DDL as an “Internalization” of Dynamic Belief Revision

Abstract

In this chapter we re-evaluate Segerberg’s “full DDL” (Dynamic Doxastic Logic) from the perspective of Dynamic Epistemic Logic (DEL), in its belief-revision-friendly incarnation. We argue that a correct version of full DDL must give up the Success Postulate for dynamic revision. Next, we present (an appropriately generalized and simplified version of) full DDL, showing that it is a generalization of the so-called Topo-logic of Moss and Parikh. We construct AGM-friendly versions of full DDL, corresponding to various revising/contracting operations considered in the Belief Revision literature. We show that DDL can internalize inside one model the “external” doxastic dynamics of DEL, as well as the evidential dynamics investigated by van Benthem and Pacuit. In our Conclusions section, we compare three styles of modeling doxastic dynamics: DDL, DEL and PDL/ETL (the Propositional Dynamic Logic approach, with its Epistemic Temporal Logic variant).

Appendix: Pictures of the Main Operations on Onions

The pictures drawn here are following Hans Rott’s presentation [18]. The spheres of the initial onion are drawn as usual, as nested circles. The numbers represent the spheres of the new onion, after the revision/expansion/contraction: e.g. all regions labeled with one form the first sphere of the new onion, the regions labeled with two form the second sphere etc. Finally, the regions labeled with $\omega$ contain the states that are outside the union of all the spheres of the onion (the “impossible states”).

Fig. 1

Radical revision ($!\varphi$)

Fig. 2

Conservative revision ($\uparrow \varphi$)

Fig. 3

Moderate revision ($\Uparrow \varphi$)

Fig. 4

Conservative expansion ($+{\uparrow }{\varphi }$)

Fig. 5

Moderate expansion ($+{\Uparrow }{\varphi }$)

Fig. 6

Radical expansion ($+!\varphi$)

Fig. 7

Severe withdrawal ($-\varphi$)

Fig. 8

Conservative contraction ($-_c\varphi$)

Fig. 9

Moderate contraction ($-_m\varphi$)