 " is a propositional function. We shall find that it is possible to incur a vicious-circle fallacy at the very outset, by admitting; as possible [(new)] arguments to a propositional function, terms which presuppose the function. This form of the fallacy is very instructive, and its avoidance leads, as we shall see, to the hierarchy of types. The question as to the nature of a function^†1 is by no means an easy one. It would seem, however, that the essential characteristic of a function is ambiguity. Take, for example, the law of identity in the form "A is A," which is the form in which it is usually enunciated. It is plain that, regarded psychologically, we have here a single judgment. But what are we to say of the object of the judgment? We are not judging that Socrates is Socrates, nor that Plato is Plato, nor any other of the definite judgments that are instances of the law of identity. Yet each of these judgments is, in a sense, within the scope of our judgment. We are in fact judging an ambiguous instance of the propositional function " A is A." We appear to have a single thought which does not have a definite object, but has as its object an undetermined one of the values of the function "A

When we say that "φx" ambiguously denotes φa, φb, φc, etc., we mean that  “φx" means one of the objects φa, φb, φc, etc., though not a definite one, but an undetermined one. It follows that "φx" only has a well-defined meaning (well-defined, that is to say, except in so far as it is of its essence to be ambiguous) if the objects φa, φb, φc, etc., are well-defined. That is to say, a function is not a well-defined function unless all its values are already well defined. It follows from this that no function can have among its values anything which presupposes the function, for if it had, we could not regard the objects ambiguously denoted by the function as definite until the function was definite, while conversely, as we have just seen, the function cannot be definite until its values are definite. This is a particular case, but perhaps the most fundamental case of the vicious-circle principle, [to be sure of our complex choices)]. A function is what ambiguously denotes some one of a certain totality, namely the values of the function; hence this totality cannot contain any members which involve the function, since, if it did, it would contain members involving the totality, which, by the vicious-circle principle, no totality can do.

It will be seen that, according to the above account, the values of a function are presupposed by the function, not vice versa. It is sufficiently obvious, in any particular case, that a value of a function does not presuppose the function. Thus for example the proposition "Socrates is human" can be perfectly apprehended without regarding it as a value of the function "x is human." It is true that, conversely, a function can be apprehended without its being necessary to apprehend its values severally and individually. If this were not the case, no function could be apprehended at all, since the number of values (true and false) of a function is necessarily infinite and there are necessarily possible arguments with which we are unacquainted. What is necessary is not that the values should be given individually and extensionally, but that the totality of the values should be given intentionally, so that, concerning any assigned object, it is at least theoretically determinate whether or not the said object is a value of the function.

It is necessary practically to distinguish the function itself from an undetermined value of the function. We may regard the function itself as that which ambiguously denotes, while an undetermined value of the function is that which is ambiguously denoted. [(fine distinctions are required grok-ing in both metaphysics AND quantum physics)] If the undetermined value is written "φx," we will write the function itself "φx^."  (Any other letter may be used in place of x.) Thus we should say " φx is a proposition," but "φx^ is a propositional function." When we say "φx is a proposition," we mean to state something which is true for every possible value of x, though we do not decide what value x is to have. We are making an ambiguous statement about any value of the function. But when we say "φx^ is a function," we are not making an ambiguous statement. It would be more correct to say that we are making a statement about an ambiguity, taking the view that a function is an ambiguity. The function itself, φx^ is the single thing which ambiguously denotes its many values; while φx, where x is not specified, is one of the denoted objects, with the ambiguity belonging to the manner of denoting [(fine distinction)].

We have seen that, in accordance with the vicious-circle principle, the values of a function cannot contain terms only definable in terms of the function. Now given a function φx^, the values for the function^†2 are all pro positions of the form φx. It follows that there must be no propositions, of the form φx, in which x has a value which involves φx^. (If this were the case, the values of the function would not all be determinate until the function was determinate, whereas we found that the function is not determinate unless its values are previously determinate.) [(he said; she said)] Hence there must be no such thing as the value for φx^ with the argument φx^, or with any argument which involves φx. That is to say, the symbol "φ(φx^) " must not express a proposition, as " φa" does if φa is a value for φx^. In fact "φ(φx^)" must be a symbol which does not express anything: we may therefore say that it is not significant. Thus given any function φx^, there are arguments with which the function has no value, as well as arguments with which it has a value. We will call the arguments with which φx^ has a value "possible values of x." We will say that φx^ is "significant with the argument x" when φx^ has a value with the argument x.

^†1 When the word "function" is used in the sequel, "propositional function" is always meant. Other functions will not be in question in the present Chapter. [(fine distinction)]

^†2 We shall speak in this Chapter of "values for φx^" and of "values of φx," meaning in each case the same thing, namely φa, φb, φc, etc. The distinction of phraseology serves to avoid ambiguity where several variables are concerned, especially when one of them is a function. [(super-fine distinction)]