Related papers: Implicative models of set theory
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
Recent advances in Bayesian probability theory and its application to cognitive science in combination with the development of a new generation of computational tools and methods for probabilistic computation have led to a 'probabilistic…
Understanding the predictions made by deep learning models remains a central challenge, especially in high-stakes applications. A promising approach is to equip models with the ability to answer counterfactual questions -- hypothetical…
We investigate different set-theoretic constructions in Residuated Logic based on Fitting's work on Intuitionistic Set Theory. We start by stating some results concerning constructible sets within valued models of Set Theory. We present two…
The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on categories with finite limits and colimits. As an…
Inductive logic programming (ILP) has been a deeply influential paradigm in AI, enjoying decades of research on its theory and implementations. As a natural descendent of the fields of logic programming and machine learning, it admits the…
By contrast wih $\mathsf{S4}$, the analysis of local tabularity above $\mathsf{IPC}$ has provided a difficult challenge. This paper studies a strengthening of local tabularity -- \textit{uniform local tabularity} -- where one demands that…
In this paper, we introduce a translation that combines the $j$-translation with Kripke forcing in the internal logic of an elementary topos. First, we show that our translation is sound for intuitionistic first-order logic and Heyting…
We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with…
In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g.…
We extend two kinds of causal models, structural equation models and simulation models, to infinite variable spaces. This enables a semantics for conditionals founded on a calculus of intervention, and axiomatization of causal reasoning for…
In the area of inductive learning, generalization is a main operation, and the usual definition of induction is based on logical implication. Recently there has been a rising interest in clausal representation of knowledge in machine…
The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with…
We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
In designing generative models, it is commonly believed that in order to learn useful latent structure, we face a fundamental tension between expressivity and structure. In this paper we challenge this view by proposing a new approach to…
Both humans and large language models are able to learn language without explicit structural supervision. What inductive biases make this learning possible? We address this fundamental cognitive question by leveraging transformer language…
We present the first steps of interaction spaces theory, a universal mathematical theory of complex systems which is able to embed cellular automata, agent based models, master equation based models, stochastic or deterministic, continuous…
We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is…
We define various type of states on implicative involutive BE algebras (Jauch-Piron state, (P)-state, (B)-state, subadditive state, valuation), and we investigate the relationships between these states. Moreover, we introduce the unital,…