Modal logics have algebraic manifestations in the form of certain classes of so-called Boolean algebras with operators (BAOs). Recent years have witnessed an increase of activity in this area, and the aim of this talk is to survey some of these developments.

More in particular, the talk will center around various notions of canonicity; this concept concerns the question, which properties are preserved when we move from an arbitrary BAO to a completion, that is, a complete superalgebra of the original one. The notion of canoniciy, which on the logical side plays an important role in modal completeness proofs, has recently revealed some of its secrets. In particular, an open problem raised by Kit Fine in the seventies has been solved just a few months ago.

How do probabilities mesh with the causal structure of space-time?

Probabilities are about chance set-ups such as the tossing of a coin. These set-ups are located spatially and temporally. Thus, it is to be expected that in some circumstances, chance set-ups "here-now" can have an influence on chance set-ups "then-there". Probability theory of itself has nothing to say on this matter. However, the theory of branching space-times (Belnap, Synthese 92, 1992) provides a framework within which causal dependencies among chance set-ups may be described.

In my talk, I will first sketch Belnap's theory of branching space-times, including his account of "originating causes". In the second part of my talk, I will introduce a general theory of probabilities for branching space-times.

Results of Rybakov, Pitts, Ghilardi, Iemhoff will be discussed and their relation to work done by the speaker in cooperation with Visser and with Bezhanishvili.

It is not immediately clear why human beings would experience time at all. Many skills do not require explicit representations of temporal succession or duration. There exists some psychological evidence, however, that conscious awareness of time comes with the capacity for planning. E.g., children's ability to use temporal expressions seems to covary with their ability to recognize plans; and planning deficits covary with temporal disintegration. If planning is an important determinant of our sense of time, one might expect that the representation of time in language, as tense and aspect, shows traces of its origin in planning. Fortunately, this idea is testable, since planning can be formalized logically, on the basis of logic programming with negation as failure. We will sketch the resulting semantics, and show what it predicts for tense and aspect.