[Vision2020] [Q2] Induction V

[Art Deco]

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This document is about some very elementary concepts in logic, hence, may appear unduly technical.  The concepts discussed are not rocket science and the mere ability to read should make them understandable.  Some would argue that the below are trivial.  Perhaps.

 

 

Michael,

 

With your kind agreement with what has so far been put forth in this thread we are nearer to discussing the problem of evil.  

 

Finally, before examining your model of the inductive process:

 

The truth of a sentence or the probability of the truth of a sentence is not always easy to determine.

 

For certain kinds of sentences, the truth of those sentences is a matter of agreement of linguistic usage, or in other words, by accepted definition and/or axioms/postulates.

 

Example:

 

A rational number is a number that can be expressed as a fraction a/b where a and b are integers and b <> 0 and where a and b have no common factors which are integers. A rational number a/b is said to have numerator a and denominator b. Numbers that are not rational are called irrational numbers.

 

For the purposes of discourse, application, and advancement of knowledge, those who work with numbers agree that the sentences above defining rational numbers are true.

 

There are many words and sentences in ordinary language whose meaning and truth depend upon such agreed usage - "all" and "none" are two such words.  The axioms of various mathematical systems such as group theory, ring theory, point-set topology, etc are also examples of truth by agreement of usage as are definitions in physics, chemistry, etc.

 

It is very important to realize contradictions can ensue and thus impossibilities proposed if some object X is defined in such a way which asserts that X can be described by a particular agreed upon definition.  In such cases it is said that it is logically impossible for X to exist.

 

Example:

 

Definition:  A Steatopygous Nilsow is a positive prime integer whose square root is a rational number.

 

Given the definition of rational number above, asserting the existence of a Steatopygous Nilsow leads to a contradiction - given the agreed upon definition of a rational number above, it is logically impossible for a Steatopygous Nilsow to exist.

 

A simple, clear, proof of this was given by Euclid and runs as follows:  (Note:  "x squared" will be written as "x^2".)

 

Euclid's proof:

 

1.    Let p be a Steatopygous Nilsow i.e. p has a rational square root.

2.    Then the square root of p =a/b where a, b are integers, have no common factors, and since p > 0, then a > b > 1.

3.    Observe:        p = a^2/b^2.

4.    Further:         pb^2 = a^2.

5.    Notice:           a divides a^2 evenly.

6.    Therefore:      a divides pb^2 evenly.

7.    Notice:           If a divides pb^2 evenly, there must be a

factor in common between a and pb^2.

8.    But:               a and b have no common factors.

9.    Also:              p is prime so a cannot divide p evenly.

10.  Hence:           a cannot divide pb^2; hence a contradiction

has been reached.

11.  Therefore:      The square root of p is not rational.  This is

a contradiction to the assertion that a

Steatopygous Nilsow exists.

 

This is a very simple illustration showing that simply because some object is defined, that does not mean that the object exists.  In the above case, the definition of the object leads to a contradiction in an axiomatic, precisely defined milieu, hence the logical impossibility that the object (a Steatopygous Nilsow) could exist is demonstrated.

 

 

The other way the truth of a sentence or the probability of the truth of a sentence is determined is by some kinds of observations.

 

The word "observation" in the previous sentence means observation in an extremely broad sense.  Examples of observations include observations of terrestrial events such as the color of smog in Los Angeles on July 12, 1999, observations of quantitative relationships, observations of the meanings of words and sentences, observations of what a book says, observations of what a person says, observations about what some alleged authority asserts, observations about internal feelings, etc, etc.

 

As we have noted above, the probability of the truth of any observation based assertion may be extremely close to one but the possibility that that further observations can change the present probability must always be considered.  Hence, any knowledge claims supported by even one observation are not completely certain although the truth of that knowledge claim for all practical purposes may be asserted as true as we noted before:  Thus we regard as true the sentence:  "Barring a high wind, a pine cone that breaks off a tree limb will always fall earthward."

 

Just as defining an object by definition in an axiomatic system (as above) can lead to a contradiction and hence the logical impossibility of that object's existence, a similar situation exists with definitions of objects whose real existence is dependent upon some kind of observations or other.  In this case, we say that the existence of the object is improbable if it is not supported by observations.

 

Common examples of the above include the unicorn, Sherlock Holmes, and Saint Puce.  In these examples, asserting the real existence of these objects would be called an extremely improbable assertion or a "false" assertion in ordinary usage.  Of course, it is logically possible for future observations to change this, but at this point "extremely improbable" is the correct description of such assertions.

 

 

Well Michael, if we can agree on the simple basics above, we are now in a position to examine your description of the induction process and perhaps make one or two very minor changes to it for the sake of clarification.

 

Please let me know at your convenience whether you agree with the above or not.  Thank you again for your patience.


Wayne A. Fox
1009 Karen Lane
PO Box 9421
Moscow, ID  83843

(208) 882-7975
waf at moscow.com

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