May
01, 2017 Monday
Bedtime
Story
Transformation Rules: Rule of Substitution and Modus Ponens
You
surely would have got the reason for the last formula of the last night to be
declared invalid.
That
is because the Formation Rule demands that a sentential connective ought to be
flanked on both its sides by a formula whereas in our case the connective ∨
After
the Formation Rules, let me give you examples of Transformation Rules (rules of
inference) which allow the formulas to be fiddled and changed.
One
of the Transformation Rule is the Rule of Substitution for sentential
variables.
The
Rule of Substitution says that another formula can be derived from a one
containing sentential variables by uniformly substituting formulas for the
variables.
Moreover,
when the variables are substituted, the same substitution must be made for each
occurrence of the variable.
Let
me give you few examples.
Suppose
the formula ‘r ⊂
Then
r can be substituted with the sentential variable p, provided that all the rs
are replaced with ps.
Hence
the rule of substitution permits us to derive ‘p ⊂
You
can even replace r with whole formulas such ‘a ∧
Hence
it is permissible to derive ‘(a ∧
It
was this Rule of Substitution that was used in the proof of the Euclid’s
theorem.
Another
Transformation Rule is the Rule of Detachment or Modus Ponens.
I
have written about Modus Ponens and other rules of inference in my previous
bedtime stories, so I shall refrain from going into it.
Now
let me state the basic axioms, the most elementary assumptions of the formal
system of Principia.
In
Principia, these axioms are labeled as primitive ideas and primitive
propositions.
These
primitive ideas and the primitive propositions are actually and
as-a-matter-of-factually primitive and basic.
It
simply cannot get more primitive than that.
I
as a student find this concept of primitive ideas and primitive propositions
very similar to those of life arising from chemistry.
Before
life (some 400 million years on earth since it formed), there was only
chemistry and certain rules for atoms and molecules to combine.
If
I recall my own bedtime stories on cosmology and astrophysics, I sense a
similar familiarity or analogy in nucleosynthesis at all the three levels, the
big bang, the stellar and the supernova nucleosynthesis with that logic of
Principia.
In
fact, I can safely say that Principia Mathematica is to mathematics (arising
out of logic) as the Alpher-Bethe-Gamow paper (April 1948, Physical review) is
to the field of cosmology.
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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