Boolean Types in Dependent Theories

Itay Kaplan, Ori Segel, Saharon Shelah

Research output: Contribution to journalArticlepeer-review


The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures-in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

Original languageAmerican English
Pages (from-to)1349-1373
Number of pages25
JournalJournal of Symbolic Logic
Issue number4
StatePublished - 26 Dec 2022

Bibliographical note

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  • Boolean algebras
  • Boolean types
  • Keisler measures
  • NIP
  • dependent theory
  • stable theories


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