August
14, 2017 Monday
Bedtime
Story
Taking Care of the Vicious Circle of Logic
Last
night we saw how according to Tarski a truth statement is best stated.
“Coal
is black” is true if and only if coal is black.
We
had labeled it as statement [A].
Many
may argue that in [A] there exists a basic flaw that goes by the name of
vicious circle in logic.
They
point out that the word “coal” is occurring both in the definiens and the
definiendum.
But
that can be explained.
The
word “coal” albeit the same on the two sides has different usage in those two
places.
On
the right side in the definiendum “coal” is syntactical where it the subject of
a sentence - coal is black.
Now
on the left hand side, the definiendum it is part of a whole phrase within
quotation marks “coal is black”.
The
whole phrase “coal is black” is being treated as one block about which
something is being said.
If
you recall, this point had come up when we were discussing Gödel’s theorems.
It
was one of those bedtime stories when I tried to explain the difference between
mathematics and meta-mathematics.
While
mathematics actually deals with signs and numbers, the meta-mathematical
statements even if they contain the numbers, signs and formulas treat them
simply as names and not anything more.
So
we can write 5 + 5 = 10 as a statement of arithmetic.
But
when we say “5 + 5 = 10” is a true mathematical statement, the equation is
being kept in quotes and is being treated as a single entity wherein each
number or sign is mathematically irrelevant.
Similarly
in our present case scenarios, the very fact that the phrase is enclosed within
quotes calls for taking it as grammatically a single entity.
This
makes the use of the word “coal” on the left hand side just a part of the whole
object “coal is black” about which something is being said.
Hence
the coal on the left hand side is not a syntactical part of the whole phrase.
The
words that are not a syntactical part of the definiendum cannot create the
vicious cycle that was being referred to and thus that fallacy is taken care
of.
Tarski
says that there is yet another way in which this vicious cycle can be taken
care of.
Have
a look at the formulation [A] once again.
[A]
“Coal is black” is true if and only if coal is black.
Here
the method deployed in naming the sentence or the expression was to put it
inside quotation marks.
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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