Logic of belief, the SPLIT principle

Abstract.  A recent paper (Ding and Holiday 2020) exhibits a fairly intuitive principle (SPLIT) and shows that it cannot be satisfied if the algebra of propositions is atomic.  That is correct, but a slight liberalization of neighborhood possible worlds semantics will accommodate it.  This another interesting point about infinities.

Reference: Ding, Yifeng and Wesley Holliday (2020) “Another problem in Possible World Semantics”. https://escholarship.org/uc/item/27k2f44p

The SPLIT principle can be stated for an arbitrary modal operator Q, but it is illustrated with the modality ‘it is queried whether‘.(but see Postscript about different examples, like belief):

          SPLIT.  If A is true then [it is possible that A is true and is queried, and it is possible that A is true and not queried)

            A → [♦(A & QA) & ♦(A & ~QA)]

The argument about atomicity is simple.  An atom is a proposition which is entailed only by the impossible proposition and itself.  (Using the same notation for propositions as for sentences:)

 therefore, if A is an atom then either (A & QA) or (A & ~QA) is just A, and the other is the impossible proposition.

In Ding and Holliday’s standard presentation of neighborhood possible world semantics, every set of worlds is a proposition. So there will definitely be atoms: if w is a world then {w} is a proposition, and includes no propositions other than itself and the empty set.

In my two previous posts on the logic of belief I presented the neighborhood semantics in a more liberal form:

  A model is an n-tuple <W, F, N, …> The worlds form a set, W.  The propositions are a Boolean algebra F of subsets of W, and F includes W. The function N assigns to each world w a set of propositions N(w), called the neighborhood of w. 

Can this generalization to propositional algebras (still algebras of sets) accommodate SPLIT?  As Ding and Holliday show, in effect, it can do so if atom-less Boolean algebras can be fields of sets. 

And that is the case by:

Stone’s Representation Theorem. Every Boolean algebra is isomorphic to a field of sets.

Note.  Stone’s representation theorem is what made possible world semantics possible in the first place.  Reference for a reader-friendly presentation: Theorem 6-15 (page 201) of James C. Abbott (1969) Sets, Lattices, and Boolean Algebras. Boston: Allyn and Bacon.

POSTSCRIPT

Belief would not yield a plausible instance of SPLIT , since it is generally a principle that it is not possible not to believe a tautology. [That is so in the quite classical logic of belief I presented in the previous two posts.] So if Q is ‘It is believed that’ and A is a tautology then SPLIT is violated no matter what.

The argument about atomicity does not refer to any features of the modal operator. But the assertion that SPLIT can be accommodated in an atom-less algebra of propositions, for a given modal operator, does depend on what features that operator has.

The argument about atomicity also goes through for the weaker WEAK SPLIT principle

         WEAK SPLIT.  If A is true and ~A is possible, then [it is possible that A is true and QA, and it is possible that A is true and not QA)

           ( A & ♦~A) → [♦(A & QA) & ♦(A & ~QA)]

That weaker principle is more aptly illustrated with is believed as modal operator, with possibility epistemic. For the argument about atomicity:  we might be in world w but not know that we are in w. Then if A = {w} then A and ♦~A are both true in w.  But then WEAK SPLIT is violated.