On some temporal logics of trees

I report on some joint work with Chantel Marais and Valentin Goranko. We take a tree to be a (strict) partial order which is left linear (any two points with a common upper bound are comparable) and connected (any two incomparable points have a common lower bound). Trees are natural representations of the flow of time: the past is determined and hence linear, while the future is open to different possibilities and so branches. A hierarchy of modal logical languages interpreted on tree-like models have been introduced, aligned, respectively, to philosophies of time put forward by William of Ockham, Charles Saunders Peirce and Arthur Prior. We axiomatize the so-called Priorean logics of several classes of trees, including the class of trees with branches isomorphic to the rational numbers and the class with branches isomorphic to the real numbers. The completeness proof for the latter introduces a number of novel model-theoretic transformations.