We propose Modal Logical Neural Networks (MLNNs), a neurosymbolic framework that integrates deep learning with the formal semantics of modal logic, enabling reasoning about necessity and possibility. Drawing on Kripke semantics, we introduce specialized neurons for the modal operators $\Box$ and $\Diamond$ that operate over a set of possible worlds, enabling the framework to act as a differentiable ``logical guardrail.'' The architecture is highly flexible: the accessibility relation between worlds can either be fixed by the user to enforce known rules or, as an inductive feature, be parameterized by a neural network. This allows the model to optionally learn the relational structure of a logical system from data while simultaneously performing deductive reasoning within that structure. This versatile construction is designed for flexibility. The entire framework is differentiable from end to end, with learning driven by minimizing a logical contradiction loss. This not only makes the system resilient to inconsistent knowledge but also enables it to learn nonlinear relationships that can help define the logic of a problem space. We illustrate MLNNs on four case studies: grammatical guardrailing, axiomatic detection of the unknown, multi-agent epistemic trust, and detecting constructive deception in natural language negotiation. These experiments demonstrate how enforcing or learning accessibility can increase logical consistency and interpretability without changing the underlying task architecture.

This paper introduces a new family of cognitive modal logics designed to formalize conjectural reasoning: a modal system in which cognitive contexts extend known facts with hypothetical assumptions to explore their consequences. Unlike traditional doxastic and epistemic systems, conjectural logics rely on a principle, called Axiom C ($φ\rightarrow \Boxφ$), that ensures that all established facts are preserved across hypothetical layers. While Axiom C was dismissed in the past due to its association with modal collapse, we show that the collapse only arises under classical and bivalent assumptions, and specifically in the presence of Axiom T. Hence we avoid Axiom T and adopt a paracomplete semantic framework, grounded in Weak Kleene logic or Description Logic, where undefined propositions coexist with modal assertions. This prevents the modal collapse and guarantees a layering to distinguish between factual and conjectural statements. Under this framework we define new modal systems, e.g., KC and KDC, and show that they are complete, decidable, and robust under partial knowledge. Finally, we introduce a dynamic operation, $\mathsf{settle}(φ)$, which formalizes the transition from conjecture to accepted fact, capturing the event of the update of a world's cognitive state through the resolution of uncertainty.

In a recent paper, Kit Fine presents some striking results concerning the logical properties of (first-order) ignorance, second-order ignorance and Rumsfeld ignorance. However, Rumsfeld ignorance is definable in terms of ignorance, which makes some existing results and the axiomatization problem trivial. A main reason is that the accessibility relations for the implicit knowledge operator contained in the packaged operators of ignorance and Rumsfeld ignorance are the same. In this work, we assume the two accessibility relations to be different so that one of them is an arbitrary subset of the other. This will avoid the definability issue and retain most of the previous validities. The main results are axiomatizations over various proper bi-frame classes. Finally we apply our framework to analyze Fine's results.

What makes a society possible at all? How is coordination and cooperation in social activity possible? What is the minimal mental architecture of a social agent? How is the information about the state of the world related to the agents intentions? How are the intentions of agents related? What role does communication play in this coordination process? This essay explores the conceptual and logical foundations of artificial social intelligence in the context of a society of multiple agents that communicate and cooperate to achieve some end. An attempt is made to provide an introduction to some of the key concepts, their formal definitions and their interrelationships. These include the notion of a changing social world of multiple agents. The logic of social intelligence goes beyond classical logic by linking information with strategic thought. A minimal architecture of social agents is presented. The agents have different dynamically changing, possible choices and abilities. The agents also have uncertainty, lacking perfect information about their physical state as well as their dynamic social state. The social state of an agent includes the intentional state of that agent, as well as, that agent's representation of the intentional states of other agents. Furthermore, it includes the evaluations agents make of their physical and social condition. Communication, semantic and pragmatic meaning and their relationship to intention and information states are investigated. The logic of agent abilities and intentions are motivated and formalized. The entropy of group strategic states is defined.

The paper presents a comprehensive analysis of the European AI Act in terms of its logical modalities, with the aim of preparing its formal representation, for example, within the logic-pluralistic Knowledge Engineering Framework and Methodology (LogiKEy). LogiKEy develops computational tools for normative reasoning based on formal methods, employing Higher-Order Logic (HOL) as a unifying meta-logic to integrate diverse logics through shallow semantic embeddings. This integration is facilitated by Isabelle/HOL, a proof assistant tool equipped with several automated theorem provers. The modalities within the AI Act and the logics suitable for their representation are discussed. For a selection of these logics, embeddings in HOL are created, which are then used to encode sample paragraphs. Initial experiments evaluate the suitability of these embeddings for automated reasoning, and highlight key challenges on the way to more robust reasoning capabilities.

This paper combines the classical model of labeled transition systems with the epistemic model for reasoning about knowledge. The result is a unifying framework for modeling and analyzing multi-agent, knowledge-based, dynamic systems. On the modeling side, we propose a process algebraic, agent-oriented specification language that makes such a framework easy to use for practical purposes. On the verification side, we define a modal logic encompassing temporal and epistemic operators.

Let $L$ be a finite lattice and $\mathcal{E}(L)$ be the set of join endomorphisms of $L$. We consider the problem of given $L$ and $f,g \in \mathcal{E}(L)$, finding the greatest lower bound $f \sqcap_{{\scriptsize \mathcal{E}(L)}} g$ in the lattice $\mathcal{E}(L)$. (1) We show that if $L$ is distributive, the problem can be solved in time $O(n)$ where $n=| L |$. The previous upper bound was $O(n^2)$. (2) We provide new algorithms for arbitrary lattices and give experimental evidence that they are significantly faster than the existing algorithm. (3) We characterize the standard notion of distributed knowledge of a group as the greatest lower bound of the join-endomorphisms representing the knowledge of each member of the group. (4) We show that deciding whether an agent has the distributed knowledge of two other agents can be computed in time $O(n^2)$ where $n$ is the size of the underlying set of states. (5) For the special case of $S5$ knowledge, we show that it can be decided in time $O(n\alpha_{n})$ where $\alpha_{n}$ is the inverse of the Ackermann function.

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reasoning in the target logic (public announcement logic), required e.g. to obtain a convincing encoding and automation of the wise men puzzle, can be realized. Key to the presented semantical embedding is that evaluation domains are modeled explicitly and treated as an additional parameter in the encodings of the constituents of the embedded target logic; in previous related works, e.g. on the embedding of normal modal logics, evaluation domains were implicitly shared between meta-logic and target logic. The work presented in this article constitutes an important addition to the pluralist LogiKEy knowledge engineering methodology, which enables experimentation with logics and their combinations, with general and domain knowledge, and with concrete use cases -- all at the same time.

An epistemic model for decentralized discrete-event systems with non-binary control is presented. This framework combines existing work on conditional control decisions with existing work on formal reasoning about knowledge in discrete-event systems. The novelty in the model presented is that the necessary and sufficient conditions for problem solvability encapsulate the actions that supervisors must take. This direct coupling between knowledge and action -- in a formalism that mimics natural language -- makes it easier, when the problem conditions fail, to determine how the problem requirements should be revised.

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reasoning in the target logic (public announcement logic), required e.g. to obtain a convincing encoding and automation of the wise men puzzle, can be realized. Key to the presented semantical embedding -- in contrast, e.g., to related work on the semantical embedding of normal modal logics -- is that evaluation domains are modeled explicitly and treated as additional parameter in the encodings of the constituents of the embedded target logic, while they were previously implicitly shared between meta logic and target logic.