44. Expressing Scope of Classes in Notation

Harold R. (Hal) Foster’s Prince Valiant

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As in the case of ƒ(ιx)(φx), so in that of ƒ{z^(φz)}, there is an ambiguity as to the scope of z^(φz) if it occurs in a proposition which itself is part of a larger proposition. But in the case of classes, since we always have the axiom of reducibility, namely

(∃ψ) : φx . ≡x . ψ!x,

which takes the place of E! (ιx)(φx), it follows that the truth-value of any proposition in which z^(φz) occurs is the same whatever scope we may give to z^(φz), provided the proposition is an extensional function of whatever functions it may contain. Hence we may adopt the convention that the scope is to be always the smallest proposition enclosed in dots or brackets in which z^(φz) occurs. If at any time a larger scope is required, we may indicate it by "[z^(φz)]" followed by dots, in the same way as we did for [(ιx)(φx)].

Similarly when two class symbols occur, e.g. in a proposition of the form ƒ{z^(φz), z^(ψz)}, we need not remember rules for the scopes of the two symbols, since all choices give equivalent results, as it is easy to prove. For the preliminary propositions a rule is desirable, so we can decide that the class symbol which occurs first in the order of writing is to have the larger scope

The representation of a class by a single letter a can now be understood. For the denotation of ⍺ is ambiguous, in so far as it is undecided as to which of the symbols z^(φz), z^(ψz), z^(χz), etc. it is to stand for, where φz, ψz, χz, etc. are the various determining functions of the class. According to the choice made, different propositions result. But all the resulting propositions are equivalent by virtue of the easily proved proposition:

┣ : φx ≡_x ψx . ⊃ . ƒ(z^(φz)} ≡ ƒ{z^(ψz)}."

Hence unless we wish to discuss the determining function itself, so that the notion of a class is really not properly present, the ambiguity in the denotation of ⍺ is entirely immaterial, though, as we shall see immediately, we are led to limit ourselves to predicative determining functions. Thus "ƒ(⍺)," where ⍺ is a variable class, is really "ƒ{z^(φz))}," where φ is a variable function, that is, it is

"(∃ψ) . φx ≡_x  ψ!x . ƒ{ψ!z^},"

where φ is a variable function. But here a difficulty arises which is removed by a limitation to our practice and by the axiom of reducibility. For the determining functions φz^, ψz^, etc. will be of different types, though the axiom of reducibility secures that some are predicative functions. Then, in interpreting ⍺ as a variable in terms of the variation of any determining function, we shall be led into errors unless we confine ourselves to predicative determining functions. These errors especially arise in the transition to total variation (cf. Post 7). Accordingly

ƒ⍺ = . (∃ψ) . φ!x ≡_x ψ!x . ƒ{ψ!z^} Df.

It is the peculiarity of a definition of the use of a single letter [viz. ⍺] for a variable incomplete symbol that it, though in a sense a real variable, occurs only in the definiendum, while "φ," though a real variable, occurs only in the definiens.

Thus "ƒ⍺^" stands for

"(∃ψ) . φ^!x ≡_x ψ!x . ƒ{ψ!z^},”

and "(⍺).ƒ⍺" stands for

"(φ) : (∃ψ) . φ!x ≡_x ψ!x . ƒ{ψ!z^}."

Accordingly, in mathematical reasoning, we can dismiss the whole apparatus of functions and think only of classes as "quasi-things," capable of immediate representation by a single name. The advantages are two-fold: (1) classes are determined by their membership, so that to one set of members there is one class, (2) the "type" of a class is entirely defined by the type of its members.

Also a predicative function of a class can be defined thus

ƒ!⍺ = . (∃ψ) . φ!x ≡_x  ψ!x . ƒ!{ψ!z^} Df.

Thus a predicative function of a class is always a predicative function of any predicative determining function of the class, though the converse does not hold.

(3) Relations. With regard to relations, we have a theory strictly analogous to that which we have just explained as regards classes. Relations in extension, like classes, are incomplete symbols. We require a division of functions of two variables into predicative and non-predicative functions, again for reasons which have been explained in Chapter II, (Post 26).  We use the notation "φ!(x,y)" for a predicative function of x and y.

We use "φ!(x^, y^)" for the function as opposed to its values; and we use "x^y^φ(x, y)" for the relation (in extension) determined by φ(x, y). We put

  ƒ{x^y^φ (x, y)} . = : (∃ψ) : φ(x, y) . ≡_x,y . ψ!(x, y) : ƒ{ψ!(x, y)}  Df.

Thus even when ƒ{ψ!(x^,y^)} is not an extensional function of ψ. ƒ{x^y^φ(x,y)} is an extensional function of φ. Hence, just as in the case of classes, we deduce

┣:.x^y^)φ(x,y) . = x^y^ψ(x,y) . ≡ : φ(x, y) . ≡_x,y . ψ(x, y),

i.e. a relation is determined by its extension, and vice versa.

On the analogy of the definition of "x ∈ ψ!z^," we put

x{ψ!(x^, y^)} y .=. ψ!(x, y)  Df*.

