Topics Logic

Question If the same proposition is derived from two different logical processes, are the answers still the same? Or to reverse the question, can the nature of the sub-premises or lower stages of logical reasoning yield the exact same conclusion? Thank you.

Accepted:
April 22, 2008

Comments

Why shouldn't two different chains of reasoning lead to one and the very same conclusion? Mathematicians often give different proofs of the same result. For example, Aigner and Ziegler's wonderful Proofs from the Book starts off with six proofs (chosen from many more) of the same proposition, i.e. Euclid's result that there is an infinite number of prime numbers. The very different routes to the same conclusion are illuminating, as they show up different connections between the fact that there is an infinite number of primes and other mathematical facts. But it is one and the same mathematical proposition that the different connecting proofs all home in on.

I've chosen a mathematical example first because of the question's emphasis on "logical reasoning". But the point generalizes to cover other sorts of grounds we might have for accepting a proposition. I take it that Jill is in the coffee bar, as she has just phoned me and told me she is waiting there for me right now. You take it that Jill is in the coffee bar as you've just glimpsed her through the window. Different routes to acquiring belief in one and the same proposition. Of course, Jill's phone call tells me other things too -- e.g. that she is cross to be kept waiting. And your glimpse tells you other things, e.g. that she is sitting in a window seat. I pick up one overall package of beliefs from my phone encounter: you pick up a different package from your glimpse. But that surely doesn't prevent one and the same proposition -- the proposition that she is in the coffee bar -- being in both packages. Why should it?

Question If the same proposition is derived from two different logical processes, are the answers still the same? Or to reverse the question, can the nature of the sub-premises or lower stages of logical reasoning yield the exact same conclusion? Thank you. Accepted:April 22, 2008

Question If the same proposition is derived from two different logical processes, are the answers still the same? Or to reverse the question, can the nature of the sub-premises or lower stages of logical reasoning yield the exact same conclusion? Thank you.

Accepted:
April 22, 2008