Wednesday, September 5, 2012

7. Propositional Functions

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.

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Logical Philosophy In American Indian thought and Perception

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