arXiv subjects

Bruno Bentzen

Publications and source records attributed to Bruno Bentzen.

8 recordsLinked to original sources

arXiv2025

Naturalistic intuitionism for physics

Recently, a novel intuitionistic reconstruction of the foundations of physics has been primarily developed by Nicolas Gisin and Flavio Del Santo drawing on naturalism. Our goal in this paper is to examine and develop the philosophical background of their naturalistic intuitionism for physics in contrast with Brouwer's defense of his intuitionistic mathematics. To be exact, we propose a systematic rearticulation of Brouwer's so-called two acts of intuitionism to serve as the self-contained philosophical framework justifying naturalistic intuitionism in physics. This revision is accompanied by an investigation of the distinctive naturalistic treatment of some central intuitionistic topics, including logic, language, time, ontology, meaning, and truth.

physics.hist-ph↗

arXiv2023

Verified completeness in Henkin-style for intuitionistic propositional logic

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional logic with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature. The full source code can be found online at https://github.com/bbentzen/ipl.

cs.LO↗

arXiv2023

BTPK-based interpretable method for NER tasks based on Talmudic Public Announcement Logic

As one of the basic tasks in natural language processing (NLP), named entity recognition (NER) is an important basic tool for downstream tasks of NLP, such as information extraction, syntactic analysis, machine translation and so on. The internal operation logic of current name entity recognition model is black-box to the user, so the user has no basis to determine which name entity makes more sense. Therefore, a user-friendly explainable recognition process would be very useful for many people. In this paper, we propose a novel interpretable method, BTPK (Binary Talmudic Public Announcement Logic model), to help users understand the internal recognition logic of the name entity recognition tasks based on Talmudic Public Announcement Logic. BTPK model can also capture the semantic information in the input sentences, that is, the context dependency of the sentence. We observed the public announcement of BTPK presents the inner decision logic of BRNNs, and the explanations obtained from a BTPK model show us how BRNNs essentially handle NER tasks.

cs.CL↗

arXiv2021

A Henkin-style completeness proof for the modal logic S5

This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover. The proof formalized is close to that of Hughes and Cresswell, but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online at https://github.com/bbentzen/mpl/ and has been typechecked with Lean 3.4.2.

cs.LO↗

arXiv2020

Frege's theory of types

It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies Church's simple theory of types. This claim roughly states that Frege presupposes a type of functions in the sense of simple type theory in the expository language of Grundgesetze. However, this view makes it hard to accommodate function names of two arguments and view functions as incomplete entities. I propose and defend an alternative interpretation of first-level function names in Grundgesetze into simple type-theoretic open terms rather than into closed terms of a function type. This interpretation offers a still unhistorical but more faithful type-theoretic approximation of Frege's theory of levels and can be naturally extended to accommodate second-level functions. It is made possible by two key observations that Frege's Roman markers behave essentially like open terms and that Frege lacks a clear criterion for distinguishing between Roman markers and function names.

math.HO↗

arXiv2020

Sense, reference, and computation

In this paper, I revisit Frege's theory of sense and reference in the constructive setting of the meaning explanations of type theory, extending and sharpening a program--value analysis of sense and reference proposed by Martin-L\"of building on previous work of Dummett. I propose a computational identity criterion for senses and argue that it validates what I see as the most plausible interpretation of Frege's equipollence principle for both sentences and singular terms. Before doing so, I examine Frege's implementation of his theory of sense and reference in the logical framework of Grundgesetze, his doctrine of truth values, and views on sameness of sense as equipollence of assertions.

math.HO↗

arXiv2019

Naive cubical type theory

This paper proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the implicit foundation, confining our presentation to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types, and the Eckmman-Hilton duality.

cs.LO↗

arXiv2018

Cubical informal type theory: the higher groupoid structure

Following a project of developing conventions and notations for informal type theory carried out in the homotopy type theory book for a framework built out of an augmentation of constructive type theory with axioms governing higher-dimensional constructions via Voevodsky's univalance axiom and higher-inductive types, this paper proposes a way of doing informal type theory with a cubical type theory as the underlying foundation instead. To that end, we adopt a cubical type theory recently proposed by Angiuli, Hou (Favonia) and Harper, a framework with a cumulative hierarchy of univalent Kan universes, full univalence and instances of higher-inductive types. In the present paper we confine ourselves to some elementary theorems concerning the higher groupoid structure of types.

cs.LO↗