Title:

Studies on modal logics of time and space

This dissertation presents original results in Temporal Logic and Spatial Logic. Part I concerns BranchingTime Logic. Since Prior 1967, two main semantics for BranchingTime Logic have been devised: Peircean and Ockhamist semantics. Zanardo 1998 proposed a general semantics, called Indistinguishability semantics, of which Peircean and Ockhamist semantics are limit cases. We provide a finite axiomatization of the Indistinguishability logic of upward endless bundled trees using a nonstandard inference rule, and prove that this logic is strongly complete. In Part II, we study the temporal logic given by the tense operators F for future and P for past together with the derivative operator < d >, interpreted on the real numbers. We prove that this logic is neither strongly nor Kripke complete, it is PSPACEcomplete, and it is finitely axiomatizable. In Part III, we study the spatial logic given by the derivative operator < d > and the graded modalities {◇nn Ε N}. We prove that this language, call it L, is as expressive as the firstorder language Lt of Flum and Ziegler 1980 when interpreted on T3 topological spaces. Then, we give a general definition of modal operator: essentially, a modal operator will be defined by a formula of Lt with at most one free variable. If a modal operator is defined by a formula predicating only over points, then it is called pointsort operator. We prove that L, even if enriched with all pointsort operators, however enriched with finitely many modal operators predicating also on open sets, cannot express Lt on T2 spaces. Finally, we axiomatize the logic of any class between all T1 and all T3 spaces and prove that it is PSPACEcomplete.
