MCT3 (fwd)
Homer Wilson Smith
HomerWSmith at lightlink.com
Thu Mar 19 13:53:33 EDT 2015
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INTRO TO LOGIC
Here follows the first broad public issue of the Machine Certainty
Theorem (MCT), one of the deeper theorems in Information Theory.
There are two fundamental aspects to any theorem or proof, the
LOGICAL FORM, and the CONTENT.
The logical form can be expressed with out the content by replacing
the various words and phrases in the proof with variables that have no
meaning. This allows the logical form of the proof to be studied
independent of its actual application.
Once the logical form is verified, then the variables can be
replaced by the meanings they stand for, and application of the proof
with its content can be studied independent of its logical form.
Any proof has at least three parts. The ASSUMPTIONS, the LOGIC,
and the CONCLUSION.
The logical form of the proof consists of all three parts in
abstract variable form, as described above. The content of the proof
also consists of all three parts in the concrete form where all
variables are replaced by their intended meanings.
The Machine Certainty Theorem states that a space-time machine
can't be certain of anything, yet a Conscious Unit can, therefore a
Conscious Unit is not a space-time machine.
Before I get on with the formal presentation of the Machine
Certainty Theory, I would like to provide a small sample proof to
explain the various parts of what you are about to see to those who have
little training in formal logic.
In this case I will work backwards from an actual argument in
concrete CONTENT FORM, to its abstract LOGICAL FORM so that you can see
how the process will be reversed when we get to the actual proof.
Consider the following argument.
1. Joe is a Christian.
2. All Christians believe in Hell.
3. Therefore, Joe believes in Hell.
Q.E.D.
Q.E.D is Latin for Quite Easily Done this is placed at the end of
the proof to demark where the proof ends and that the conclusion has
been proved. (Actually QED stands for Quod Erat Demonstrandum, 'that
which was to be demonstrated'.)
All proofs contain three parts, the ASSUMPTIONS, the LOGIC and the
CONCLUSION. The conclusion is true if and only if the assumptions are
true AND the logic is valid. If either the assumptions are false or the
logic is invalid, then the conclusion may be false (it could still be
true though, you don't know.)
For example, it is clear from the argument above, that if Joe is
not a Christian, or if some Christians don't believe in Hell, then the
conclusion that Joe necessarily believes in Hell becomes indeterminate,
he may or may not.
A properly presented proof would show all three parts, assumptions,
logic, and conclusion, clearly marked so that no confusion could result.
The purpose of first presenting the proof in logic form devoid of
meaningful content is to verify or validate the LOGIC part of the proof.
Once that is accomplished, then the proof must be presented for a
second time in CONTENT form, so that the assumptions and conclusion can
first be UNDERSTOOD and then their truth verified or argued. One first
verifies each of the assumptions in turn. If all of the assumptions
check out to be true, then the conclusion must be true if the logic is
also valid.
One then looks to see if the conclusion actually fits with
actuality. If it does you are finished for the moment. If it turns out
the conclusion is observably false, then either the logic was invalid or
one or more of the assumptions was false.
In the above example, there are two assumptions.
1. Joe is a Christian.
2. All Christians believe in Hell.
There is one conclusion,
3. Joe believes in Hell.
Normally in a more complex proof there would be more statements
in between 2 and 3 which would be partial conclusions on the way to the
final conclusion, but in this case the logic is so simple we go directly
from lines 1 and 2 to line 3 with a logical form called Modus Ponens.
Modus Ponens is a fancy Latin phrase meaning 'If A implies B, and A
is true, then B is true too.' (Actually Modus Ponens means 'Mode that
affirms')
For example, 'If being a dog implies being an animal, and Joey is a
Dog, then Joey is an animal.
Modus Ponens can be compared to Modus Tolens, another fancy Latin
phrase meaning 'If A implies B and B is false, then A is false.'
(Actually Modus Tolens means 'mode that denies'.)
For example, "If being a dog implies being an animal, and Jane is
not an animal, then Jane is not a dog."
Notice that if Jane IS an animal, one cannot tell if Jane is a dog
or not, because it is NOT true that if A implies B, then B implies A!
1. "Joe is a Christian" can be symbolized as "J -> C" which says
"If it's Joe, then it's a Christian", or "Being Joe implies being a
Christian", or more simply, "Joe implies Christian".
2. "All Christians believe in Hell" can be symbolized as "C -> H"
which says, "If it's a Christian then it believes in Hell", or "Being a
Christian implies Believing in Hell", or just "Christian implies Hell".
3. "Joe believes in Hell" can be symbolized as "J -> H" which
says, "If it's Joe, then it believes in Hell" or "Being Joe implies
Believing in Hell", or "Joe implies Hell".
We can thus symbolize the entire argument as follows, and this is
its logical form.
We explain each part in the section below the proof.
