Standpoint logics offer unified modal logic-based formalisms for representing multiple heterogeneous viewpoints. At the same time, many non-monotonic reasoning frameworks can be naturally captured using modal logics - in particular using the modal logic S4F. In this work, we propose a novel formalism called S4F Standpoint Logic, which generalises both S4F and propositional standpoint logic and is therefore capable of expressing multi-viewpoint, non-monotonic semantic commitments. We define its syntax and semantics and analyze its computational complexity, obtaining the result that S4F Standpoint Logic is not computationally harder than its constituent logics, whether in monotonic or non-monotonic form. We also outline mechanisms for credulous and sceptical acceptance and illustrate the framework with an example.

Artificial Intelligence (AI) is a powerful new language of science as evidenced by recent Nobel Prizes in chemistry and physics that recognized contributions to AI applied to those areas. Yet, this new language lacks semantics, which makes AI's scientific discoveries unsatisfactory at best. With the purpose of uncovering new facts but also improving our understanding of the world, AI-based science requires formalization through a framework capable of translating insight into comprehensible scientific knowledge. In this paper, we argue that logic offers an adequate framework. In particular, we use logic in a neurosymbolic framework to offer a much needed semantics for deep learning, the neural network-based technology of current AI. Deep learning and neurosymbolic AI lack a general set of conditions to ensure that desirable properties are satisfied. Instead, there is a plethora of encoding and knowledge extraction approaches designed for particular cases. To rectify this, we introduced a framework for semantic encoding, making explicit the mapping between neural networks and logic, and characterizing the common ingredients of the various existing approaches. In this paper, we describe succinctly and exemplify how logical semantics and neural networks are linked through this framework, we review some of the most prominent approaches and techniques developed for neural encoding and knowledge extraction, provide a formal definition of our framework, and discuss some of the difficulties of identifying a semantic encoding in practice in light of analogous problems in the philosophy of mind.

Automated deduction lies at the core of Artificial Intelligence (AI), underpinning theorem proving, formal verification, and logical reasoning. Despite decades of progress, reconciling deductive completeness with computational efficiency remains an enduring challenge. Traditional reasoning calculi, grounded in binary resolution, restrict inference to pairwise clause interactions and thereby limit deductive synergy among multiple clauses. The Contradiction Separation Extension (CSE) framework, introduced in 2018, proposed a dynamic multi-clause reasoning theory that redefined logical inference as a process of contradiction separation rather than sequential resolution. While that work established the theoretical foundation, its algorithmic realization remained unformalized and unpublished. This work presents the Extended Triangular Method (ETM), a generalized contradiction-construction algorithm that formalizes and extends the internal mechanisms of contradiction separation. The ETM unifies multiple contradiction-building strategies, including the earlier Standard Extension method, within a triangular geometric framework that supports flexible clause interaction and dynamic synergy. ETM serves as the algorithmic core of several high-performance theorem provers, CSE, CSE-E, CSI-E, and CSI-Enig, whose competitive results in standard first-order benchmarks (TPTP problem sets and CASC 2018-2015) empirically validate the effectiveness and generality of the proposed approach. By bridging theoretical abstraction and operational implementation, ETM advances the contradiction separation paradigm into a generalized, scalable, and practically competitive model for automated reasoning, offering new directions for future research in logical inference and theorem proving.

Trustworthy AI requires reasoning systems that are not only powerful but also transparent and reliable. Automated Theorem Proving (ATP) is central to formal reasoning, yet classical binary resolution remains limited, as each step involves only two clauses and eliminates at most two literals. To overcome this bottleneck, the concept of standard contradiction and the theory of contradiction-separation-based deduction were introduced in 2018. This paper advances that framework by focusing on the systematic construction of standard contradictions. Specially, this study investigates construction methods for two principal forms of standard contradiction: the maximum triangular standard contradiction and the triangular-type standard contradiction. Building on these structures, we propose a procedure for determining the satisfiability and unsatisfiability of clause sets via maximum standard contradiction. Furthermore, we derive formulas for computing the number of standard sub-contradictions embedded within both the maximum triangular standard contradiction and the triangular-type standard contradiction. The results presented herein furnish the methodological basis for advancing contradiction-separation-based dynamic multi-clause automated deduction, thereby extending the expressive and deductive capabilities of automated reasoning systems beyond the classical binary paradigm.

