Logics of Infinite Depth

Consider a definition of depth of a logic as the supremum of ordinal types of wellordered descending chains. This extends the usual definition of codimension to infinite depths. Logics may either have no depth, or have countable depth in case a maximal well-ordered chain exists, or be of depth !1. We shall exhibit logics of all three types. We show in particular that many well-known systems, among them K, K4, G, Grz and S4, have depth !1. Basically, if a logic is the intersection of its splitting logics and has finite model property, then either the splitting logics have an infinite antichain (and the depth is therefore !1) or the splitting logics form a well-partial order whose supremum type is realised and therefore countable, though it may be di↵erent from the supremum type of the splitting logics alone.