September
13, 2017 Wednesday
Bedtime
Story
Deriving True Sentences
Just
like mathematical theorems, if a sentence is derived from another true sentence
using the simple laid out rules of inference then it has to be true as well.
Using
the rules of proof based on an adequate definition of truth, a formidable
system can be created that can be virtually infallible.
The
best example of a rule of proof is the rule of detachment known as modus ponens.
We
had discussed it a couple of times earlier.
It
is one of the accepted mechanisms for the construction of deductive proofs.
Just
as in mathematical logic, similarly here it can be used to derive true
sentences based on a conditional sentence.
So
if there are two sentences “a” and “b” and there exists a condition or a rule
which states that if sentence “a” is true, then so is “b”.
If
the truth value of sentence “a” has been established then in that case it can
be said that “b” is a true sentence.
Formal
proof of any given sentence is not very different from mathematical proofs.
One
has to start from axioms and using rules of proof derive new sentences.
Then
those same rules can be applied to those new sentences or perhaps jointly to
axioms and new sentences and obtain further sentences.
This
process can go on endlessly.
Any
sentence that was derived in this manner could be said to have been formally
proved.
All
this can be summed up with great pithy in the following manner.
One
can formally prove any given sentence by constructing a series of finite number
of sentences such that:
(a)
The first sentence of the sequence is an axiom
(b)
The sentences that follow are either also axioms or they have been derived
sequentially from their predecessors using any of the rules of proof
(c)
The last sentence of the entire sequence is the sentence to be proved
You
are well versed by now with the idea of the axiomatic theory.
When
this system gets formalized, it becomes a formalized theory.
Such
a formalized theoretical system will only contain either axioms or sentences
that have been proved.
Tarski
claims that in such a formalized system with well established axioms, the
sentences that are the theorems of this system have an appearance of a theorem.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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