Since probability is the completion—we do not say “branch”—of logic, and logic is the science of the relations between propositions, probability is also the study of the relations between propositions.
A (provable) rule of logic is that you cannot prove a contingent proposition starting from a necessary truth. Since it is a rule of logic, it is a rule of probability, too. Contingent propositions are the ones most familiar to us. “The car will start”, “The price of lemons will increase”, “A proton is made from quarks”, and on and on. Propositions which are themselves not necessarily true, which even if they are true contingently (lemon prices have risen in my neighborhood), don’t have to be true. The world would continue were lemon prices to remain flat, and the universe could have been built with quarkless protons.
In shorthand, if we have some proposition P and evidence E, Pr(P|E) = Pr(P|ET), where T is a necessary truth. T does nothing. It also does nothing when P is contingent and there is no other evidence than T. Thus Pr(P|T) is not a unique number, but because a tacit premise is P’s contingency we know that 0 < Pr(P|T) < 1. Note that the inequalities are not strict.
I think reader JH pointed to whathisname who wrote a book defending Objective Bayesianism (names escape from me faster than female interns from the late Ted Kennedy’s office). Williamson, maybe. Anyway, he and even subjective Bayesians insist on precise probabilities and so would pick 1/2 for Pr(P|T), which is screwy, as an example will prove.
The one that came up in class was the proposition P = “The Patriots win next year’s Super Bowl.” The necessary truth was the tautology T = “The Patriots might win”, which only acknowledges the contingency of P. T is equivalent to T = “The Patriots might win or they might lose.”
It is well at this point to remind that tacit premises about word definitions and grammar always accompany all our written evidence.
Now T is true no matter what, if by “lose” we mean not that they entered the final competition and lost, but that they do not win, such as is the natural state of the Detroit Lions. Any team which does not take the trophy (or whatever it is; I don’t follow football) “loses.”
Students have a difficult time wrapping their heads around (logical, not grammatical) tautologies, not being familiar with them. T is true no matter what, even if the Patriots disassemble their team tomorrow, even if the NFL goes bust or the universe ends.
Again, many like to say Pr(P|T) = 1/2. But we could have also said T2 = “The Lions might win” or T2 = “The Cowboys might win” and it cannot be true that the probability of each of these is 1/2.
Williamson (I think that’s the name; I’ll look it up later) wants probability to be a unique number and says so. He wants that unique number to be, in the face of uncertainty, the maximally equivocal. That’s what he forces the 1/2. But I could have also used the tautology T3 = “Tomorrow it will rain or it won’t” and surely “Pr(P|T) = 1/2” makes no kind of sense.
Indeed, let Q = “I wore an orange pocket square”, which is true based on observation yesterday. P is the same. Then Pr(P|Q) is undefined or is still the same open interval if we insist the tacit premise of P’s contingency. Q gives no evidence of P, and so there is no possibility of the probability being 1/2.
And there is no problem in asking “What is Pr(P|Q)?” just as there is no problem asking “Given x + y = 7 and w = 13, solve for x.” This is not a typo. The w is not probative or x or y; it adds no information. We cannot come to a unique single number of x, and there is no reason in the world we should insist on one.
That public house, where we usually spend our first night, was mobbed with USA-Ghana watchers, so it was off to Ruloff’s, an indifferent and non-air-conditioned bar in College Town. Greasiest appetizers (a contradiction in terms if ever there was one) you can find. But convenient. They had a small badly tuned television where we were able to celebrate the victory.
I’ll also note for Luis, if he’s reading this—Luis is from Portugal—that suicide is never an option.
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