Probability

Does it make sense to talk of "probability" with regard to existential claims? Consider the following propositions: (1) Rolling snake eyes is improbable. (2) The existence of Big Foot is improbable. Though I can't quite finger the distinction, it seems to me that the notion of probability is being used very differently in (1) and (2).

Comments

Yes, different notions are indeed at stake here. We need to distinguish physical probabilities from evidential probabilities.

Physical probabilities, also known as chances, are what are involved when we say, for example, that

An atom of plutonium 238 has a 50/50 chance of decaying within 88 years.
Smokers have a greater chance of getting lung cancer than non-smokers.
The chance of rolling 1-1 with a particular throw of a pair of fair dice is 1/36.

Note, the half-life of a plutonium atom is an objective physical property of it (a property it has independently of our beliefs about it). Likewise the probability of rolling "snake eyes" is a physical property of the chance set-up. And physical chance is related to another kind of physical property, namely the long-run frequency with which certain events turn up in a sufficient number of trials. For example, in the long-run, about 1 throw in 36 will turn up snake eyes. But philosophers argue over the relationship between the chance of a particular event and the long-run frequency in a sequence of trials.

Epistemic probabilities, by contrast, are measures of the weight of evidence. These are involved when we say, for example,

Given the evidence, it is very probable that human activity is responsible for the rate of global warming.
The absence of any large footprints shows it is very unlikely that the butler did it.
The evidence (or rather lack of evidence) makes it improbable that Big Foot exists.

When in a civil court, you are asked to judge "on the balance of probabilities", again you are being asked to weigh the evidence.

To help see the difference between weight of evidence and chance, note that you might well have evidence that makes it (epistemically) highly probable that a highly improbable (low-chance) event occurred. In the casino, we all seem to witness someone fairly draw the 10, J, Q, K, A of spades in sequence. A very low-chance event that on the balance of probabilities -- if we'd seen that enough checks against cheating were in place -- almost certainly happened! Note too that getting evidence for something doesn't make it more likely in the sense of increasing its chances. As we seek more evidence that you threw snake eyes three times in a row yesterday, we aren't changing the chances of that past event because it is over and done with: we can't change the physical properties of the past!

The degree to which evidence does support some conclusion is not just a matter of mere personal subjective decision. Perhaps though it is in some sense an intersubjective issue, a matter of how idealized unprejudiced rational thinkers would tend to weigh the evidence. Or perhaps not! Philosophers argue over this too.

Anyway, in sum, the two cases mentioned in the question do indeed involve two different notions of probability.

And in fact there's a third important notion of probability I should mention too, namely subjective probabilities. Subjective probabilities or credences are measures of personal degrees of confidence. Rationally or otherwise, I hold some beliefs much more confidently than others. My degree of confidence is reflected in how I behave on the basis of those beliefs. For example, I only risk crossing the road if I am very sure there is no oncoming traffic. I'll only take on a 60-40 bet that it will rain if I am that degree more confident that it will rain than that it won't. And so on.

Of course, if I'm rational and well-informed, I will apportion my degrees of belief to the available evidence, including evidence about objective chancces. My degree of belief that you will throw snake eyes on the next throw should ideally correspond to my best estimate of the objective chance of getting thatoutcome (if I'm overconfident, you'll be able to exploit that andextract money from me in unwise bets!). So our various notions of probability hang together more than just because chance, degree of evidential support, and degree of belief all obey (or at least under idealized conditions) the laws of the probability calculus. However, although they are linked, they are importantly different notions.

For a very lucid discussion of these things, see D.H. Mellor's short book, Probability, A Philosophical Introduction.

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