Tense Information Logic
Timo Niek Franssen

Abstract:
This thesis studies tense information logic (TIL), an extension of modal information logic (MIL). MIL was introduced by van Benthem [2] to model information flow using possible worlds semantics by adding a binary modality ⟨sup⟩ to the language of propositional logic, interpreted via the supremum of two states; TIL adds a second binary modality ⟨inf⟩ interpreted via the infimum of two states.
We give a sound and complete axiomatization of TIL on posets, extending Knud- storp’s [17] axiomatization of MIL. As a corollary, we obtain completeness of TIL on preorders. We also show that TIL has the finite model property with respect to a generalized class of structures, thereby establishing its decidability.
Beyond completeness and decidability, we develop a Stone–Jónsson–Tarski duality for TIL, show that interpreting the modalities via minimal and maximal bounds leaves the logic unchanged, and construct two translations between weak positive logic (WPL) and an extended version of TIL containing Kleene star-like versions of the ⟨sup⟩ and ⟨inf⟩ modalities.