Logic is the study of relationships between propositions, and probability, which is the same, is thus the completion of logic to instances where the relationships are not certain.
From this it follows (by a long path which we’ll skip over today) that not all probabilities are calculable. And anyway, it is proved easily by example. Here’s is one which I heisted from Henry Kyburg Jr (Epistemology and Inference; chapter on “Chance”). The proposition of interest is:
q = “This (really quite tasty) bottle of 2009 Muga Rioja Reserve will break into N pieces when struck by a hammer”,
and we want to know the probability of q. The answer is that there is no answer: there is no “probability of q.” This non-answer answer holds for all q which are not self-referential, and the reason is that, as said, probability is a measure between propositions. Here there is only q; there is no between. This is the end of the lesson. If you’re in a hurry, there’s no need to read further.
Because the question makes some kind of sense, though, it encourages (what we can call) the Subjective heresy, which is the belief that probabilities are subjective “feelings.” The most virulent form of this heresy is the diversity-loving academic who spouts “all truth is relative” or “there is no ‘truth’.” Skip that. Subjective probabilists presented with q and asked its probability are tempted beyond resistance to provide an answer. The move requires them to provide unstateable evidence so that q can be put in a relation. Such evidence is in the form of a complex proposition, which might look like this:
e = “I feel this and that.”
Given e, the Subjectivist can say “the probability of q given e is X”, where X is usually stated with astonishing precision.1 The danger is e is likely to vary based on the Subjectivist’s last meal (fast-food Chinese produces very small Xs). Of course, if e is rigorously specified, then it is quite possible for “Pr(q|e) = X” to exist. The only problem is that this is not the answer to the question; instead, it is the answer to “Given e, what is the probability of q?” And nobody asked that.
Relative frequency is the number of times a proposition is true (given a set of observations) divided by the number of times a proposition is true or false (given the same set of observations). Relative frequency is not probability; further, relative frequencies often do not exist. Counterfactuals do not have relative frequencies, nor do hypotheticals (Martians wearing hats). Worse, followers of the Frequentist heresy (which is the claim that probabilities are RFs) are also prone to the Subjective heresy.
Notice that q does not specify how and under what circumstances the bottle is to be struck. If we are to keep q, unadulterated as logic requires,2 then these informational tidbits must remain forever unknown. The Frequentist, like the Subjectivist, is tempted beyond all ability to withstand and invents this information. “Suppose,” he might say, “we set the bottle on a concrete pavilion and have a drunken one-eyed orangutan belt it with a 20 OZ Solid Steel Nail Eagle Claw Hammer (my father’s and therefore my choice) whilst I try to feed it bananas.” This—and it must be something like this, with the experiment precisely defined—becomes the e. Or, rather, the e’, because it is still impossible to compute Pr(q|e).
The Subjectivist (as above) agglomerates emotions to e’ but the Frequentist must insist that the experiment he imagined be carried out. The results from these experiments (call them r) are tacked onto e’ with the result that, after a very long time, just after the last trump sounds to be exact, he finally can state Pr(q|e’r). Never wait on a Frequentist.
We’re back at the deeply unsatisfactory result that q has no probability, not in isolation, not without respect to some other proposition. Since we do not know what this other proposition is, we are unable to compute. Computing is scientific; science is numbers. Perhaps the “real life” quality of q is what causes people to invent information. They see the question (reminder: “What is the probability of q?”) as a scientific one, which it is not. It is logical and logic is not a branch of any empirical science. The tendency to invent is yet another symptom of the rabid scientism which permeates our culture.
1There are times when the Subjectivist heresy isn’t quite: these are when you have actual bottles and hammers in front on you and an experiment will be carried out. “What is the chance?” is then tangible and a tacit invitation to supply missing information. People may use their knowledge of hammers and bottles to hone in on a probability, though unless a person’s knowledge of the physics of breaking bottles is magisterial, this probability should be no more than a wide interval (yes, interval). It only becomes a fixed number when this person makes a bet (perhaps only a mental wager with himself), which is a decision to act in a certain way. A decision or bet is not a probability, but an act. And anyway, probabilities are immaterial, like logical statements, and are not acts.
2Believing you have God-like control over specified propositions is like telling your calculus teacher that you decided not to answer the questions he gave you, but ones you invented based on his questions.
Technical notes: Suppose we agree to amend our question with an e which describes an experiment, like with the orangutan. We are still ignorant of N. A secondary temptation arises to let N be any number from 0 (the bottle might not break) to infinity. Bottles cannot break into infinitely many pieces, but supposing they might allows mathematics to enter, and allowing mathematics to enter is what scientism demands. This in fine if the real experiment was of definite interest, but the approximation necessarily implies the calculations will be too certain.