THE ALGEBRAS of LEWIS'S COUNTERFACTUALS: DUALITY THEORY Giuliano Rosella, Sara Ugolini Review of Symbolic Logic, 2026 Abstract This paper explores the mathematical connections between the algebraic and relational semantics of Lewis’s logics for counterfactual conditionals. Specifically, we introduce topological variants of Lewis’s well-known possible-worlds semantics—based on spheres, selection functions, and orders—and establish duality results with respect to varieties of Boolean algebras equipped with a counterfactual operator, which serve as the equivalent algebraic semantics of Lewis’s main systems. These results aim to provide a solid mathematical foundation for the study of Lewis’s logics, and offer a new perspective on the most well-known possible worlds-based models. In particular, we write explicit proofs for several results that are often assumed without proof in the literature. Leveraging these duality results, we also derive alternative proofs of strong completeness for Lewis’s variably strict conditional logics with respect to their intended models, and clarify the role of the limit assumption in sphere semantics.
Modal Weak Kleene Logics Through Variables Inclusion Giuliano Rosella Journal of Philosophical Logic, 2025 This paper presents a novel internal modal weak Kleene semantics and its derived logics. Our approach offers an intuitive understanding of modal operators as first-order weak Kleene quantifiers, drawing inspiration from the standard translation of classical modal logic. We explore the properties of this semantics and its associated logics. Our primary contribution lies in characterization theorems for some of these modal weak Kleene logics, leveraging classical modal logic, augmented with a refined variables inclusion requirement. These results not only extend the established characterization of non-modal weak Kleene logics but also provide fresh insights into the interpretations of our modal weak Kleene logics. Specifically, building on these technical findings, we propose philosophical interpretations for our logics and their semantics that coherently extend those of non-modal weak Kleene logics.
THE ALGEBRAS OF LEWIS’S COUNTERFACTUALS: AXIOMATIZATIONS AND ALGEBRAIZABILITY GIULIANO ROSELLA, SARA UGOLINI Review of Symbolic Logic, 2025 The logico-algebraic study of Lewis’s hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work starts filling this gap by providing a logico-algebraic analysis of Lewis’s logics. We begin by introducing novel finite axiomatizations for Lewis’s logics on the syntactic side, distinguishing between global and local consequence relations on Lewisian sphere models on the semantical side, in parallel to the case of modal logic. As first main results, we prove the strong completeness of the calculi with respect to the corresponding semantical consequence on spheres, and a deduction theorem. We then demonstrate that the global calculi are strongly algebraizable in terms of a variety of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local ones are generally not algebraizable, although they can be characterized as the degree-preserving logic over the same algebraic models. This yields the strong completeness of all the logics with respect to the algebraic models.
Conditionals Based on Selection Functions, Modal Operators and Probabilities Tommaso Flaminio, Lluis Godo, Gluliano Rosella Electronic Proceedings in Theoretical Computer Science Eptcs, 2025 Methods for probability updating, of which Bayesian conditionalization is the most well-known and widely used, are modeling tools that aim to represent the process of modifying an initial epistemic state, typically represented by a prior probability function P, which is adjusted in light of new information. Notably, updating methods and conditional sentences seem to intuitively share a deep connection, as is evident in the case of conditionalization. The present work contributes to this line of research and aims at shedding new light on the relationship between updating methods and conditional connectives. Departing from previous literature that often focused on a specific type of conditional or a particular updating method, our goal is to prove general results concerning the connection between conditionals and their probabilities. This will allow us to characterize the probabilities of certain conditional connectives and to understand what class of updating procedures can be represented using specific conditional connectives. Broadly, we adopt a general perspective that encompasses a large class of conditionals and a wide range of updating methods, enabling us to prove some general results concerning their interrelation.
Possibility of Conditionals and Conditional Possibilities: From a Triviality Result to Possibilistic Imaging Proceedings of the International Conference on Knowledge Representation and Reasoning, 2024
Counterfactuals as modal conditionals, and their probability Giuliano Rosella, Tommaso Flaminio, Stefano Bonzio Artificial Intelligence, 2023 In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.
