Bit of logic (a favorite example) from our teacher Lewis Carroll: given “All cats understand French and that some chickens are cats”, what can we say about the proposition “Some chickens understand French”?
We can say that it follows, that its probability is 1, that it is proved, that it is deduced, true. True conditionally, true given the premises, locally true.
But we also know it is false. Universally false, false because we observe that no cats understand French and that no chickens are cats. Universally or necessarily false, but conditionally true. And also conditionally false, conditional on different premises (such as “No cats understand French” and “No chickens are cats” and, also necessary, “No non-human animals, and very few human animals, understand French”).
Conditional truth is no trouble for logic, because logic is merely the study of the relation between propositions. Logic has nothing to say about the propositions themselves, not formally. Not about where they come from, their utility, their practicality or their lyricism. This is why the premises do not need to be empirical for logic to “work”.
For instance, given “All Martians wear hats and George is a Martian” it is conditionally true that “George wears a hat”—even though we know there are no Martians. Examples abound. See this site for a wealth of Carroll’s fun logic puzzles.
Probability, part of logic, is also concerned only with the relation between propositions and not the origin or usefulness of those propositions. We deduced “George wears a hat” had probability 11, just as we deduce the probability of “George wears a hat” is 3/4 given the premise “Three-quarters of Martians wears hats and George is a Martian.”
Probability, like logic, has no difficulty with non-empirical propositions. This is how we know it is (conditionally) true that “One creature must come out” given the premises “A gnome, fairy, and Godzilla are in a room and only one must come out”, and it is how we know the probability of “Godzilla comes out” is 1/3 given those same premises. We deduced the certainty of both conclusions (we are certain the probability is 1/3 in the second case).
Probability got its start answering empirical questions, mostly about gambling and people’s guilt (see Jim Franklin’s The Science of Conjecture: Evidence and Probability Before Pascal). Aristotle also relied on many empirical questions when delineating logic, but somehow logic came to be seen to be general, whereas probability hasn’t. Probability is empirical to many.
The empirical bias in probability is strong. It accounts for the frequentist fallacy that no probability exists except limiting frequencies of empirical events. A frequentist must remain mute about hat-wearing Martians, and about every other non-empirical proposition, including counterfactual ones. For example, historians often debate about whether there would have been a World War II in Europe if America had not joined World War I. Germany would have won WWI, these historians say, thus ensuring a kind of peace, or at least a Germany victory would not have created the conditions necessary for the National Socialists in that country and the International Socialists in Russia to come into existence.
Now those are perfectly understandable (and debatable) propositions, as are most “What if?” questions. And probability (and logic) can handle them, but no system which is empirically based can. Given (Stove’s example) “Bob is a winged horse” it follows that “Bob is a horse”, a statement which makes sense in logic and probability-as-logic, but a frequentist must pretend it is incomprehensible because there is no way to construct any empirical relative frequency.
More subtly, the empirical bias accounts for de Finetti’s and Ramsey’s error of defining probability as coherent gambles. This is backward. That coherent gambles are good decisions is result of probability, not its definition. (See this about Dutch Books.) There will be no “payoffs” for “events” which will never occur (or never not occur).
Besides, coherence is a weak criterion and mixes up consequences of evidence with evidence itself. A subjectivist can say the probability Godzilla comes out is 0.015764 (or any 0-1 number). As long as he also insists the probability Godzilla does not comes out is 1 – 0.015764, he is coherent. And you cannot prove his probability is wrong if probability is a gamble. No empirical evidence will ever be available.
1In measure theory “with probability 1” has a technical meaning which I do not here use. I mean this phrase in plain English.