Dov Gabbay

In this paper, we introduce the concept of logemes, a novel framework for reasoning that bridges nontrivial fragments of formal logic and topological spaces. Logemes are defined as logic diagrams that capture logical relationships relevant to specific datasets or problems. Unlike standard symbolic logic, which emphasizes completeness and formal validation within an algebraic framework, logemes focus on pragmatic, nontrivial subsets of inference rules tailored to real-world reasoning tasks. This approach builds on historical tools like logic diagrams (e.g. the square of opposition) while leveraging the combinatorial and topological richness of simplicial complexes. The paper explores the interplay between syntax and semantics through logemes, using topology to compare logical fragments across different systems. By focusing on structural properties rather than syntactic or semantic details, logemes reveal deeper similarities and differences between reasoning patterns, even when derived from distinct logical frameworks. The narrative alternates between bottom-up and top-down approaches: starting with basic components to build complex systems and simplifying complete systems into manageable fragments. Finally, logemes are connected to homotopy types, positioning them as a bridge between reasoning and topology. This unified framework offers a modular, flexible tool for studying logical fragments. By grounding reasoning in the topology of simplicial complexes, logemes provide a novel paradigm for analysing datasets and logical systems.

Legal theory, political sciences, sociology, philosophy, logic, artificial intelligence: there are many approaches to legal argumentation. Each of them provides specific insights into highly complex phenomena. Different disciplines, but also different traditions in disciplines (e.g. analytical and continental traditions in philosophy) find here a rare occasion to meet. The present book contains contributions, both historical and thematic, from leading researchers in several of the most important approaches to legal rationality. One of the main issues is the relation between logic and law: the way logic is actually used in law, but also the way logic can make law explicit. An outstanding group of philosophers, logicians and jurists try to meet this issue. The book is more than a collection of papers. However different their respective conceptual tools may be, the authors share a common conception: legal argumentation is a specific argumentation context.

We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix $${\mathbb {A}}$$ with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a i,j =?. The method allows us to decide whether to declare a i,j = 0 or a i,j = 1 or a i,j =? undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences.

This paper examines the deontic logic of the Talmud. We shall find, by looking at examples, that at first approximation we need deontic logic with several connectives: O T A Talmudic obligation F T A Talmudic prohibition F D A Standard deontic prohibition O D A Standard deontic obligation. In classical logic one would have expected that deontic obligation O D is definable by $O_DA \equiv F_D\neg A$ and that O T and F T are connected by $O_TA \equiv F_T\neg A$ This is not the case in the Talmud for the T (Talmudic) operators, though it does hold for the D operators. We must change our underlying logic. We have to regard {O T, F T } and {O D, F D } as two sets of operators, where O T and F T are independent of one another and where we have some connections between the two sets. We shall list the types of obligation patterns appearing in the Talmud and develop an intuitionistic deontic logic to accommodate them. We shall compare Talmudic deontic logic with modern deontic logic.

We consider conditionals of the form A ⇒ B where A depends on the future and B on the present and past. We examine models for such conditional arising in Talmudic legal cases. We call such conditionals contrary to time conditionals.Three main aspects will be investigated: Inverse causality from future to past, where a future condition can influence a legal event in the past (this is a man made causality).Comparison with similar features in modern law.New types of temporal logics arising from modelling the Talmudic examples. We shall see that we need a new temporal logic,which we call Talmudic temporal logic with linear open advancing future and parallel changing past, based on two parameters for time.