May
07, 2017 Sunday
Bedtime
Story
Tautology in Formal Logic
In
the last night’s bedtime story, we had seen what tautology is concerning the
art of discourse.
Today
we shall consider tautology from the perspective of formal logic.
Let
me provide you with two simple tautological formulas of formal logic:
(A ∨
Since
A is the only sentential variable and A can be assigned one of the truth
values, true or false, then its negation A has to be assigned the opposite
value.
The
second example is more intricate and far more interesting.
((¬A →
→
A
If
not-A implies B and not-A implies not-B, then not-A must be false and A must be
true
This
simple logical tautology in rhetoric logic manifests as reductio ad absurdum.
Reductio
ad absurdum is a form of argument in which a statement is disproved either by
showing that it leads to an absurd conclusion or a statement is proved by
showing that if it were not true, it would lead to an impossible conclusion.
In
formal logic and mathematics, reductio ad absurdum manifests as proof by
contradiction.
As
is seen from the example, since both the proposition B and its negation ¬B are
derivable from the premise ¬A, then ¬A premise has to false.
So
now let us get back to Gödel and the testing of consistency.
We
were looking for a structural property that would be capable of passing from
one formula to another like a genetic trait.
How
about tautology as that trait?
Bizarre,
right!
Let
us at least see and try it out.
Have
a look at the very first axiom that we had proposed for our formal system.
It
is (p ∨
Or
in other words, if either p or p, then p.
This
is most clearly a tautology.
It
becomes more evident if you replace p with any ordinary English statement.
Let
us replace the sentential variable p with ‘Bombay is an overpopulated city’.
Then
the first axiom would read something like this:
‘If
either Bombay is an overpopulated city or Bombay is an overpopulated city, then
Bombay is an overpopulated city’.
It
is plainly obvious that this is a tautology.
In
the nights to come, we shall consider the other three axioms and see whether
they are tautologies or not.
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.
Advertisements
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:
No comments:
Post a Comment