Script: ? Pencils and Inks: Sid Check
© Respective copyright holders.
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. The 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” is an ambiguous “value” of a propositional function.
When we wish to speak of the propositional function corresponding to “x is
hurt,” we shall write “x^” is hurt” (*I've solved most of my font/symbol problems, except how to put the dern function symbol (^) directly over the x, I won't let this stop me now, I'll figure it out by 'n by.)
Thus “x^”
is hurt” is the propositional function and “x is hurt” is the ambiguous value
of that function.
Accordingly though “x
is hurt” and “y is hurt” occurring in the same context can be distinguished, “x^
is hurt” and y^ is hurt” this will convey no distinction of meaning at
all.
More generally φx is an
ambiguous value of the propositional function φx^, and when a definite
signification a is substituted for x,
φa is an unambiguous
value of φx^.
Derivative functions:
“sin x” or “log x” or “the father of x.”
The range of values and
total variation
Thus corresponding to
any propositional function φx, there a range or collection of values consisting
of all the propositions (true or false) which can be obtained by giving every
possible determination of x in φx.
A value of x for which φx
is true will be said to satisfy “φx^.”
Now, as for the truth
or falsehood of propositions in this range three important cases must be noted
and symbolized. These cases are given by three propositions of which at least
one must be true. Either
(1)
all propositions in the range are true, “(x)
. φx”
(2)
some propositions in the range are true, “($x)
. φx” or
(3)
no proposition in the range is true.
The symbol “(x) . φx” may
be read “φx always,” or “φ is always
true,” or “φx is true for all possible values of x”
The symbol “($x)
. φx” may be read “There exists an x for which φx is true,” or “there exists an
x satisfying φx^,” and thus conforms to the natural form of expression of
thought.
Proposition (3) can be expressed in terms of the fundamental ideas
now at hand.
~φx stands for the contradictory of φx, accordingly ~φx^ is
another propositional function such that each value of φx^ contradicts a value
of ~φx, and vice versa.
Hence “(x) . ~φx” symbolizes the proposition that every value of
~φx^ is untrue.
Script:
Robert Kanigher Pencils and Inks:
Harry G. Peter Wonder Woman™ © Respective
copyright/trademark holders.
Our introduction here to propositional
functions is the key foundation to the author’s system of logical philopophy,
and our first step in the process of understanding Indian thought as it existed
before first European contact, as your shaman blogger understands the case to
exist. To be clear, Alfred North Whitehead and Bertrand Russell are “the
authors” meant anywhere in the text that term is used, while the “blogger” is
to infer Comic Book Shaman, who publishes this blog.
One pleasant convention used by the
authors is to use terms like “we” and “our” in reference to expanding and
explaining parts of the text in such a way as may be taken as including the
reader in an 'ownership of understanding' the system described. I find this very
appropriate, because understanding this system is to adopt basic metaphysical
changes in our thought process. Starting above, with learning the crazy fine
distinctions of functions; so much of life is filled with contradictions and as
we move ahead we learn to sort random functions into appropriate groups, types
and classes that allow us to solve problems that appeared to be unsolvable
contradictions.
William Overgard's Rudy © Respecive copyright/trademark holders.
han
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laka
Logical Philosophy In American Indian
thought and Perception
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