October 16, 2005

Some Unnecessary Formalization involving Deontic and Epistemic Logic

[UPDATE: Fixed some egregious typos.]

So maybe Mark A.R. Kleiman's point about Cliff May's silliness can be formalized like this, where O is the operator for obligation, K is the knowledge operator, and both operators are relativized to a subject:

Secrets are not secrets tout court, but secrets that one person ought to keep from another. So let Sa, b(p) be, roughly, "p is a secret that a ought to keep from b."

Sa, b(p) =df Oa~Kb(p)

Let q stand for "Valerie Plame is a CIA officer."

For Valerie Plame to be a secret CIA officer is for q to be a secret that must be kept by those with security clearances, say Lewis Libby, from those without, say Judy Miller. (Assuming she didn't have one--May doesn't try to invoke the possibility that she might have.)

In other words, the secret can be expressed is SLibby, Miller(q).

May points out that, though Libby brought about KMiller(q), arguably ~KMiller(SLibby, Miller(q)).

But, expanding the definition of SLibby, Miller(q), we get


which, combined with KMiller(q), entails that Libby has violated an obligation.

This has been "Unnecessarily formal and complicated explanations of things a three-year-old should be able to understand," part I. Why Mr. May doesn't understand things a three-year-old should be able to understand is an interesting question, but its analysis awaits a satisfactory formalization of the operators "understand" and "pathetic hack."

[In fact, what is relevant is that whether b knows that p, but whether a brought it about that b knows that p. So we might want to reanalyze Oa in terms of Belnap/Perloff/Xu's "see to it that" operator: Oa(p) =df ([a stit: ~p] -> X), where X is a sentential constant standing for "wrongdoing has occurred," and the -> is something stronger than material implication. Then Sa, b(p) becomes ([a stit: Kbp] -> X). Kleiman's analysis could easily be reframed in this system.]

Posted by Matt Weiner at October 16, 2005 04:06 PM