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While explaining why a function cannot take itself as an argument, I realize that the vicious-circle principle says nothing about why a function cannot take one of its values as argument.

The vicious-circle principle says F(Fŷ) is meaningless - please notice the hat - where Fŷ is the function denoting a totality. This I totally understand.

But the vicious circle principle didn't say anything about F(Fy). Fy - no hat here - is one of the values of the function Fŷ.

In other words, by what principle is "Socrates is a man is a man" considered false or nonsense?

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  • @JohnAm - Invoking rules of syntax to dispel a paradox is a proof of logical incapacity, because philosophy asks why there is such rule in the first place.
    – George Chen
    Mar 18, 2017 at 12:06
  • By the theory of types, "'Socrates is a ma' is a man" is nonsense, and the the examples you gave are "typically ambiguous", i.g. "a sentence" should be spelt out as "a 2nd-order sentence," "a 3rd-order sentence," etc. The theory of types rises from the avoidance of vicious-circles, but vicious-circle principle says nothing about why a function cannot take one of its values as argument.
    – George Chen
    Mar 18, 2017 at 13:12
  • @MauroALLEGRANZA - Theory of types specifically. I think the Theory of Types is stricter than what the vicious-circle principle can warrant. But I don't want to set the tone of attacking ToT, there is no shortage of attackers on this site. Based on my personal experience, chances are I do not fully understand ToT yet.
    – George Chen
    Mar 18, 2017 at 13:27
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    John makes a fair enough point. The title of the question is "By what principle is "'Socrates is a man' is a man" considered false or nonsense?" and the last sentence in the question is "In other words, by what principle is "'Socrates is a man' is a man" considered false or nonsense?" and the vicious cycle principle isn't a topic that is exclusive to PM. Nothing about this question screams "I only want answers having to do with type-theory" or "this should only be in regards to type-theory." What it screams is "by what principle is "'Socrates is a man' is a man" considered false or nonsense?"
    – Not_Here
    Mar 18, 2017 at 16:38
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    Why is this nonsense, instead of being false? Seems like it is trivially false because no statement is a man.
    – user9166
    Jul 17, 2017 at 23:54

4 Answers 4

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We can see: Alfred North Whitehead & Bertrand Russell, Principia Mathematica to #56, Cambridge UP (2nd ed, 1927), Introduction: Ch.II THE THEORY OF LOGICAL TYPES, page 39-40:

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.

It is necessary practically to distinguish the function itself from an undetermined value of the function.[...] If the undetermined value is written "ϕy," we will write the function itself "ϕŷ."

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 ϕŷ, the values for the function [we shall speak of "values for ϕŷ" and of "values of ϕy," meaning in each case the same thing, namely ϕa, ϕb, ϕc, etc.] are all propositions of the form ϕy. It follows that there must be no propositions, of the form ϕy, in which y has a value which involves ϕŷ. [...] Hence there must be no such thing as the value for ϕŷ with the argument ϕŷ, or with any argument which involves ϕŷ.

We have to be careful with the similar (but different) symbols:

Socartes is a man

is a proposition.

The (propositional) funcion is: ŷ is a man.

The expression y is a man stay for a "generic" value of the function ŷ is a man, i.e. for: Socartes is a man, Plato is a man, etc.

The vicious circle principle forbid to form a proposition of the form y is a man with some value of y that involves ŷ is a man. A fortiori, we cannot use ŷ is a man itself as value for y, i.e. we cannot write:

(ŷ is a man) is a man.

What about y is a man ? It is a "generic" name for the (meaningful) expressions: Socartes is a man, etc.

So the question amounts to: why (Socartes is a man) is a man is forbidden by the vicious circle principle ?

It is not; it is forbidden by the (not so clearly stated) syntax rules.

ŷ is a man is a first-order functions [see page 51], i.e. a function that involves no variables except individuals.

Thus, the possible value for its argument must be (names of) individuals [the type of the argument of the function must be the "lowest" one], like Socrates, Plato, etc. and not (names of) propositions, like: Socrates is bald.

In conclusion, with an "abuse of terminology" with respect to PM, we have that (Socartes is a man) is a man is ill-formed with respect to the (not clearly stated) PM syntax rules, irrespective of the vicious circle principle.

Compare with Mathematical logic as base on the theory of types (1908):

every propositional function has a certain range of significance, within which lie the arguments for which the function has values. Whitin this range of arguments, the function is true or false.; outside this range it is nonsense.

