Deductive reasoning defined

By
Waldo
on December 23, 2004 1:00 PM | | Comments (0)

Deductive reasoning refers to the process where one derives a conclusion (C) starting with a known (or assumed) set of premises (P). An example may illustrate this better:

 P: Assume all men will die someday
 P: Assume bin Laden is a man
 C: bin Laden will die someday

This deductive step was based on the logical principle that if A implies B, and A is true, then B is true.

Deductive reasoning can fail. This is seen in the following famous paradox:

Let

a = b

Thus,

a2 = ab
a2 + a2 = a2 + ab
2a2 = a2 + ab
2a2 - 2ab = a2 + ab - 2ab
2a2 - 2ab = a2 - ab

Rewrite this as:

2(a2 - ab) = 1(a2 - ab)

Dividing both sides by (a2 - ab) gives:

2 = 1

QED.

See the problem with this line of reasoning? This is a very good example that one cannot apply deductive reasoning blindly for mathematical proofs.

Hint: Substitute the original assumption back into the numerator...

Another, much more subtle example is given below. See if you can find the error made in the reasoning:

\begin{displaymath}\frac{-1}{1} = \frac{1}{-1}\end{displaymath} (1)
\begin{displaymath}\sqrt{\frac{-1}{1}} = \sqrt{\frac{1}{-1}} \qquad \textrm{(taking the square root on both sides)}\end{displaymath} (2)
\begin{displaymath}\frac{\sqrt{-1}}{\sqrt{1}} = \frac{\sqrt{1}}{\sqrt{-1}} \qquad \textrm{(Simplifying)}\end{displaymath} (3)
\begin{displaymath}\textrm{In other words,} \quad \frac{i}{1} = \frac{1}{i}\end{displaymath} (4)
\begin{displaymath}\textrm{Therefore,} \quad \frac{i}{2} = \frac{1}{2i} \qquad \textrm{(Dividing both sides by 2)}\end{displaymath} (5)
\begin{displaymath}\frac{i}{2} + \frac{3}{2i} = \frac{1}{2i} + \frac{3}{2i} \qquad \textrm{(Adding }\frac{3}{2i}\textrm{ to both sides)}\end{displaymath} (6)
\begin{displaymath}i\left(\frac{i}{2} + \frac{3}{2i}\right) = i\left(\frac{1}{2i} + \frac{3}{2i}\right) \qquad \textrm{(Multiplying both sides with } i\textrm{)}\end{displaymath} (7)
\begin{displaymath}\frac{i^2}{2} + \frac{3i}{2i} = \frac{i}{2i} + \frac{3i}{2i} \qquad \textrm{(Multiplying out with } i\textrm{)}\end{displaymath} (8)
\begin{displaymath}\frac{-1}{2} + \frac{3}{2} = \frac{1}{2} + \frac{3}{2} \qquad \textrm{(Simplifying)}\end{displaymath} (9)
Which reduces to $1 = 2$ (Adding terms)
QED.

Hint: Do not make any assumptions about algebra other than those you have explicitly seen proofs for. Or mail and ask me :)

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