************************************************************************
LOGICAL FORM OF THE PROOF
1. J -> C (Being Joe implies being Christian)
2. C -> H (Being Christian implies Believing in Hell)
(1,2)[A] 3. J -> H (Being Joe implies Believing in Hell)
Q.E.D
(M.P.) A. (A -> B) and (B -> C)) -> (A -> C)
************************************************************************
In the above example there are two assumptions, lines 1 and 2, and
one conclusion, line 3.
The '(1,2)[A]' to the left of line 3 denotes that line 3 was
derived from lines 1 and 2 using Logical Form A which is shown at the
bottom of the proof below the Q.E.D. The particular Logical Form in
this case is Modus Ponens, which is denoted by (M.P.) to the left of the
same line.
Not all logical forms have formal names, and if not, the name or
its abbreviation is left out.
So how does one go about checking this proof out?
1.) Well the first thing that needs to be done is to check out and
verify all the Logical Forms shown below the Q.E.D, as these are the
extracted GENERALIZED statements of the LOGIC part of the proof that
gets you from the assumptions to the conclusion.
2.) The next thing to do is to familiarize yourself with the
assumptions and the conclusion.
3.) The next thing to do is to verify each step between the
assumptions and the conclusion to see that indeed the GENERAL Logical
Forms stated below Q.E.D are used correctly in their SPECIFIC
application to each step of the proof between the assumptions and the
conclusion.
The GENERAL Logical Forms will usually be stated in generic
variables like A, B and C which have nothing to do with the proof.
The assumptions and the conclusion and the SPECIFIC USES of the
General Logical Forms will usually be stated in letters that relate to
their content, such as J, C and H (Joe, Christian and Hell).
Thus one needs to be able to see that the SPECIFIC use of a
particular Logical Form parallels the GENERAL use of the same form to
know that the general form has been used correctly.
For example,
GENERAL ((A -> B) and (B -> C)) -> (A -> C)
SPECIFIC ((J -> C) and (C -> H)) -> (J -> H)
Where ever there is an A in the general form there had better be a
J in the specific form. Where ever there is a B in the general form
there had better be an C in the specific form. And where ever there is
a C in the general form there had better be an H in the specific form.
Don't get the C in the GENERAL form confused with the C in the
SPECIFIC form. They are unrelated and are the same letter only by
coincidence. In the general form the C doesn't stand for anything, it
is merely a place holder. In the specific form the C stands for
Christian and corresponds to the PLACE HOLDER B in the general form!
Now at this point it should be possible to say with perfect
certainty that the proof is either logically valid or not.
There is no such thing as an uncertain proof. Either it is valid
or it is not valid. This can be determined with perfect certainty
before anything else is known about the meaning of the variables in the
proof.
Remember though that just because a proof has been prooven valid,
this does not mean that the conclusion is necessarily true. This would
also depend on the assumptions being true, and determining the truth of
the assumptions, not the validity of the logic, comprises the main body
of work in verifying the conclusion of a proof.
Verifying the validity of the logic of the proof is the first and
easiest step and by this time in the analysis should be satisfactorily
completed.
So that was a lot of work, no? But, as I said, we are not done
yet.
Once the logic form of the proof has been verified completely as we
have just done, you next need to verify the CONTENT form of the proof.
This is done by replacing each specific variable in the proof with
its English equivalent so that you can see what each of the assumptions
and the conclusion actually say.
This is done first by providing a little table that shows what
each variable means, like so.
J = Joe
C = Christian
H = Hell
Then you plug them in and you get the following.
************************************************************************
CONTENT FORM OF THE PROOF
J = Joe
C = Christian
H = Hell
1. Joe -> Christian
2. Christian -> Hell
(1,2)[A] 3. Joe -> Hell
Q.E.D
(M.P.) A. ((A -> B) and (B -> C)) -> (A -> C)
************************************************************************
This provides a rather sparse and pared down version of what the
proof is about, but it serves to convey the meaning of each of the
lines.
The last step would be to take up each line of the proof and expand
it into a grammatically correct full English sentence and discuss it at
length.
Discussion of the assumptions would involve not only their
meaning, but also evidence that they are true.
In general there are 4 kinds of assumptions.
1.) Logical Tautologies.
2.) Definitions
3.) Observations
4.) Intuitions
LOGICAL TAUTOLOGIES are always true because of their inherent
logical structure. An example of a logical tautology would be,
1.) Christian or not Christian
A full english expansion of this might be,
1.) Joe is either a Christian or not a Christian.
You have to be careful when presenting such tautologies to make
sure that your words are defined in such a way that the tautology is
true. If someone has a sloppy or fuzzy definition of what it means to
be a Christian, then it might be possible to be both a Christian and not
a Christian! But really he would be changing meanings in mid sentence,
so its a good idea to set rigorous definitions of your words that
everyone can agree on before you start an argument or proof like this
one.
DEFINITIONS are statements that are true by definition.
An example might be,
1. All Christians believe in Christ, if they don't believe in
Christ then they are not real Christians.