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  • At least the order of presupposition is clear. It seems that the authors supposed that there is book that contains all the values of "x is a man" - either true or false, but all significant - but by what principle this book is compiled, the authors didn't say.
    – George Chen
    Mar 18, 2017 at 14:54
  • @GeorgeChen - The "ontology" of Principia is not clearly stated and it is often "intermingled" with the syntactical rules: thus the well-known charge against it (inter alia: Godel) of lack of a clear syntax-semantics discrimination. On this issue, PM must be supplemented with PoM.
    – Mauro ALLEGRANZA
    Mar 18, 2017 at 15:00
  • Having said that, see page 14: "Propositional functions. Let ϕx be a statement containing a variable x and such that it becomes a proposition when x is given any fixed determined meaning. Then ϕx is called a "propositional function"; it is not a proposition, since owing to the ambiguity of x it really makes no assertion at all. Thus "x is hurt" really makes no assertion at all, till we have settled who x is."
    – Mauro ALLEGRANZA
    Mar 18, 2017 at 15:01
  • page 15: "The range of values and total variation. Thus corresponding to any propositional function ϕŷ, there is a range, or collection, of values, consisting of all the propositions (true or false) which can be obtained by giving every possible determination to y in ϕy. A value of y for which ϕŷ is true will be said to "satisfy" ϕŷ.
    – Mauro ALLEGRANZA
    Mar 18, 2017 at 15:04
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    Page 15 and 42 are illuminating. Rage of values in an integral part of a propositional function, which is common sense in mathematics but implicit in ordinary language.
    – George Chen
    Mar 18, 2017 at 16:41
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  1. When you put a word or phrase between quotes,

'James'

'The man who sold the world'

conventionally you are saying that you are not referring to the referrent of those words or phrases, but to the words or phrases themselves.

So, by 'Socrates is a man', between quotes, we mean the sentence, so that the following propositions are true:

'Socrates is a man' has four words

'Socrates is a man' begins with 'S'

'Socrates is a man' is a proposition

However, the following proposition is usually accepted as true:

A - A sentence is not a man.

So, when you wrap the words within quotes, you are implicitely stating that

B - 'Socrates is a man' is a sentence.

From A and B,

C - 'Socrates is a man' is not a man.

So, it isn't the case that "'Socrates is a man' is a man" is nonsense, but the case that it is false, for its negation is true.

As you see,

'Dances with wolves' was not a man, but

Dances with wolves was a man.

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Here is how the philosophical minds work:

First, they discovered that some sentences are neither true nor false but meaningless.

Then they pointed out that the root of the problem is that the range of significance is not clearly defined.

Then they began to speculate the boundaries of the range of significance.

Finally, they worked out a theory that clearly bars some terms from entering into the range of significance but says little about what constitutes significance: the Theory of Types judges "Socrates is a man is a man" as nonsensical but says nothing about whether "quadrupliciy is a man" is false or meaningless.

Stopping short of being precise is typical of philosophical work: when one does not know the precise distinction between bald and not bald, he can work out a theory that says, give a square inch of scalp, less than 5 hair on it definitely constitutes baldness, more than 200 is definitely not bald.

Significance is more fundamental than the theory of types. In mathematics, the range of significance is part of the definition of a function and is clearly defined; in the ordinary language the range of significance should have been clearly defined but, at its current stage, it is not. The Theory of Types partially addressed this problem, and, by classifying arguments in terms of types, greatly extended the realm of mathematics, which is certain and precise, into the former domain of philosophy, which was mostly speculative.

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Here is a long comment:

  1. The Theory of Types is consistent with both the vicious-circle principle and the range of significance principle.

    1.1 All the arguments that are significant by the Theory of Types are also within the range of significance, although sometimes regarding whether the argument is forbidden or not the Theory of Types is mute.

    1.2 When an argument is explicitly forbidden by the Theory of Types, it is also beyond the range of significance.

    1.3 Some arguments, which the Theory of Types does not explicitly forbid or permit, are forbidden by the range of significance.

  2. The range of significance is an integral part of a propositional function's definition.

    2.1 In the case of such functions as "arcsine(x) = 0", the range of significance is explicit, [-1, 1].

    2.2 In the case of ordinary language, the range of significance is often unclear, which is the cause of many confusions.

    2.3 The Theory of Types greatly improved upon the clarity of ordinary language by explicitly forbids a range of arguments from being part of the range of significance and by pointing out that the source of this class of confusion is the lack of a clear definition of the range of significance.

    2.4 When the range of significance is unclear, the Theory of Types greatly narrows the grey area between significant and insignificant values; when the range of significance is clearly defined, the theory of types is redundant.

  3. Where the significance of an argument is dubious, there are simple tricks to test it without invoking the theory of types:

    3.1 By looking at what the resulting value of a propositional function implies and see if it is capable of being either true or false:

    For example, "x is a man" implies "x is lighter than the earth." (This is not a purely logical implication; there are empirical premises.)

    If "quadruplicity is heavier than the earth" is false, then the negation would be true, which is not the case. Thus "quadrupliciy" is beyond the range of significance.

    Same can be said about why the proposition "Socrates is a man" cannot be a meaningful argument to "x is a man."

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