April 23, 2017 Sunday
Bedtime Story
Logical Truths and Rules of Inference
All these logical constants that I had listed last night can
either be logical connectives or quantifiers.
Here for instance the symbol
and
are logical connectives.
On the other hand the symbols
and
are quantifiers; the former
being the universal quantifier and the latter being the existential quantifier.
So logical truths are analytical statements that are considered to
be necessarily true.
The term “necessarily true” implies that the statements considered
so simply cannot be untrue and there can be no possible situation wherein a
need would arise to reject the logical truth.
“A positive integer can either be a prime or not be a prime” is an
example of logical truth.
It is a necessary truth that cannot be debated.
Yet even this does not explain the assumption that was used for
the statement:
“Then q can either be a
prime or not be a prime”.
The point being, how did q, an unknown variable also known as
sentential variable in the language of formal logic, replace positive integer.
Here in kicks in yet another basic premise of logic that goes by
the name of rules of inference.
Rules of inference in logic is a kind of transformation that
preserves the truth.
I had written about the most common rule of inference that goes by
the name of modus ponens.
It is quite simple and yet very surprisingly I nowhere during my
whole life of education ever recall having been taught formally.
What is says is this, “If P implies Q, and if P is asserted to be
true, then Q must be true”.
Formally,
P →
Q, P ⊢
The other common rule of inference in logic is the modus tollens
or denying the consequent.
What is says is this:
If P then Q; and if not Q then not P.
Formally, it can be written as:
P →
This rule of inference in logic is attributed to the Greek Stoics.
Stoicism is one of those philosophies much like Buddhist
philosophy (Note that I refrain from using the word Buddhism as it has a
religious connotation), that has greatly appealed to me.
I will talk a little bit of stoicism in the nights to come and
whence its appeal (to me) comes from.
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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