The logics $\mathbf{R}\mathbf{\text{Ł}}$, $\mathbf{RP}$, and $\mathbf{RG}$ have been obtained by expanding Łukasiewicz logic $\mathbf{\text{Ł}}$, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in $\mathbf{\text{Ł}}$, P, and G. Namely, $\mathbf{R}\mathbf{\text{Ł}}$ is hereditarily structurally complete. $\mathbf{RP}$ is algebraized by the variety of rational product algebras that we show to be $\mathcal{Q}$-universal. We provide a base of admissible rules in $\mathbf{RP}$, show their decidability, and characterize passive structural completeness for extensions of $\mathbf{RP}$. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of $\mathbf{RP}$, and this is also the case for extensions of $\mathbf{RG}$, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of $\mathbf{RG}$, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is $\mathcal{Q}$-universal.

We analyze the causal-observational languages that were introduced in Barbero and Sandu (2018), which allow discussing interventionist counterfactuals and functional dependencies in a unified framework. In particular, we systematically investigate the expressive power of these languages in causal team semantics, and we provide complete natural deduction calculi for each language. Furthermore, we introduce a generalized semantics which allows representing uncertainty about the causal laws, and we analyze the expressive power and proof theory of the causal-observational languages over this enriched semantics.

This paper motivates the logic PCON, an extension of connexivity to conjunction and disjunction, called poly-connexivity. The motivation arises from differences in intonational stress patterns due to focus, where PCON turns out to be a logic of intentionally stressed connectives in focus.

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks, we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fraïssé theory.

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then $G^{\mathrm{st}}=G^{00}$ (where $G^{\mathrm{st}}$ is the smallest ∧-definable subgroup with $G∕G^{\mathrm{st}}$ stable, and $G^{00}$ is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion $T^{\mathrm{heq}}$. The second result is an example of a group G definable in a nondistal NIP theory for which $G=G^{00}$, but $G^{\mathrm{st}}$ is not an intersection of definable groups. Our example is a saturated extension of $(\mathbb{R},+,[0,1])$. Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic.

We show that any formula with two free variables in a Vapnik–Chervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where ${\mathrm{acl}}^{\mathrm{eq}}={\mathrm{dcl}}^{\mathrm{eq}}$, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).

This paper starts the systematic study of inconsistency measures for propositional logics enriched with operators involving time. We use Prior’s operators for tense logic: H, G, P, and F; however, we apply different semantics to them. We define two logics. The first one, $\mathit{ATPL}$, allows formulas with the application of any of the four operators any number of times to propositional logic formulas. The semantics is given in terms of TPL structures. We then show how to measure the inconsistency of a set of $\mathit{ATPL}$ formulas using appropriately modified versions of standard inconsistency measures for propositional logic. Additionally, these new measures are checked for the satisfaction of various properties. We also investigate a new concept of inconsistency that cannot be defined for propositional logic. Then, $\mathit{ATPL}$ is extended to $\mathit{CTPL}$ by applying the propositional connectives to $\mathit{ATPL}$ formulas and measuring inconsistency is extended to sets of $\mathit{CTPL}$ formulas.