We study the mathematical properties of bilateral state-based modal logic (BSML), a modal logic employing state-based semantics (also known as team semantics), which has been used to account for free choice inferences and related linguistic phenomena. This logic extends classical modal logic with a nonemptiness atom which is true in a state if and only if the state is nonempty. We introduce two extensions of BSML and show that the extensions are expressively complete, and develop natural deduction axiomatizations for the three logics.

There are two distinct forms of responsibility that can be found in literature: counterfactual responsibility and responsibility for “seeing to it that.” It has been previously observed that, in the case of strategic games, the counterfactual form of responsibility can be defined through responsibility for “seeing to it that,” but not the other way around. We consider these two forms of responsibility in the case of infinite extensive form games. The main technical result is that, in this new setting, neither of the two forms of responsibility can be defined through the other. Some preliminary results for finite extensive form games are also given.

The aim of this paper is to give a mathematical account of an argument of David Lewis in Parts of Classes in defense of universalism in mereology. Specifically, we study how to extend models of core mereology (following Achille Varzi’s terminology) to models in which every collection of parts can be composed into another part. We focus on the two main definitions for mereological compositions and show that any model can be extended to satisfy universalism. We explore which are the “most economical” ways of extending models under various conditions. Remarkably, we show that if the principle of strong supplementation is assumed, there is a unique mereological completion, up to isomorphism.

Logics of significance have been proposed in an attempt to overcome the shortcomings of classical logic as a model of reasoning in the presence of nonsignificant (e.g., meaningless, ill-formed, unverifiable) sentences. Many-valued logicians have addressed this problem by introducing logics with infectious truth values. Cases in point are the weak Kleene logics ${\mathrm{B}}_3$ (paracomplete weak Kleene logic) and $\mathrm{PWK}$ (paraconsistent weak Kleene logic). Over time, it has become clear that the valid entailments of these significance logics obey variable inclusion patterns that link them to other, usually better known, logics—such patterns, however, allow for disturbing exceptions. Logics of pure (left or right) variable inclusion have been introduced with an eye to removing these exceptions. In this paper, we consider the pure left variable inclusion companion of classical logic and give a complete description of its subclassical extensions. We also provide relative axiomatizations and characteristic (sets of) matrices for each one of these extensions, as well as syntactic descriptions (in terms of variable inclusion criteria) for the valid entailments of some of them, and determine in each case the algebra reducts of the Suszko-reduced matrix models.

We generalize the combinatorial principles $C_n(\mathrm{\kappa })$, $C_n^s(\mathrm{\kappa })$, and $Princ(\mathrm{\kappa })$ introduced by various authors, and prove some of their properties and connections between them. We also answer a question asked by Juhász and Kunen about the relation between these principles, by showing that $C_n(\mathrm{\kappa })$ does not imply $C_{n+1}(\mathrm{\kappa })$ for any $n>2$. We also show the consistency of $C(\mathrm{\kappa })+\neg C^s(\mathrm{\kappa })$.