What is an argument?
It is basically a series of statements that
comprises the premises that you would have
to assume, and a conclusion which you would
have to evaluate as true or false.
The argument becomes valid when the conclusion
logically follows from the premises.
But what if this argument is invalid?
This is what you call a fallacy.
Let's take a look at two fallacies that assume
“if p then q”.
The first is the fallacy of the inverse.
It works by rejecting the "if" part in order
to conclude the "then" part.
Normally, you would have a valid argument
by affirming p in order to affirm q.
But in the fallacy of a converse, you do it
the other way around.
It is illegal to affirm q to conclude p; just
because p implies q does not always mean the reverse.
Let's compare modus ponens with the fallacy
of the inverse.
Modus ponens is valid when you have the conditional
"if p then q" and you affirm p,
then by modus ponens, you affirm q.
But, in the fallacy of the inverse, you reject
p in order to reject q.
This argument is invalid, and the conclusion
"not q" is not always logically true.
We can compare the fallacy of the converse
with the valid statement modus tollens.
Modus tollens is valid when you're given the
conditional "if p then q" then, it is legal
to reject q in order to reject p.
But instead, when you affirm q to affirm p,
now you have an invalid argument.
This is the fallacy of the converse; just
because p implies q doesn't mean that it's
converse will hold.
Here's an example of a fallacy of the inverse
compared with modus ponens.
In modus ponens, you would normally say "All
dogs are hairy.
My pet cotton is a dog.
Therefore, Cotton is hairy."
Now, that is a valid argument.
However, it is invalid to say "All dogs
are hairy.
My pet cotton is not a dog.
Therefore, cotton is not hairy."
You can verify this by drawing an Euler diagram.
In the fallacy of the inverse, it is invalid
because cotton is not a dog doesn't mean that
cotton will not necessarily be hairy.
Here's an example for the fallacy of the converse.
It is valid to say "All dogs are hairy.
My pet cotton is not hairy.
Therefore, cotton is not a dog".
In the fallacy of the converse, it is invalid
to say "All dogs are hairy.
My pet cotton is hairy.
Therefore, cotton is a dog"
This is invalid because just because she said
that cotton is hairy doesn't mean
it's going to be a dog.
It might be a hairy cat.
So, here are the fallacies of the inverse
and the fallacy of the converse when displayed
in a truth table.
Unlike valid arguments where there's only
one truth value, mainly true, these two fallacies
are not always true.
In the highlighted case where p is false and
q is true, both fallacies become false.
So, the fallacies are not tautologies, and
that is just the tip of the iceberg.
There are many other fallacies that have defects
in content.
Even though there may not be a problem with
the logical structure.
For example, the fallacy Argumentum Ad Hominem
has its conclusion justified by attacking
the arguer, whether to its character or to
its personality.
Argumentum Ad Populum appeals to the majority
of the audience.
It's like the bandwagon principle.
A product is good because 90% of the consumers
agree that it is good.
This is an example of Ad Populum—an appeal
to popularity.
Another fallacy is Appeal to Authority.
Historically, there is an argument that you
should take lots of vitamin C.
That's why we have daily supplements and that is because someone of a reasonable [or a high] authority rank
said that.
Just because someone who occupy—who has
won a Nobel Prize said that “vitamin C is good for you”
doesn't necessarily mean that—doesn’t necessarily agree with the scientific evidence of the effects
of vitamin C.
Another fallacy is False Cause.
It's like saying “X is correlated to Y so,
X causes Y”.
An example: whenever I wake up, it will rain
today.
So, my waking up caused the rain.
This is a false cause.
And, there's also the fallacy of Hasty Generalization.
The conclusion is justified by very few supporting
examples.
A famous example links autism with vaccination,
despite this not being true.
The argument states that due to a small number
of cases that seem to link autism with vaccinations,
then we would say that autism should be linked
to vaccination.
That is a hasty generalization.
It does not consider a reliable and sufficient evidence.
And here's one more: Argumentum Ad Baculum,
when in doubt, use force.
This comic is self explanatory.
There are some video resources on fallacies
and their examples—two, to name a few.
Now, I would like to talk about deductive
and inductive reasoning.
Consider the following classical statement:
"All men are evil.
Socrates is a man.
Therefore, Socrates is evil."
This is a classical example of inductive reasoning.
You start with something general “all men
are evil”, and you come up with a specific case:
“Socrates being a man must be evil”.