Such a statement is true only because we say it is true, it has no
other basis. There may be other people who don't believe in Christ who
none the less wish to be called Christians. This is not a problem, you
have the right to define your words how ever you wish, just remember
that what you are calling a Christian may not include others who call
themselves Christians. They will no doubt complain, but their
complaints will be irrelevant to your proof, because YOUR proof has to
do with Christians as YOU have defined the word.
If you wish to define your words in some other way, that is fine,
just make sure that everyone knows what YOUR definitions are before you
proceed.
No one can ever say your proof about Christians is wrong because
your DEFINITION of 'Christian' is wrong.
OBSERVATIONS are statements that are true by observation.
1. Some Christians go to Church on Sunday.
It's true because it's true, go out and LOOK for yourself. It's
not true by LOGICAL TAUTOLOGY, and it's not true by definition, it's
true because someone went out and measured the phenomenon and reported
back what he found.
The certainty level of a observation is dependent on how many vias
you use to make that observation, how many levels of symbols referring
to referents before you come to the actual thing being observed. A
person who is using radio telescope data to determine the temperature of
some planet circling a sun 4 galaxies away, is on far less certain
grounds, than someone looking at a thermometer in his back yard.
Someone who goes out and just feels that it is hot outside is in even
more direct contact.
Observations of the external physical universe however can never be
perfectly certain because all observers are using effects in themselves
as observers to make conclusions about what must be out there.
In this sense, 'making an observation' means 'to be the effect of
an external cause' and THEN to logically compute back in time to what
that cause might be like in order to have had the effect that one
received.
The fact that one received an effect might be a certainty, but the
nature of what caused that effect can not be determined from the nature
of the effect alone.
This 'computing back from later effects to earlier causes' is
always an uncertain process, because effects 'here' do not prove
anything about causes 'there'. One can merely create a 'causal model'
and hope for a dependable but uncertain world view.
Observations of one's own conscious color forms, though, CAN be
perfectly certain. If you see a color form mockup of red and green in
front of you, there can be no denying that you see it. Anything it
might be USED TO REPRESENT to you in the external universe might be
uncertain, but the existence of the color form itself is certain.
INTUITIONS are statements which one feels to be true because it
violates some inner sense of propriety to think they aren't. This of
course doesn't mean that they are true, but it does mean that if you can
get agreement among a number of people who have the same sense of
intuition, then you can proceed with your proof as if your intuitions
were true, recognizing that the truth of the conclusion is only as as
certain as the truth of your intuition.
Even if you can't get agreement among others about intuitions, you
can still have your proof to yourself and be satisfied with it as far as
it goes.
As an example of an intuition,
Something can't come from nothing.
Any given proof will have assumptions that consist of mixtures of
the above 4 kinds of 'truths'. It is often enlightening to actually
state next to each assumption which kind of truth it is.
For example,
A something is an object with a non empty quality set. DEFINITION
An nothing is an object with an empty quality set. DEFINITION
0.) An object is either a something or a nothing LOGICAL
1.) Something can't come from nothing INTUITION
2.) Something exists now. OBSERVATION
Q.E.D. 3.) Something must have always existed. (conclusion)
In closing I would like to add that it is not clear that every
argument can be put into such simple terms as I have laid out here, or
that every assumption can be divided into the above 4 categories.
Sometimes its takes an enormous reworking of the WORDING of an argument
to make it conform to the simpler rules of logic. The English language
is very complex and the simple Logical Form is often lost in more poetic
forms of argument.
People in fact will often try to hide bad logic in the complex
nuances of the language, which is why it is important to break arguments
down into raw logical form.
However for the purposes of the Machine Certainty Theorem, the
above discussion is relatively complete and satisfactory.
The Machine Certainty Theory is VERY SIMPLE, so simple in fact that
once you get it, it will be a BIG DOWN, because you will have been
expecting all these fireworks to go off in your brain once you realize
this 'Great Eternal Truth' of the ages.
Your actual reaction will be more like, 'Duh, so what else is new.'
However it is the application of the MCT to consciousness that will
give you something to think about.
Machines can't be certain of anything, consciousness can.
Finally, I would like to remind you of a wise old saying.
"At first they said it wasn't true.
Then they said it wasn't important.
Then they said they knew it all along.
Which was true."
Well, the guy who said that, was talking about the Machine
Certainty Theorem, which is the grand daddy of all truths that people
argue about with you until they convince themselves they showed it to
YOU in the first place!
At that point you know they got it.
Homer
================ http://www.clearing.org ====================
Thu Mar 19 03:06:02 EDT 2015
ftp://ftp.lightlink.com/pub/archive/homer/mct3.memo
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================== http://www.lightlink.com/theproof ===================
Learning implies Learning with Certainty or Learning without Certainty.
Learning across a Distance implies Learning by Being an Effect.
Learning by Being an Effect implies Learning without Certainty.
Therefore, Learning with Certainty implies Learning, but
not by Being an Effect, and not across a Distance.
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