Perfect Induction

The problem of induction is very significant, and it pertains to a great deal of our experiences.  Inductive reasoning, as opposed to deductive reasoning, extrapolates from one instance or case of something to another using fallacious means.  The resulting problem of induction acknowledges that inductive reasoning cannot establish the certainty of a future event purely from past occurrences.  For instance, the sun having risen each past morning of my life does not mean it will rise tomorrow morning.  Likewise, just because most people have eventually died does not mean that I myself will one day die.  What is true of the part is not always true of the whole.

Let's say that I'm trying to figure out if all of the boxes on a specific ship are red.  I see three boxes laying out on deck, but the storage areas below contains another one.  The boxes that I see are indeed red.  Does this mean that the other box down below is the same color as the ones I can see?  By no means!  Normal induction is incapable of demonstrating that the other one is red, and deductive reasoning dismembers any claim that because some of the boxes are red all of the rest must also be.

This is where perfect induction (which is sometimes alternatively called proof by exhaustion) is useful.  Perfect induction is when all members of a set can be examined one by one to find if they all meet a certain criterion.  When the boxes are spread out in multiple places, induction is unreliable, because the fact that one or several boxes are red does not mean the others are as well.  If I have every single box from the ship in front of me, though, I can see if they are all red.  In fact, this is the only way I could ever legitimately say that all the members (boxes) in this particular set (the group of boxes on the ship) are red.  I could otherwise never know if that is the case from only seeing some of them.

Induction is utterly unreliable as a method of securing knowledge--unless it is perfect induction and all members of a set are known.  Short of proof by exhaustion, induction inherently relies on the fallacy of composition and the non sequitur fallacy and thus is irrational, as it does not follow from one member of a set being a certain way that all other members of that set must be the same way.  The only thing that by necessity is true of all members of a set is that logical truths govern them (each set member is what it is and so on).  The unfortunate reality about perfect induction, though, is that it is not always usable because every member of a set is not always available for examination.