April 25, 2017 Tuesday
Bedtime Story
The Third Rule of Inference: Contraposition
The third rule of inference is called contraposition.
In general English, the word ‘contrapositive’ refers to either
converse or inverse.
In logic, contraposition applies to conditional statements.
For instance,
If the sky is dark and cloudy, then it is bound to rain.
Conditional statements have the “if…then…” relationship.
In other words, it has an antecedent and consequent.
In our example, dark and cloudy sky is the antecedent condition
and the rain is the consequence.
The contrapositive of a conditional statement has its antecedent
and consequent inverted and flipped.
So the contrapositive of the statement that we took as an example
would be:
If it is raining, then the sky ought to be dark and cloudy.
The third rule of inference i.e. contraposition states that a
conditional statement is logically equivalent to its contrapositive.
So if we again refer back to our example, what the contraposition rule
says is this:
The statement ‘If the sky is dark and cloudy, then it is bound to
rain’ is logically equivalent to ‘If it is raining, then the sky ought to be
dark and cloudy’.
All these examples should suffice to give a general idea of the
rules of inferences that are used in logic.
You will notice that all of these were never taught yet they were
all implied throughout our education in nearly all the subjects, except may be
in the Bible class that I had the misfortune to painfully endure.
Let us now go back to the Euclid’s theorem on the infinity of
primes.
In the proof, if you seek to go back few nights ago, was a
statement:
“Then q can either be a prime or not be a prime.”
The rule of inference that was used to get this statement is known
as the “Rule for Substitution for Sentential Variables”.
According to this rule, a statement can be derived from another
that contains a variable by substituting any statement for each occurrence of a
distinct variable.
In our case, the variable was p and the distinct variable that we
replaced it with was q.
I know it is banal and may be overdone, but advances in formal
logic brought out into open the rules that were being applied almost
subconsciously for ages.
It can even be said that mathematicians and logicians of ages have
been using reason for thousands of years without explicitly being aware of the
underlying principles.
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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