I simultaneously got behind and ahead of myself with my last post. Thankfully, some folks have responded to it, and it’s become clear that I need to spend a bit more time talking about logic and Coq rather than just diving right into problems in analysis. Before even doing that, thought, I want to address this tweet from Raul Miller. On my urging, Raul’s taken the challenge to pursue Jaynes, and has posted an initial sharply-observed critique of some material from the first chapter:

“Good mathematicians see analogies between theorems; great mathematicians see analogies between analogies”
…
“To state these ideas more formally, we introduce some notation of the usual symbolic logic, or Boolean algebra, so called because George Bool (1854) introduced a notation similar to the following… These symbols are only a shorthand way of writing propositions, and do not stand for numerical values.”

The sentence which bothers me is that last sentence that I quoted, and it bothers me because of the first sentence that I quoted.

Here’s the thing: in George Boole’s work, these statements represent numerical results. George Boole started out by expressing his concepts in terms of axioms, but those axioms have a well understood correspondence to numerical operations. Boolean multiplication is the operation we know as “Least Common Multiple” and Boolean addition is the operation we know as “Greatest Common Denominator”… So, possibly he was saying that boolean algebra does not represent numbers to avoid confusion between the notation he is going to introduce and the notation he is using here.

Raul’s right: Jaynes is speaking prescriptively, rather than descriptively, here. It’s not that Boole doesn’t intend the symbols to represent numerical values; it’s that Jaynes doesn’t. It becomes clear after a few more paragraphs that Jaynes is quite familiar with Boole’s work—two editions of it, even!

Evidently, then, it must be the most primitive axiom of plausible reasoning that two propositions with the same truth value are equally plausible. This might appear too trivial to mention, were it not for the fact that Boole himself (Boole, 1854, p. 286) fell into error on this point, by mistakenly identifying two propositions which were in fact different—and then failing to see any contradiction in their different plausibilities. Three years later, Boole (1857) gave a revised theory which supersedes that in his earlier book; for further comments on this incident, see Keynes (1921, pp. 167-168); Jaynes (1976, pp. 240-242).

So one problem that Jaynes observes is that Boole 1854, at least, is inconsistent. Since we’re still just talking about logic rather than probability proper, this is unacceptable. You might argue that we should give Boole a break, since it was early days. But Boole antedates Laplace, and in fact, as we’ll see later in Jaynes, Boole’s work on probability was written in reaction to Laplace.

In addition, though, it makes no sense to ascribe numerical values to propositions! Consider: $$A = \verb|Christopher Columbus discovered America in 1492.|$$ $$B = \verb|It will rain today.|$$ $$AB$$ The conjunction, using Boole’s notation, reads “Christopher Columbus discovered America in 1492 and it will rain today.” That’s a perfectly sensible conjunction that we may wish to determine the probability of. Boole’s algebra won’t get us there in either its 1854 or 1857 incarnations, but as we’ll see later, Richard Cox’s sum and product rules will.