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:
Which reduces to 1 = 2 (Adding terms)
Hint: Do not make any assumptions about algebra other than those you have explicitly seen proofs for.