FOM: Boolean algebra vs Boolean ring
pratt at cs.Stanford.EDU
Tue Mar 10 14:31:48 EST 1998
(My apologies for the very delayed response to the following, which I
noticed in passing at the time but was too preoccupied with other
matters to respond to.)
From: Stephen G Simpson 1/31/98 4:04PM
>Note first that Boolean algebras are not isomorphic to Boolean rings.
This is only true for those who find the distinction useful, which
many people don't. For one thing the Boolean algebra operations (those
composable from whatever basis you choose to start from) are identical
to the Boolean ring operations. And the equations governing those
operations are also identical. In the respects of language and theory
therefore, Boolean algebras are not just isomorphic to Boolean rings,
they *are* Boolean rings. The distinction is at best "psychological"
as Sol Feferman allowed in a November posting.
Marshall Stone initially regarded rings as a helpful analogy in viewing
logic algebraically. But halfway through writing his celebrated 1936
paper on what is now known as the Stone duality of Boolean algebras and
Stone spaces, it dawned on him that Boolean algebras *were* Boolean rings.
This realization led him to rewrite a large portion of his paper. In a
subsequent paper he treated the corresponding duality for intuitionistic
It is understandable, in fact I suppose necessary, that those failing to
appreciate intuitionism from the algebraic point of view will also fail
to appreciate it as seen through the "mirror" of Stone duality. I have
found both the algebraic view and its "geometric" dual a great source of
insight into both classical and intuitionistic logic.
>Historically, where did "Boolean algebra" and "Boolean ring" come
>from? Boolean algebra was invented by George Boole in order to
>analyze logical reasoning, what we now call propositional logic. The
>notion of Boolean ring was invented much later, by nameless ring
>theorists, for their own purposes. One of their purposes was to
>subsume Boolean algebra under ring theory, in order to be able to say
>that "logic is just a branch of ring theory", as one prominent
>algebraist told me. This is incredibly crass, because it ignores
>Boole's original motivation.
Marshall Stone is not nameless, nor was he (at least initially) writing
as a ring theorist when developing Stone duality.
Whenever I hear X claiming that Y wrote Z for X's own purposes, I now
automatically translate it to "X doesn't understand Z." In 1979 the
extant completeness proofs for Segerberg's axiomatization of propositional
dynamic logic (including my own, "A Near-Optimal Method for Reasoning
About Actions," JCSS, 20:2, 231-254, April, 1980) all struck me as
too baroquely complex to understand. I tried translating the gist of
the argument into other frameworks, and eventually, following up tips
from Jonas Makowsky and Dexter Kozen, found that the universal algebra
setting permitted reducing most of the lemmas to well known algebraic
facts, leaving only half a page of argument as the novel part of the
proof ["Dynamic Algebras and the Nature of Induction," Proc. 12th ACM
Symposium on Theory of Computing, 22-28, May 1980]. This was something of
an epiphany for me, so I was greatly distressed when some of my closest
colleagues declined to accept this as a proof of the theorem, viewing it
instead as an exercise in algebra having no significant bearing on the
question I claimed to have answered. As far as I was concerned it was
the first proof of the theorem short and clear enough for me to believe.
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