Inductive reasoning goes in the other direction.
So, in a deductive argument, the premises
yield a logical conclusion.That is, you start
with something—you assume a general statement
and you come up with a specific case.
In an inductive argument, the premises merely
support a plausible conclusion.
Here's a mathematical example comparing deductive
argument and inductive argument regarding
even numbers.
In a deductive argument, you would start by
saying that “All multiples of two are even.
12 is a multiple of two.
Therefore, 12 is even.”
In an inductive argument, the idea of an even
number is based on examples.
We know that 2 is an even number, 12 is an
even number, 22 is also an even number.
So, inductively you can say that all numbers
ending in 2 are even.
Although, we still need to show if this conclusion
is indeed true.
And it is indeed true; all numbers ending
in 2 are multiples of 2.
Therefore, they are even.
Here's our famous example: If p is prime consider
the number 2 to the power p minus 1.
Do you think it's always prime?
Now, we can verify for a few examples of p.
It is true when p is 2, 3, and 5.
So by inductive reasoning, we believe that
2 to the power p minus 1 is prime for any
prime number p.
However, the following counter example, 2
to the power of 11 minus one, states that
the conclusion must be false.
So, you see, in an inductive reasoning, you
only get a plausible conclusion that is supported
by a few or enough premises.
If even just one counter example can be found
then the conclusion is definitely false.
Let me introduce you to Pierre de Fermat,
one of the famous mathematicians in the 17th century.
He was interested in a problem that basically
extends the Pythagorean theorem.
Now, I would like to bring your attention
to a very famous quote.
He wrote in a margin of a mathematics textbook, "It is impossible to separate a cube into two cubes, or fourth
power into two fourth powers, or in general
any power higher than the second into two
like powers.
I have discovered a marvelous proof of this,
which this margin is too narrow to contain."
This is a historical example of an inductive
argument, called Fermat's Last Theorem.
Now, think of the theorem as a major discovery
or the logical conclusion of a deductive argument.
Historically, at the time when Fermat's Last
Theorem was first stated, it was not actually
a theorem, but it was rather, a conclusion
by inductive argument.
We can dissect Fermat's quotes in the following
manner and you can see how that translates
to an inductive argument.
Consider the statement that there are no natural
numbers a, b and c such that a to the power
of n plus b to the power of n equals c to
the power of n.
Now, this is true when n equals 3, also true
when n equals 4.
Mathematicians have also verified this for
other values of p.
But at the time, it was not sure when the
statement p is true for all n greater than 2.
This is what you call Fermat's Last Theorem.
And it gets a little bit interesting.
The good news is Fermat's Last Theorem is
now a theorem. Statement q is a logical conclusion
from a deductive argument.
However, it took 358 years for mathematicians
to figure out that Fermat was right all along.
Sir Andrew Wiles provided the first successful
verification of Fermat's Last Theorem.
A little story about Andrew Wiles.
He did publish—he did announce the proof
of Fermat's Last Theorem, however, some mathematicians
later realized that there was a problem with
his arguments—a hole in his proof.
So, it took a few more years and the help
of one of Wiles' students in order to finally
plug that hole.
So, it was really a concerted effort of mathematicians
around the world who are interested in number
theory to show that Fermat was right all along,
and that took 358 years.
I would like to show you how logic can be
used to solve mathematical problems.
Actually, there is an algorithm or a series
of steps popularized by George Polya in his book:
"How to Solve It".
You can consider this as a manual for solving
all of your mathematical problems.
It's basically a four-step process.
The very first thing you need to do is to
understand the problem.
Of course, you're faced with a problem, so
it is natural to understand it.
Once you have an understanding, then you need
to come up with the appropriate strategy.
And it turns out that inductive and deductive
reasoning are two of many strategies.
You can also do pattern searching or reducing
the problem into smaller bite-sized pieces
or finding a similar problem.
Once you have a strategy, carry out the plan,
see if it works, look back and see how this
might be applicable to other related problems.
So, you've learned what it's like for an argument
to be invalid and you have encountered fallacies
of the converse, fallacies of the inverse,
and other real-life fallacies.
You've also learned how to argue in a deductive
reasoning—using deductive reasoning—and
inductive reasoning.
And, you have also encountered some historical
examples of inductive reasoning.
In particular are the 350 years old mathematical
problem.
So, I hope that this helps you think better
and think more logically.
I wish you all the best.
Thank you very much.
