[Vision2020] [Q2] Induction IV

[Michael]

Wayne,

 

All in your latest installment seems fairly orthodox to me.  Any minor
qualifications I might make with any of the details are most likely going to
be peripheral to the goal of our discussion.

 

Thanks

Michael Metzler

 

 

 

 

 

 

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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.

 

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