Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.
Ambiguous
assertion and the real variable
Any
value "x" of the function φx^ can be asserted. Such an
assertion of an ambiguous member of the values of φx is symbolized by
“├ .
φx”
Ambiguous assertion of
this kind is a primitive idea, and cannot be defined in terms of the assertion of
propositions. This primitive idea is the one which embodies the use of the
variable. Apart from ambiguous assertion, the consideration of “φx,” which is an ambiguous
member of the values of φx^,
would be of little consequence. When we are considering or asserting "φx," the variable x is called a "real variable."
Take, for example, the law of excluded middle in the form which it has in
traditional formal logic:
" a is either b or
not b."
Here a and b are real
variables: as they vary, different propositions are expressed, though all of
them are true. While a and b are undetermined, as in the above enunciation, no
one definite proposition is asserted, but what is asserted is any value of the
propositional function in question. This can only be legitimately asserted if,
whatever value may be chosen, that value is true, i.e. if all the values are
true. Thus the above form of the law of excluded middle is equivalent to
" (a, b). a is either
b or not b,"
i.e. to "it is always
true that a is either b or not b." But these two, though equivalent, are
not identical, and we shall find it necessary to keep them distinguished.
When we assert something
containing a real variable, as in e.g.
"├ .
x=x,
we are asserting any value of a propositional function.
When we assert something containing an apparent variable, as in
"├. (x) . x=x" or
" ├ .
($x) . x = x,
we are asserting, in the
first case all values, in the second
case some value (undetermined), of
the propositional function in question.
It is plain that we
can only legitimately assert "any
value" if all values are
true; for otherwise, since the value of the variable remains to be determined,
it might be so determined as to give a false proposition. Thus in the above
instance, since we have
├ . x=x we may infer
├ . (x) . x=x.
And generally, given an
assertion containing a real variable x,
we may transform the real variable into an apparent one by placing the x in brackets at the beginning, followed
by as many dots as there are after the assertion sign.
When we assert something
containing a real variable, we cannot strictly be said to be asserting a
proposition, for we only obtain a definite proposition by assigning a value to
the variable, and then our assertion only applies to one definite case, so that it has not at all the same force as
before. When what we assert contains a real variable, we are asserting a wholly
undetermined one of all the propositions that result from giving various values
to the variable. It will be convenient to speak of such assertions as asserting
a propositional function. The
ordinary formulae of mathematics contain such assertions; for example
" sin2 x + cos2 x = 1"
does not assert this or
that particular case of the formula, nor does it assert that the formula holds
for all possible values of x, though it is equivalent to this
latter assertion; it simply asserts that the formula holds, leaving x wholly undetermined; and it is able to
do this legitimately, because, however x
may be determined, a true proposition results. Although an assertion
containing a real variable does not, in strictness, assert a proposition, yet
it will be spoken of as asserting a proposition except when the nature of the
ambiguous assertion involved is under discussion.
Script, Pencils and Inks: Henry
Boltinoff Superturtle™ © Respective
copyright/trademark holders.
This post goes right to the heart of
the difficulty of language and rhetoric in modern life, that of being sure of
the truth of what we are expressing, or even thinking, when it comes to complex
and abstract thought and ideas. Even combining a number of simple, elemental
ideas everyone can agree are true, one soon reaches complexities of divergence,
where agreement becomes problematic. As we wish to examine problems that are
regarding one, or some number of complex propositions; ones that may be general
or specific, or a combination of both, this task is quite possible for an ordered
mind to undertake. A more difficult undertaking is to express any results of
our calculations to others, perhaps even to ourselves, in a concise, meaningful
and coherent fashion. Verbally or even in writing, the use of language is
always painfully inadequate to the purpose of expressing, let alone proving
complex abstract propositions to another person.
In the end, any acceptance or rejection
of plans based on such propositions is, in the end, partially or mostly based
on emotion, intuition, affection, dislike or any number of other reasons
unrelated to the data presented. Not that the audience is incapable of
understanding the plan, or even that the author is inadequate in expressing it;
the problem lies with the nexus of the meeting of expression and comprehension
by means of language.
We are taking in the precepts that will
in total allow the consideration and expression of any complexity or multiple wildly
diverse complexities to any purpose in equation-al form that clearly displays
any error in their constitution for all to see, the author of the plan
included; and by this means others can benefit by understanding the equations,
and they in turn may act as a guide to translate the complexity accurately into
language.
Script: Carl Wessler Pencils and Inks: Bernard Krigstein © Respective copyright/trademark holders.
Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.
hanBLOGlaka
Logical Philosophy In American Indian
thought and Perception
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