On the share of closed IL formulas which are also in GL

Normal forms for wide classes of closed IL formulas were given in [4]. Here we quantify asymptotically, in exact numbers, how wide those classes are. As a consequence, we show that the "majority" of closed IL formulas have GL-equivalents, and by that, they have the same normal forms as GL formulas. Our approach is entirely syntactical, except for applying the results of [4]. As a byproduct we devise a convenient way of computing asymptotic behaviors of somewhat general classes of formulas given by their grammar rules. Its applications do not require any knowledge of the recurrence relations, generating functions, or the asymptotic enumeration methods, as all these are incorporated into two fundamental parameters.Comment: 23 pages; v3: title changed, logical and combinatorial parts separated, more general alphabets considere

An overview of Generalised Veltman Semantics

Interpretability logics are endowed with relational semantics \`a la Kripke: Veltman semantics. For certain applications though, this semantics is not fine-grained enough. Back in 1992, in the research group of de Jongh, the notion of generalised Veltman semantics emerged to obtain certain non-derivability results as was first presented by Verbrugge ([76]). It has turned out that this semantics has various good properties. In particular, in many cases completeness proofs become simpler and the richer semantics will allow for filtration arguments as opposed to regular Veltman semantics. This paper aims to give an overview of results and applications of Generalised Veltman semantics up to the current date