28. Possible Arguments for Functions

Harold R. (Hal) Foster’s Prince Valiant

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Possible arguments for functions   

When it is said that e.g. "φ(φz^)" is meaningless, and therefore neither true nor false, it is necessary to avoid a misunderstanding. If “φ(φz^)" were interpreted as meaning "the value for φz^ with the argument φz^ is true," that would be not meaningless, but false. It is false for the same reason for which "the King of France is bald" is false, namely because there is no such thing as "the value for φz^ with the argument φz^." But when, with some argument a, we assert φa we are not meaning to assert "the value for φx^ with the argument a is true"; we are meaning to assert the actual proposition which is the value for φx^ with the argument a. Thus for example if φx^ is "x^ is a man," φ(Socrates) will be "Socrates is a man,” not "the value for the function ‘x^ is a man,' with the argument Socrates, is true." Thus in accordance with our principle that "φ(φz^)" is meaningless, we cannot legitimately deny "the function 'x is a man' is a man," because this is nonsense, but we can  legitimately deny " the value for the function 'x^ is a man' with the argument 'x^ is a man' is true," not on the ground that the value in question is false, but on the ground that there is no such value for the function.

We will denote by the symbol "(x). φx" the proposition " φx always*," i.e. the proposition which asserts all the values for φx^. This proposition involves the function φx^, not merely an ambiguous value of the function. The assertion of φx, where x is unspecified, is a different assertion from the one which asserts all values for φx^ for the former is an ambiguous assertion, whereas the latter is in no sense ambiguous. It will be observed that "(x) . φx" does not assert     "φx with all values of x," because, as we have seen, there must be values of x with which "φx " is meaningless. What is asserted by "(x) . φx" is all propositions which are values for φx^; hence it is only with such values of x as make "φx" significant, i.e. with all possible arguments, that φx is asserted when we assert "(x) . φx." Thus a convenient way to read "(x) . φx" is "φx is  true with all possible values of x." This is, however, a less accurate reading than " φx always," because the notion of truth is not part of the content of what is judged. When we judge "all men are mortal," we judge truly, but the notion of truth is not necessarily in our minds, any more than it need be when we judge "Socrates is mortal."

* We use "always" as meaning "in all cases," not "at all times." Similarly "sometimes" will mean "in some cases."

'Wringle Wrangle" - Pencils and Inks: Jesse Marsh   © Respective copyright/trademark holders.

Studying Principia Mathematica is as a path to ambiguity. The system that we may learn is individual, and I infer it to be of various use to each who attempts to learn it. It is rater unique because the authors Alfred North Whitehead and Bertrand Russell saw that the system was a vehicle rather than a destination, and that some readers would be defeated, some readers would be right with them, and some readers would continue on with modifications they hadn’t anticipated. Each of the posts up to here, and also those to come, are layers of fine distinction in both thought (perception) and expression, by whatever form it takes. There are aspects, such as the vicious-circle principle, that are challenging to conscious perception. I have experienced great feelings of joy and amusement in reading and re-reading various passages of this text, delighted at the amazing crazy fine distinctions beyond anything I expected.

The nature of taking in our system cannot be subject to conscious recitation, instead a better gauge is to turn back and read the posts again, and when you find that you can breeze thru them easily and understand them as you do a comic strip that is the key to this bus. There are two aspects at play in our text, ‘the system of symbols and their form in use’ and ‘the fine distinctions that are the reasons for the necessity of the system.’ The forms allow the construction of any sort of propositions, functions, objects, classes and more in a form that immediately (visually) reveals any errors within the structure. This is only possible by the systematic application of this metaphysical system.

Pencils and Inks: Ken Bald © Respective copyright holder.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.