May
08, 2017 Monday
Bedtime
Story
Demonstrating Consistency of a Formal System
If
you care to glance back and look back at the other three axioms of the formal
system, they too turn out to be tautologies.
All
the four axioms are tautologies!
It
is possible to prove that this tautology character is capable of getting
transmitted from formulas to theorems very much like a genetic trait.
Then
it follows that any formula or a theorem that is derived from the system must
be a tautology.
That
is great.
We
have now two conditions fulfilled to test consistency of our formal system.
We
have found out a structural property (tautology) that is common to all the four
axioms.
We
have agreed that this property can be and will be retained by the derived
formulas and theorems.
Now
there remains that last point, a formula that is derived from these axioms that
is not tautological.
That
did not seem to be a very difficult task.
If
one tries, it would not be too difficult to derive this formula from the above
axioms:
‘v ∨
This
implies ‘v or q’ where both v and q are different sentential variables.
This
surely is not a tautology.
Consider
the English variant of this formula.
‘Navin
is an average ape or America is a mighty superpower’
Moreover,
this statement is not a logical truth or a necessary truth even if either of
the statements that constitute it be true or false.
So
then now we have derived (not really but agreed that it is derivable) a formula
that is not a theorem as it is not a tautology.
This
is the demonstration of absolute consistency of our formal system.
This
in turn also proves that from this formal system it is not possible to derive
both a formula and its formal negation by using the Transformation rules and
Rules of Inference.
This
whole sequence of logical thought is quite convoluted and puts great strain on
the mental capacity of an average ape like me.
So
in the nights to come, I shall once again go over the entire chain of reasoning
used to prove the consistency of a formal logical system.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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