This definition, like that of "x ∈ ψ!z^," is not introduced for its own sake, but in order to give a meaning to

x{ψ!(^x, y^)} y.

This meaning, in virtue of our definitions, is

(∃ψ) : φ(x, y) . ≡_x,y . ψ!(x, y) : x{ψ!(x^, y^)} y,

i.e.  (∃ψ) : φ(x, y) . ≡_x,y . ψ!(x, y) : ψ!(x, y),

and this, in virtue of the axiom of reducibility

“(∃ψ) : φ(x, y) . ≡_x,y . ψ!(x, y)

is equivalent to φ(x, y).

Thus we have always

┣: x{x^, y^φ (x, y)} y . ≡ . φ(x, y).

Whenever the determining function of a relation is not relevant, we may replace x^y^φ (x, y) by a single capital letter. In virtue of the propositions given above,

┣:. R = S . ≡ : xRy . ≡_x,y . xSy,

≡:. R = x^y^φ(x, y) ≡ : xRy . ≡_x,y . φ(x, y),

and   ┣. R = x^y^(xRy).

Classes of relations, and relations of relations, can be dealt with as classes of classes were dealt with above.

Just as a class must not be capable of being or not being a member of itself, so a relation must not be referent or relatum with respect to itself. This turns out to be equivalent to the assertion that φ!(x^, y^) cannot significantly be either of the arguments x or y in φ!(x, y). This principle, again, results from the limitation to the possible arguments to a function explained at the beginning of Chapter II (see Post 26).

We may sum up this whole discussion on incomplete symbols as follows.

The use of the symbol "(⍳x)(φx)" as if in "ƒ(⍳x)(φx)" it directly represented an argument to the function ƒz^ is rendered possible by the theorems

┣:. E!(⍳x)(φx) . ⊃ : (x) . ƒx . ⊃ . ƒ(⍳x)(φx),

┣:(⍳x)(φx) = (⍳x)(ψx) . ⊃ . ƒ(⍳x)(φx) ≡ ƒ(⍳x)(ψx),

┣: E! (⍳x)(φx) . ⊃ . (⍳x)(φx) = (⍳x)(φx),

┣:(⍳x)(φx) = (⍳x)(ψx) . ≡ (⍳x)(ψx) = (⍳x)(φx),

┣:(⍳x)(φx) = (⍳x)(ψx) . (⍳x)(ψx). = (⍳x)(χx) . ⊃ . (⍳x)(φx) = (⍳x)(χx).

The use of the symbol "x^(φx)" (or of a single letter, such as ⍺, to represent such a symbol) as if, in "ƒ{x^(φx)}," it directly represented an argument ⍺ to a function ƒ⍺^, is rendered possible by the theorems

┣: (⍺). ƒ⍺ . ⊃ . ƒ{x^(φx)},

┣: x^(φx) = x^(ψx) . ƒ{x^(φx)} ≡ ƒ{x^(ψx)},

┣. x^(φx) = x^(φx)

┣:x^(φx) = x^(ψx) . ≡ . x^(ψx) = x^(φx),

┣: x^(φx) = x^(ψx) . x^(ψx) = x^(χx) . ⊃ . x^(φx) = x^(χx).

Throughout these propositions the types must be supposed to be properly adjusted, where ambiguity is possible.

The use of the symbol " x^y^{φ(x, y)}” (or of a single letter, such as R, to represent such a symbol) as if, in "f/{ x^y^ φ (x, y)}," it directly represented an argument R to a function ƒR^, is rendered possible by the theorems

┣: (R) . ƒR . ⊃ . ƒ{x^y^ φ(x, y)},

┣: x^y^ φ(x, y) = x^y^ ψ(x, y) . ⊃ . ƒ{ x^y^ φ(x, y)} ≡ ƒ{ x^y^ ψ(x, y)},

┣. x^y^ φ(x, y) = x^y^ φ(x, y),  

┣: x^y^ φ(x, y) = x^y^ ψ(x, y) . ≡ . x^y^ ψ(x, y) = x^y^ φ(x, y),

┣: x^y^ φ(x, y) = x^y^ φ(x, y) . x^y^ χ(x, y) = x^y^ χ(x, y).

⊃ .  x^y^ φ(x, y) = x^y^ χ(x, y).

Throughout these propositions the types must be supposed to be properly adjusted where ambiguity is possible.

It follows from these three groups of theorems that these incomplete symbols are obedient to the same formal rules of identity as symbols which directly represent objects, so long as we only consider the equivalence of the resulting variable (or constant) values of propositional functions and not their identity. This consideration of the identity of propositions never enters into our formal reasoning.

Similarly the limitations to the use of these symbols can be summed up as follows. In the case of (⍳x)(φx), the chief way in which its incompleteness is relevant is that we do not always have

(x) . ƒx . ⊃ . ƒ(⍳x)(φx),

i.e. a function which is  always true may nevertheless not be true of (⍳x)(φx). This is possible because ƒ(⍳x)(φx) is not a value of ƒx^, so that even when all values of ƒx^ are true, ƒ(⍳x)(φx) may not be true. This happens when (⍳x)(φx) does not exist. Thus for example we have (x) . x = x, but we do not have

the round square = the round square.

