Frame definability is a well-studied notion for a plethora of logics: a class of frames is defined by a set of formulas if and only if such formulas are validities for all and only the frames in such class.

In this talk, I will present a series of results about frame definability in a fragment of conditional logic. Conditional logic is an extension of propositional logic that includes a binary “counterfactual” conditional operator \(\leadsto\). We specifically make the following choices: (i), we consider the non-nested fragment of conditional logic (i.e., where conditionals are not allowed in the scope of other conditionals), and (ii), we consider only finite posets as the underlying structure of the models of our logic. Given these assumptions, a formula of the form \(\varphi \leadsto \psi\) is intuitively true if the consequent \(\psi\) is true in all the minimal (with respect to the partial order of the poset) worlds where the antecedent \(\varphi\) is true. Via these semantics, this fragment of conditional logic becomes relevant in several fields, such as default reasoning and belief revision.

I will present the following results: (i) a set of non-nested conditional formulas each defining a different class of finite posets; (ii) a characterization theorem identifying the classes of finite posets that can be defined by a non-nested conditional formula; and (iii), a Sahlqvist-like theorem proving that every class of finite posets definable via a non-nested conditional formula is definable also by a first-order formula.

This talk is based on the paper Frame Definability in Conditional Logic, a joint work with Damiano Fornasiere and Johannes Marti, which was published at AiML 2024.