RECENT SCHOLAR PUBLICATIONS
The algebras of Lewis’s counterfactuals: duality theory G Rosella, S Ugolini The Review of Symbolic Logic 19 (1), 46-80 , 2026 2026.0
Modal Weak Kleene Logics Through Variables Inclusion: G. Rosella G Rosella Journal of Philosophical Logic, 1-27 , 2025 2025.0
On Measuring the Possibility of Selection Function-Based Conditionals, General Updates, and Qualitative Capacities T Flaminio, L Godo, G Rosella European Conference on Symbolic and Quantitative Approaches with Uncertainty … , 2025 2025.0
Updates, and Qualitative Capacities L Godo¹, G Rosella Symbolic and Quantitative Approaches to Reasoning with Uncertainty: 18th … , 2025 2025.0
The algebras of Lewis’s counterfactuals: axiomatizations and algebraizability G Rosella, S Ugolini The Review of Symbolic Logic 18 (2), 563-588 , 2025 2025.0 Citations: 2
Conditionals Based on Selection Functions, Modal Operators and Probabilities T Flaminio, L Godo, G Rosella Proceedings Twentieth Conference on Theoretical Aspects of Rationality and … , 2025 2025.0
On Measuring the Possibility of Selection Function-Based Conditionals, General Updates, and Qualitative Capacities G Rosella, T Flaminio, L Godo 2025.0
Possibility of conditionals and conditional possibilities: from the triviality result to possibilistic imaging T Flaminio, L Godo, G Rosella Proceedings of the International Conference on Principles of Knowledge … , 2024 2024.0 Citations: 3
Causal modeling semantics for counterfactuals with disjunctive antecedents G Rosella, J Sprenger Annals of Pure and Applied Logic 175 (9), 103336 , 2024 2024.0
The Algebras of Lewis's Counterfactuals G Rosella, S Ugolini arXiv preprint arXiv:2407.11740 , 2024 2024.0
Wanna bet? Pascal e la Logica della Scommessa su Dio G Rosella Res Cogitans , 2023 2023.0
Counterfactuals as modal conditionals, and their probability G Rosella, T Flaminio, S Bonzio Artificial Intelligence 323, 103970 , 2023 2023.0 Citations: 10
Counterfactuals 2.0: Logic, Truth Conditions, and Probability G Rosella 2023.0
Logical Form: Between Logic and Natural Language , by Andrea Iacona G Rosella Philosophical Inquiries 10 (1), R7-R13 , 2022 2022.0
Truthmakers, incompatibility, and modality V Saitta, M Plebani, G Rosella The Australasian Journal of Logic 19 (5), 214-253 , 2022 2022.0 Citations: 15
A truthmaker semantics approach to modal logic G Rosella 2019.0 Citations: 1
Conditional Possibilities, Possibilistic Imaging and Boolean Algebras of Conditionals G Rosella
The algebras of Lewis’s counterfactuals and their duality theory G Rosella, S Ugolini
Modal Algebraic Models of Counterfactuals S Bonzio, T Flaminio, G Rosella
Algebras of Counterfactual Conditionals G Rosella, S Ugolini
MOST CITED SCHOLAR PUBLICATIONS
Truthmakers, incompatibility, and modality V Saitta, M Plebani, G Rosella The Australasian Journal of Logic 19 (5), 214-253 , 2022 2022.0 Citations: 15
Counterfactuals as modal conditionals, and their probability G Rosella, T Flaminio, S Bonzio Artificial Intelligence 323, 103970 , 2023 2023.0 Citations: 10
Possibility of conditionals and conditional possibilities: from the triviality result to possibilistic imaging T Flaminio, L Godo, G Rosella Proceedings of the International Conference on Principles of Knowledge … , 2024 2024.0 Citations: 3
The algebras of Lewis’s counterfactuals: axiomatizations and algebraizability G Rosella, S Ugolini The Review of Symbolic Logic 18 (2), 563-588 , 2025 2025.0 Citations: 2
A truthmaker semantics approach to modal logic G Rosella 2019.0 Citations: 1
The algebras of Lewis’s counterfactuals: duality theory G Rosella, S Ugolini The Review of Symbolic Logic 19 (1), 46-80 , 2026 2026.0
Modal Weak Kleene Logics Through Variables Inclusion: G. Rosella G Rosella Journal of Philosophical Logic, 1-27 , 2025 2025.0
On Measuring the Possibility of Selection Function-Based Conditionals, General Updates, and Qualitative Capacities T Flaminio, L Godo, G Rosella European Conference on Symbolic and Quantitative Approaches with Uncertainty … , 2025 2025.0
Updates, and Qualitative Capacities L Godo¹, G Rosella Symbolic and Quantitative Approaches to Reasoning with Uncertainty: 18th … , 2025 2025.0
Conditionals Based on Selection Functions, Modal Operators and Probabilities T Flaminio, L Godo, G Rosella Proceedings Twentieth Conference on Theoretical Aspects of Rationality and … , 2025 2025.0
On Measuring the Possibility of Selection Function-Based Conditionals, General Updates, and Qualitative Capacities G Rosella, T Flaminio, L Godo 2025.0
Causal modeling semantics for counterfactuals with disjunctive antecedents G Rosella, J Sprenger Annals of Pure and Applied Logic 175 (9), 103336 , 2024 2024.0
The Algebras of Lewis's Counterfactuals G Rosella, S Ugolini arXiv preprint arXiv:2407.11740 , 2024 2024.0
Wanna bet? Pascal e la Logica della Scommessa su Dio G Rosella Res Cogitans , 2023 2023.0
Counterfactuals 2.0: Logic, Truth Conditions, and Probability G Rosella 2023.0
Logical Form: Between Logic and Natural Language , by Andrea Iacona G Rosella Philosophical Inquiries 10 (1), R7-R13 , 2022 2022.0
Conditional Possibilities, Possibilistic Imaging and Boolean Algebras of Conditionals G Rosella
The algebras of Lewis’s counterfactuals and their duality theory G Rosella, S Ugolini
Modal Algebraic Models of Counterfactuals S Bonzio, T Flaminio, G Rosella
Algebras of Counterfactual Conditionals G Rosella, S Ugolini