The inference (x) . ƒx . ⊃ . ƒ(⍳x)(φx)

is only valid when E! (⍳x)(φx). As soon as we know E! (⍳x)(φx), the fact that (⍳x)(φx) is an incomplete symbol becomes irrelevant so long as we confine ourselves to truth-functions of whatever proposition is its scope. But even when E! (⍳x)(φx), the incompleteness of (⍳x)(φx) may be relevant when we pass outside truth-functions. For example, George IV wished to know whether Scott was the author of Waverley, i.e. he wished to know whether a proposition of the form " c = (⍳x)(φx)" was true. But there was no proposition of the form "c =y" concerning which he wished to know if it was true.

In regard to classes, the relevance of their incompleteness is somewhat different. It may be illustrated by the fact that we may have

z^(φz) = ψ!z^ . z^(φz) = χ!z^

without having  

ψ!^ = χ!z^.

For, by a direct application of the definitions, we find that

┣: z^(φx) = ψ!z^ . ≡ . φx ≡_x ψ!x.

Thus we shall have

┣: φx ≡_x  ψ!x . φx ≡_x  χ!x . ⊃ . z^(φz) = ψ!z^ . z^(φz) = χ!z^,

but we shall not necessarily have ψ!z^ = χ!z^ under these circumstances, for two functions may well be formally equivalent without being identical; for example,

x = Scott . ≡_x . x = the author of Waverley,

but the function "z^ = the author of Waverley" has the property that George IV

 

wished to know whether its value with the argument "Scott" was true, whereas the function "z^= Scott" has no such property, and therefore the two functions are not identical. Hence there is a propositional function, namely

x = y . x = z . ⊃ . y = z,

which holds without any exception, and yet does not hold when for x we substitute a class, and for y and z we substitute functions. This is only possible because a class is an incomplete symbol, and therefore "z^(φz) = ψ!z^" is not a value of "x = y.

"It will be observed that "θ!z^ = ψ!z^" is not an extensional function of  ψ!z^. Thus the scope of z^(φz) is relevant in interpreting the product

z^(φz) = ψ!z^ . z^(φz) = χ!z^.

If we take the whole of the product as the scope of z^(φz), the product is equivalent to

(∃θ) : φx ≡_x θ!x . θ!z^ = ψ!z^ . θ!z^ = χ!z^,

and this does imply   ψ!z^ = χ!z^.

We may say generally that the fact that z^(φz) is an incomplete symbol is not relevant so long as we confine ourselves to extensional functions of functions, but is apt to become relevant for other functions of functions.

* This definition raises certain questions as to the two senses of a relation, which are dealt with in *21.

Script: Stan Lee  Pencils Pencils: Jack Kirby Inks: Don Heck 

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This post brings to a conclusion my intention of closely examining the three introductory chapters of Bertrand Russell and Alfred North Whitehead’s Principia Mathematica. My feeling is that this ‘introduction’ is the meat of their work, and the main text is in fact a practical demonstration of the use and value of the system. The authors might well have chosen another subject, but mathematics was the best subject to use as a demonstration, because it was and still is the most well-known and widely understood one imaginable.

 

As we see in the above posts the real value in our system is not mathematical, but in the field of logical philosophy. To take ideas, functions, propositions, relationships, groups and classes and form equations with them in practical ways. With care they can be notated in ways that are expressed compactly and reveal any errors within the structure. Whitehead seems to have used this system to plot his great work of Cosmology ‘Process and Reality’, first as a series of lectures, then eventually a book. First the equational outline, then transposed into spoken and then written language.

 

I would like to thank all of you who have been or are still using this blog for whatever you may find in it. Although the number of hits is small, only a few hundred per month, and I have no doubt many only stumble upon me searching for cartoons or Indian stuff; still I like to think a few are looking and thinking about my subject. Thank you for looking and reading; I am proud of the readers from many, many countries around the world who have visited. Perhaps I am most satisfied by the fact that in 43 posts up to now, no one has commented. I take that as an affirmation that no one yet has had cause for complaint.

 

I have contemplated stopping here or continuing with an examination of Mathematical logic. In any case I am going to change gears and do a more free form blogging. There is a long way to go to see Val and Aleta through the winter with the Indians in the New World and back home to Camelot. Also I have been collecting images of Indians depicted from all perspectives; negative, demeaning, positive, heroic, accurate, absurd drawings, photographs of Indians, Indian art, Indian Artists and examples of Indians in popular culture. I’m thinking of putting them up here to share the Red Man’s burden.

 

Finally I have some thoughts of revising some of the earliest posts here in hopes of better clarity. I would really appreciate any comments on this blog, any errors spotted or questions for clarification.

 

Dan DeCarlo The greatest of all time comic book artist!

Archie Andrews Gang   © Respective copyright/trademark holders.



Harold R. (Hal) Foster’s Prince Valiant © Respective copyright/trademark holders.