40. Defeating Ambiguity by Notation



Harold R. (Hal) Foster’s Prince Valiant

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The proposition "a = (⍳x)(φx)" is easily shown to be equivalent to "φx . ≡_x . x = a." For, by the definition, it is

         (∃c) : φx . ≡_x. x = c : a = c,

i.e. "there is a c for which φx . ≡_x . x = c, and this c is a," which is equivalent to "φx. ≡_x. x = a." Thus "Scott is the author of Waverley" is equivalent to:

 "'x wrote Waverley' is always equivalent to 'x is Scott,’"

i.e. "x wrote Waverley" is true when x is Scott and false when x is not Scott.

Thus although “(⍳x)(φx)" has no meaning by itself, it may be substituted for y in any propositional function fy, and we get a significant proposition, though not a value of fy.

When ƒ{(⍳x)(φx)}, as above defined, forms part of some other proposition, we shall say that (⍳x)(φx) has a secondary occurrence. When (⍳x)(φx) has a  secondary occurrence, a proposition in which it occurs may be true even when (⍳x)(φx) does not exist. This applies, e.g. to the proposition: "There is no such person as the King of France." We may interpret this as

          ~{E!(⍳x)(φx)},

or as      ~{(∃c) . c = (⍳x)(φx)},

if "φx" stands for "x is King of France." In either case, what is asserted is that a proposition p in which (⍳x)(φx) occurs is false, and this proposition p is thus part of a larger proposition. The same applies to such a proposition as the following: "If France were a monarchy, the King of France would be of the House of Orleans."

It should be observed that such a proposition as

         ~ƒ{(⍳x)(φx)}

is ambiguous; it may deny ƒ{(⍳x)(φx)}, in which case it will be true if  (⍳x)(φx) does not exist, or it may mean

         (∃c) : φx . ≡_x. x = c : ~ƒc,

in which case it can only be true if (⍳x)(φx) exists. In ordinary language, the latter interpretation would usually be adopted. For example, the proposition "the King of France is not bald" would usually be rejected as false, being held to mean "the King of France exists and is not bald," rather than "it is false that the King of France exists and is bald." When (⍳x)(φx) exists, the two interpretations of the ambiguity give equivalent results; but when (⍳x)(φx) does not exist, one interpretation is true and one is false. It is necessary to be able to distinguish these in our notation; and generally, if we have such propositions as,

         y(⍳x)(φx). ⊃ . p,

         p . ⊃ . y(⍳x)(φx),

         y (⍳x)(φx) . ⊃ . c (⍳x)(φx),

and so on, we must be able by our notation to distinguish whether the whole or only part of the proposition concerned is to be treated as the "ƒ(⍳x)(φx)" of our definition. For this purpose, we will put "[(⍳x)(φx)]" followed by dots at the beginning of the part (or whole) which is to be taken as ƒ(⍳x)(φx), the dots  being sufficiently numerous to bracket off the ƒ(⍳x)(φx); i.e. ƒ(⍳x)(φx) is to be everything following the dots until we reach an equal number of dots not signifying a logical product, or a greater number signifying a logical product, or the end of the sentence, or the end of a bracket enclosing "[(⍳x)(φx)]."

Thus

           [(⍳x)(φx)]. y(⍳x)(φx) . ⊃ . p,

 will mean      (∃c) : ≡_x . x = c : yc: ⊃ . p,

but       [(⍳x)(φx)] : y(⍳x)(φx)) . ⊃ . p

will mean      (∃c): φx . ≡_x . x = c : yc . ⊃ . p.

It is important to distinguish these two, for if (⍳x)(φx) does not exist, the first is true and the second false.  Again

         [(⍳x)(φx) . ~ y(⍳x)(φx)

will mean       (∃c) : φx . ≡_x. x = c : ~ yc,

while           ~{[(⍳x)(φx)] . y(⍳x)(φx)}

will mean       ~{(∃c) : φx . ≡_x . x = c : yc}.

Here again, when (⍳x)(φx)  does not exist, the first is false and the second true.

 In order to avoid this ambiguity in propositions containing (⍳x)(φx), we amend our definition, or rather our notation, putting

[(⍳x)(φx)] . ƒ(⍳x)(φx) . = : (∃c) : φx . ≡_x . x = c : ƒc  Df.

By means of this definition, we avoid any doubt as to the portion of our whole asserted proposition which is to be treated as the "ƒ(⍳x)(φx)" of the definition. This portion will be called the scope of (⍳x)(φx).   Thus in

        [(⍳x)(φx)] . ƒ(⍳x)(φx) . ⊃ . p

the scope of     (⍳x)(φx) is ƒ(⍳x)(φx);  but in

        [(⍳x)(φx)] : ƒ(⍳x)(φx) . ⊃ . p

the scope is (⍳x)(φx) . ⊃ . p;

in      ~{[(⍳x)(φx)] . ƒ(⍳x)(φx)}   

the scope is ƒ(⍳x)(φx) but in

        [(⍳x)(φx)] . ~ ƒ(⍳x)(φx)

the scope is       ~ƒ(⍳x)(φx).