Neurosymbolic AI focuses on integrating learning and reasoning, in particular, on unifying logical and neural representations. Despite the existence of an alphabet soup of neurosymbolic AI systems, the field is lacking a generally accepted formal definition of what neurosymbolic models and inference really are. We introduce a formal definition for neurosymbolic AI that makes abstraction of its key ingredients. More specifically, we define neurosymbolic inference as the computation of an integral over a product of a logical and a belief function. We show that our neurosymbolic AI definition makes abstraction of key representative neurosymbolic AI systems.

We define several canonical problems related to contrastive explanations, each answering a question of the form ''Why P but not Q?''. The problems compute causes for both P and Q, explicitly comparing their differences. We investigate the basic properties of our definitions in the setting of propositional logic. We show, inter alia, that our framework captures a cardinality-minimal version of existing contrastive explanations in the literature. Furthermore, we provide an extensive analysis of the computational complexities of the problems. We also implement the problems for CNF-formulas using answer set programming and present several examples demonstrating how they work in practice.

The extent to which neural networks are able to acquire and represent symbolic rules remains a key topic of research and debate. Much current work focuses on the impressive capabilities of large language models, as well as their often ill-understood failures on a wide range of reasoning tasks. In this paper, in contrast, we investigate the generalization behavior of three key neural architectures (Transformers, Graph Convolution Networks and LSTMs) in a controlled task rooted in propositional logic. The task requires models to generate satisfying assignments for logical formulas, making it a structured and interpretable setting for studying compositionality. We introduce a balanced extension of an existing dataset to eliminate superficial patterns and enable testing on unseen operator combinations. Using this dataset, we evaluate the ability of the three architectures to generalize beyond the training distribution. While all models perform well in-distribution, we find that generalization to unseen patterns, particularly those involving negation, remains a significant challenge. Transformers fail to apply negation compositionally, unless structural biases are introduced. Our findings highlight persistent limitations in the ability of standard architectures to learn systematic representations of logical operators, suggesting the need for stronger inductive biases to support robust rule-based reasoning.

Knowledge representation and reasoning in neural networks have been a long-standing endeavor which has attracted much attention recently. The principled integration of reasoning and learning in neural networks is a main objective of the area of neurosymbolic Artificial Intelligence (AI). In this chapter, a simple energy-based neurosymbolic AI system is described that can represent and reason formally about any propositional logic formula. This creates a powerful combination of learning from data and knowledge and logical reasoning. We start by positioning neurosymbolic AI in the context of the current AI landscape that is unsurprisingly dominated by Large Language Models (LLMs). We identify important challenges of data efficiency, fairness and safety of LLMs that might be addressed by neurosymbolic reasoning systems with formal reasoning capabilities. We then discuss the representation of logic by the specific energy-based system, including illustrative examples and empirical evaluation of the correspondence between logical reasoning and energy minimization using Restricted Boltzmann Machines (RBM). Learning from data and knowledge is also evaluated empirically and compared with a symbolic, neural and a neurosymbolic system. Results reported in this chapter in an accessible way are expected to reignite the research on the use of neural networks as massively-parallel models for logical reasoning and promote the principled integration of reasoning and learning in deep networks. We conclude the chapter with a discussion of the importance of positioning neurosymbolic AI within a broader framework of formal reasoning and accountability in AI, discussing the challenges for neurosynbolic AI to tackle the various known problems of reliability of deep learning.

However, to deal with ambiguity and partial information, new approache s have emerged - examples of which are fuzzy logic, probabilistic modal logic, Bayesian networks and belief-based systems. Even though progress has been made, these approaches genera lly have a limitation: the probability or degree of belief, in general, being kept out of the l ogical semantics, remaining at another level of interpretation on a deterministic model. In other w ords, maintaining the binary characteristic of truth - true or false, with uncertainty being treate d as associated with models, rather than a property of logical language in itself. The proposed logic will be used to solve the problem of tackling certain types of uncertainty and imprecision with Bayesian Networks. The aim is to take advantage of the conceptual and practical benefits of this sy stem in practical situations that have not yet been adequately explored.

This paper considers the complexity and properties of KLM-style preferential reasoning in the setting of propositional logic with team semantics and dependence atoms, also known as propositional dependence logic. Preferential team-based reasoning is shown to be cumulative, yet violates System~P. We give intuitive conditions that fully characterise those cases where preferential propositional dependence logic satisfies System~P. We show that these characterisations do, surprisingly, not carry over to preferential team-based propositional logic. Furthermore, we show how classical entailment and dependence logic entailment can be expressed in terms of non-trivial preferential models. Finally, we present the complexity of preferential team-based reasoning for two natural representations. This includes novel complexity results for classical (non-team-based) preferential reasoning.