Related papers: The Stratified Foundations as a theory modulo

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a…

In this paper we give an ordinal analysis of the theory of second order arithmetic. We do this by working with proof trees -- that is, "deductions" which may not be well-founded. Working in a suitable theory, we are able to represent…

There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…

We introduce stratified toposes, which are toposes that are stratified by a suitable hierarchy of universes. The term `stratified topos' recalls the notion of stratified pseudotopos of Moerdijk and Palmgren (2002). However, the details of…

Prototypical Learning is based on the idea that there is a point (which we call prototype) around which the embeddings of a class are clustered. It has shown promising results in scenarios with little labeled data or to design explainable…

Justification theory is a general framework for the definition of semantics of rule-based languages that has a high explanatory potential. Nested justification systems, first introduced by Denecker et al. (2015), allow for the composition…

Pearl observes that causal knowledge enables predicting the effects of interventions, such as actions, whereas descriptive knowledge only permits drawing conclusions from observation. This paper extends Pearl's approach to causality and…

Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed-point logic. Inclusion logic is not axiomatizable in full, but its first-order consequences can be axiomatized. In…

The success of denoising diffusion models raises important questions regarding their generalisation behaviour, particularly in high-dimensional settings. Notably, it has been shown that when training and sampling are performed perfectly,…

Foundation models have achieved remarkable success across diverse machine-learning domains through large-scale pretraining on large, diverse datasets. However, pretraining on such datasets introduces significant challenges due to…

The main aim of this paper is to promote a certain style of doing coinductive proofs, similar to inductive proofs as commonly done by mathematicians. For this purpose, we provide a reasonably direct justification for coinductive proofs…

Dependence logic provides an elegant approach for introducing dependencies between variables into the object language of first-order logic. In [1] generalized quantifiers were introduced in this context. However, a satisfactory account was…

We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely…

In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…

Ill-founded (or non-wellfounded) proof systems have emerged as a natural framework for inductive and coinductive reasoning. In such systems, soundness relies on global correctness criteria, such as the progressivity condition. Ensuring that…

A common method to study deep learning systems is to use simplified model representations--for example, using singular value decomposition to visualize the model's hidden states in a lower dimensional space. This approach assumes that the…

First-order logic is typically presented as the study of deduction in a setting with elementary quantification. In this paper, we take another vantage point and conceptualize first-order logic as a linear space that encodes "plausibility".…

We prove a general decomposition theorem for the modal $\mu$-calculus $L_\mu$ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two…

We introduce `atomic flows': they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations.…

This article examines the notion of invariance under different kinds of permutations in a milieu of a theory of classes and sets, as a semantic motivation for Quine's new foundations "NF". The approach largely depends on interpreting a…