It will be seen that when (⍳x)(φx) has the whole of the proposition concerned for its scope, the proposition concerned cannot be true unless E!(⍳x)(φx); but when (⍳x)(φx) has only part of the proposition concerned for its scope, it may often be true even when (⍳x)(φx) does not exist. It will be seen further that  when E!(⍳x)(φx), we may enlarge or diminish the scope of (⍳x)(φx) as much as we please without altering the truth-value of any proposition in which it occurs.

If a proposition contains two descriptions, say (⍳x)(φx) and (⍳x)(yx), we have to distinguish which of them has the larger scope, i.e. we have to distinguish  

(1) [(⍳x)(φx)] : [(⍳x)(yx)] . ƒ{(⍳x)(φx), (⍳x)(yx)},

(2) [(⍳x)(yx)] :  [(⍳x)(φx)] . ƒ{(⍳x)(φx), (⍳x)(yx)}.  

  The first of these, eliminating (⍳x)(φx), becomes

(3) (∃c) :.  φx . ≡_x . x = c : [(⍳x)(yx)]. ƒ{c, (⍳x)(yx)},

which, eliminating (⍳x)(yx), becomes

(4) (∃c):. φx . ≡_x . x = c :. (∃d) : yx ≡_x . x = c : ƒ(c, d),

and the same proposition results if, in (1), we eliminate first (⍳x)(yx) and then (⍳x)(φx). Similarly (2) becomes, when (⍳x)(φx)  and (⍳x)(yx) are eliminated,

  (5) (∃d) :. yx. ≡_{x .} x = d :. (∃c) :. φx . ≡_x . x = c : ƒ(c, d).

(4) and (5) are equivalent, so that the truth-value of a proposition containing two descriptions is independent of the question which has the larger scope. It will be found that, in most cases in which descriptions occur, their scope is, in practice, the smallest proposition enclosed in dots or other brackets in which they are contained. Thus for example

[(⍳x)(φx)] . y (⍳x)(φx) . ⊃ . [(⍳x)(φx)] . c (⍳x)(φx)

will occur much more frequently than

    [(⍳x)(φx)] : y (⍳x)(φx) . ⊃ . c(⍳x)(φx).

For this reason it is convenient to decide that, when the scope of an occurrence of (⍳x)(φx)  is the smallest proposition, enclosed in dots or other brackets, in   which the occurrence in question is contained, the scope need not be indicated by "[(⍳x)(φx)]." Thus e.g.

             p . ⊃ . a = (⍳x)(φx)

will mean   p . ⊃ . (⍳x)(φx)] . a = (⍳x)(φx);

and           p. ⊃ . (∃a) . a = (⍳x)(φx)

will mean      p . ⊃ . (∃a) [(⍳x)(φx)] . a = (⍳x)(φx);

and            p. ⊃ . a ≠ (⍳x)(φx)

will mean    p . ⊃ . [(⍳x)(φx)] . ~ {a = (⍳x)(φx)};

but             p . ⊃ . ~ {a = (⍳x)(φx)}

will mean    p . ~ {[(⍳x)(φx)] . a = (⍳x)(φx)}.

This convention enables us, in the vast majority of cases that actually occur, to dispense with the explicit indication of the scope of a descriptive symbol; and it will be found that the convention agrees very closely with the tacit conventions of ordinary language on this subject. Thus for example, if "(⍳x)(φx)” is "the so-and-so," "a ≠ (⍳x)(φx)" is to be read "a is not the so-and-so," which would ordinarily be regarded as implying that "the so-and-so" exists; but "~{a = (⍳x)(φx)}" is to be read "it is not true that a is the so-and-so," which would generally be allowed to hold if "the so-and-so" does not exist. Ordinary language is, of course, rather loose and fluctuating in its implications on this matter; but subject to the requirement of definiteness, our convention seems to keep as near to ordinary language as possible.

Sheldon Mayer’s Sugar and Spike^TM  © Respective copyright/trademark holders.

 

This post digs in and offers real clarity to the ambiguity in everyday language. We can begin to see here the structure to conversation and written language’s undoing. These aspects are mostly or completely unknown in Indian languages in North America. It is only in the shaman spirit aspect that metaphysical thought or communication is required; these served the function of healing, sciences, religion, agriculture and even a very useful and developed sort of computer technology very similar to memory based techniques used by the greatest engineers, architects, artists and clerics in the West for centuries.

 

Indian cultures were concerned with life, a life connected to the land and the seasons and human concerns in which physical possessions were minor, trivial matters to most people. Society structure was complex and tribal to be sure; but everyone had their place, and these roles were nearly always voluntary and clearly evident. In these societies languages were rudimentary by Western standards, but they held, and still hold great variety and complexity for native speakers. They are brilliantly functional because they are entirely separate from the metaphysical, grammatical and rhetorical aspects that are the province of Chiefs and shamans. A final important aspect to help understand how this worked is to recall that these cultures were non-material and that money was unknown.


Script: Elliot S! Maggin  Pencils and Inks: Kurt Schaffenberger

Mary Marvel ^TM © Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.