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:

    a = b  
    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  

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:

Complex equation
Complex equation

Which reduces to 1 = 2 (Adding terms)


Hint: Do not make any assumptions about algebra other than those you have explicitly seen proofs for.