We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev's logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces) [2]. In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic.

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This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

We present the use of a novel spatio-temporal model-checker to detect problems in the data and operation of a collective adaptive system. Data correctness is important to ensure operational correctness in systems which adapt in response to data. We illustrate the theory with several concrete examples, addressing both the detection of errors in vehicle location data for buses in the city of Edinburgh and the undesirable phenomenon of "clumping" which occurs when there is not enough separation between subsequent buses serving the same route. Vehicle location data is visualised symbolically on a street map, and categories of problems identified by the spatial part of the model-checker are rendered by highlighting the symbols for vehicles or other objects that satisfy a property of interest. Behavioural correctness makes use of both the spatial and temporal aspects of the model-checker to determine from a series of observations of vehicle locations whether the system is failing to meet the expected quality of service demanded by system regulators.

Classical first-order logic FO is commonly used to study logical connections between statements, that is sentences that in every context have an associated truthvalue. Inquisitive first-order logic InqBQ is a conservative extension of FO which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for InqBQ relative to inquisitive disjunction and inquisitive existential quantifier ∃. Moreover we extend these results to several families of theories, among which the one in the language of FO. To this end, we initiate a model-theoretic approach to the study of InqBQ. In particular, we develop a toolkit of basic constructions in order to transform and combine models of InqBQ.