[Vision2020] [Q2] Induction IV

[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,

 

We progress laboriously toward a meaningful discussion of the problem of evil.  Since arguments can fail because of faulty structure or because one or more of the premises of the argument are false or as we will see below, one or more of the premises is more improbable than probable, it is important for us to agree upon a few elementary things about truth and probability.

 


C.    Brief but Important Remarks on Arguments, Truth, Part II

 

What about the truth or probability of sentences used as premises in an argument which result/depend upon in some way on some kind of observation?  Can it be said that such sentences are true?

 

It is always possible for observations and conclusions from obser­vations to be false for a wide variety of reasons, hence the truth of sentences supported by observation can never, in theory, be apodictically asserted, but only the probability range of the truth of the sentence can be given.  I hope we can accept this without going through all the traditional arguments supporting problems of observations - prejudice/impairment/ignorance of the observer, conditions attendant, perspective, etc.

 

Therefore, very technically and very strictly speaking, it would not be proper to assert that observation based sentences are strictly true or false.  However, an examination of the use of the words "true" and "false" in "ordinary language" can perhaps steer us around this difficulty.

 

The sentence:

 

Barring a high wind, a pine cone that breaks off a tree limb will always fall earthward.

 

is a sentence that supported by the theory of gravity, and based on (trillions and trillions of) observations and continuing observations, hence the probability of its truth is regarded as extremely close to 1.00 (but not exactly 1.00).  Hence in ordinary language, we regard such sentences as true.  In fact, were someone to assert the above sentence is false, that person would be regarded as delusional.

 

Therefore, for the sake of proceeding without greatly expanding the wordage in our discussion, I hope you will henceforth agree to regard as true those observation based sentences whose probability of truth is extremely close to 1.00 and regard as false those observation based sentences whose probability is extremely close to 0.00.  Of course, the status of any particular sentence in this regard is always open to discussion.

 

 

The truth or falsity of some sentences can be sometimes quite easily determined.

 

If you say:

 

"It will not rain today",

 

but it does rain very hard for ten minutes at 2:00pm, your assertion was false.

 

Of course, there may be a disagreement of the meaning of the word "rain".  You might argue your original sentence is true by asserting:

 

"That wasn't rain at 2:00pm, it was only drizzle."

 

What the above kind of counterargument illustrates is:

 

That it is important for the participants in a discussion to agree on the meanings of words as close as is necessary.

 

That is why we are going through this laborious and boring set of exercises preliminary to discussing the truth of the knowledge claims and other problems found in the consideration of problem of evil and related problems.

 

 

Likewise, suppose someone asserts that:

 

"All prunes contain a single pit."

 

The falsity of that sentence is established by one confirmed observation of a prune without a pit or a prune with two or more pits.

 

Likewise, in ancient times when it was asserted that:

 

"All numbers are rational." or "No numbers are irrational."

 

the falsity of those sentences was established by Euclid's proof that the square root of two is an irrational number, a single counterexample.

 

In general:  The falsity of general sentences such as:

 

All X are Y.

No X is a Y.

 

can be established by just one confirmed observation/counter­example to the contrary.

 

 

More subtly, there are many general sentences whose truth are carelessly asserted and/or taken for granted which are actually quite false.

 

Suppose someone was under oath on the witness stand in a secular court.  Suppose under cross examination they were asked:

 

"Is the sky blue?"

 

How should they answer?  Many would answer "Yes."

 

They would be perjuring themselves!

 

If you observe the sky on even on a clear midday, there are parts of the sky that are not blue.  For example, the part occupied by and around the sun and the parts nearest some portions of the horizon is not blue.  Those parts are variations of white without any visible tinge of blue.  On a day with a few clouds in the sky, there are even more parts of the sky that are not blue.  Normally the sky is not blue at night.  Skillful attorneys often make use of such erroneous assertions by a witness to mischaracterize what a witness has said and/or to discredit the witness.

 

The above example again illustrates how important for the participants in a discussion to agree as exactly as possible or necessary on the meaning of words and sentences they are using, if that discussion is to progress toward a truthful and reliable conclusion.

 

 

Building on the above, let's see if we can agree on some other basic items.

 

Showing that an argument is not supportive of its conclusion(s) can be done by:

 

[1]   Showing the premises of the argument are false or improbable; or

[2]   Showing the structure of the argument is defective.

 

Obvious examples of [1] and [2]:

 

[P1] All true evangelists are lean and selfless.

[P2] Douglas James Wilson is a true evangelist.

 

Therefore,

 

[C]  Douglas James Wilson is lean and selfless.

 

 

[P1] All mathematicians can count.

[P2] All those who practice the rhythm method of contraception can count.

 

Therefore,

 

[C]  All mathematicians practice the rhythm method of contraception.

 

 

In practice, determining whether an argument is correctly structured is sometimes quite difficult.

 

Gödel's proof of the incompleteness of the unrestricted predicate calculus is beyond the understanding of many, and thus beyond their ability to determine if that proof is a valid deductive argument.

 

The correctness of even simple inductive arguments is not so easily determined since the addition of other premises to the argument may cause the probability of the outcome to change radically.  Many inductive arguments can be described as incomplete by nature - open ended.

 

There are methods of decidability of those arguments which can be translated into the sentential calculus without loss of fundamental cognitive import.  Likewise, there is a method of decidability for those syllogistic arguments for which Aristotle designed a test for validity (and which method was corrected in the 19th century by George Boole).

 

Hence, in our discussion we must try our best to determine the structural correctness of the simple arguments that we will be using, whether they be deductive or inductive.  This may not always be simple.  However, it can be simplified by noting that certain class statements can be recast as statements amenable to treatment in the sentential calculus or even by much simpler means.

 

Example:

 

All pencil sharpeners are really secret spy devices.

 

can be recast as:

 

If X is a pencil sharpener, X is really a secret spy device.

 

 

A further issue:  What can be said about the conclusions of valid deductive arguments which would be true if the premises are strictly true, but the probability of the truth of one or more the premises is less than extremely close to 1.00?

 

Example:

 

[P1] The next President of the United States will visit Meander River, Alberta.

[P2] Bill Frist will be the next President of the United States.

 

Therefore,

 

[C]  Bill Frist will visit Meander River, Alberta.

 

Suppose, the probability of P1 is 0.05 and the probability of P2 is 0.33.

 

[1]   If the events described by P1 and P2 are independent of each other, then the probability of C is (0.05)(0.33) = 0.0165.  [Notice that if two numbers are between 0 and 1, their product will always be smaller than either number.]

 

[2]   If the events described by P1and P2 are not independent, then the probability of C is always less than or equal to the smallest of the probabilities of P1, P2, in this case the probability of C is less than or equal to 0.05.

 

A convenient way to say this is:

 

In general, the probability of the truth of the conclusion of a valid deductive argument in the context of the argument is less than or equal to the smallest probability of the probabilities associated with each premise.

 

 

Please let me know whether or not you are in agreement with the very basic fundamentals expressed above.  If so, then I have only one other small, but important issue to seek your agreement upon before we examine your succinct, charming characterization of the inductive process.

 

Thank you again for your patience and forbearance with an aged, slow-witted codger.


Art Deco (Wayne A. Fox)
deco at moscow.com

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