I'm asked to Deduce the alternate form of PMI from WO as a homework problem. To me, this sounds as if I should be doing some form of direct proof of its existence, however, every proof I see that the basic form of PMI follows from WO uses a proof by contradiction to establish its validity.
My question is, is this generally how its done? Is there a way to do it directly instead of by contradiction?
Sometimes when you want to prove something is true, you can approach it from many different angles, and both direct and indirect (contradiction) methods are possible.
But the PMI and WO are core primal concepts; each principle when held up to a 'carnival math mirror' looks like the other statement. The PMI is about 'higher-higher-higher' while the WO is about 'lower-lower-lower'.
Think of a penny as representing a truth. You can see a penny by looking at it face up - heads, or face down - tails. Maybe we are introduced to the 'penny truth' by looking at it heads-up. When we look at the tail side, we want to know if it is a penny. So, suppose it is not. Then when flipped over it will still not be a penny. But when you do flip it, you see the heads side. A contradiction.
You might find the link, minimal counterexample, of interest.
