Sept. 6, 2002, 3:46 p.m.

## Deductive reasoning defined

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: