The Nature Of Probability And Statistics. Ithaca Teaching Journal, Day 1

Ithaca was once voted the “Most Enlightened City.” I can’t say whether this is true, but I can tell you that the Cornell’s Statler hotel’s refrigerator contain a placard, which reads

SUSTAINABILITY at Cornell [strangely, there is no &tm;]

This absorption refrigerator is eco-friendly and therefore may take longer to cool items.

I at least can verify the truth of that warning—except that I would insert the modifier much before longer. I am still waiting for the items I placed in the refrigerator last night to cool.

Incidentally, and before we start, it might help to know that there is no difference between probability and statistics. The later is the practical version of the former, but properly speaking, they are identical. Further, both—or rather, the one—are no different than epistemology, which is the philosophy of knowledge. You might say that probability is the mathematical branch of epistemology concerned with logic, but since logic is most of epistemology, so is probability.

The divisions of logic are there for human convenience. They are: deduction, induction, and non-deductive logic. The first you probably know, the search for valid conclusions from fixed premises. The second are those cases like flames which are hot, i.e. inference from observed cases which contain no counter-example to some conclusion and that there is no counter example to this conclusion. To clarify, we say the next flame we meet will be hot because all the flames we have met before are. (Importantly, we do not say that it is because most other flames were hot, but all.)

Non-deductive logic contains all the other cases, what is thought of as traditional probability, i.e. dice games, medical trials, which is to say, the parts of empirical science that are not deductive or inductive. It also contains so-called paradoxes, like the Liar, and everything else that won’t fit neatly into a bin.

All epistemology—hence all science, logic, and mathematics—has ultimate foundations in intuition. This is obvious in the case of logic and mathematics, both of which rest of unproved and unprovable axioms, i.e. beliefs which are assumed true on faith, on our intuitions. That is, there are no observations that can be made that would verify or invalidate an axiom. Thus, any science which uses logic and mathematics (in any way) always—as in always—forms a chain which anchors to intuition.

Of course, some so-called axioms are eventually proved true or are invalidated by empirical observations. But that only means that that axiom wasn’t. There is nothing in epistemology that says we won’t make mistakes.

I’ll use Stove’s simple demonstration that argument schema barbara must rest on intuition (this proof works for any argument schema, or any unique argument, too). Barbara is this:

(A) For all x, all F, all G, either ‘x is F and all F are G’ is false, or ‘x is G’ is true.

Empiricists claim that we learn the truth of logical arguments like (A) from observations such as

(B) Abe is black and Abe is a person now in this room and all persons now in this toom are black.

But that means that we must already know that (B) confirms (A). That is, we had to learn from somewhere that (B) is positive evidence for (A). How could you have learned that? Well, you might have learned from some observation (C) that observations like (B) confirms (A).

But how did you learn from (C) that (B) confirms (A)? You might posit a (D) where you learn that (C) that (B) confirms (A). You can see where this is going. Empiricism, a.k.a. experience, is not enough. You must have a start in intuition. This goes for math, logic, induction, and non-deductive logic. All your knowledge begins within intuition.

The numerical probability for all validly derived conclusions is 1, just as it is 0 for all conclusions which are proved false. All arguments begin with a list of premises which are assumed true—they need not be true with respect to some external evidence. For example

(P1) All men are mortal
(P2) Socrates is a man
——————————————————————
(C) Socrates is mortal

We can write Pr( C | P1 & P2) = 1. But (P1) itself is only assumed true. Whether or not it is depends on other premises.

The important thing to take away is that all statements of knowledge (math, logic, probability) are conditional. You cannot say whether a conclusion is true without always saying what the premises are, just as you cannot say what the probability of some conclusion is without first saying what the premises are. The only (seeming) exception is intuition: there are no premises which we can articulate for intuitive beliefs. (And properly, in the notation above, we should put Pr( C | P1 & P2 & I) = 1, where I represents how we know that (P1) and (P2) give (C).)

So much for foundations. The nature of probability next.