# Christoph BenzmüllerOtto-Friedrich-Universität Bamberg · Department of Applied Computer Sciences

Christoph Benzmüller

Prof. Dr.

Artificial intelligence, computer science, philosophy, mathematics

## About

448

Publications

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Introduction

My research activities are interfacing the areas of artificial intelligence, philosophy, mathematics, computer science, and natural language. I am particularly interested in the use of classical higher-order logic (HOL) as a universal meta-logic to automate various non-classical logics and to utilise them in topical application areas, including machine ethics & machine law (responsible AI), metaphysics, mathematical foundations (e.g. category theory) and rational argumentation.

Additional affiliations

February 2022 - present

July 2018 - present

March 2017 - June 2018

## Publications

Publications (448)

A framework and methodology—termed LogiKEy—for the design and engineering of ethical reasoners, normative theories and deontic logics is presented. The overall motivation is the development of suitable means for the control and governance of intelligent autonomous systems. LogiKEy's unifying formal framework is based on semantical embeddings of deo...

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is co...

Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep l...

A simplified variant of Gödel's ontological argument is presented. The simplified argument is valid already in basic modal logics K or KT, it does not suffer from modal collapse, and it avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by Gödel. The variant presented has been obtained as a side result of a...

A simplified variant of Gödel’s ontological argument is presented. The simplified argument is valid already in basic modal logics K or KT, it does not suffer from modal collapse, and it avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by Gödel. The variant presented has been obtained as a side result of a...

Written reflective practice is a regular exercise pre-service teachers perform during their higher education. Usually, their lecturers are expected to provide individual feedback, which can be a challenging task to perform on a regular basis. In this paper, we present the first open-source automated feedback tool based on didactic theory and implem...

Higher-order logic HOL offers a very simple syntax and semantics for representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a rich type system, but has rather substantial conceptual differences to HOL, as well as comparatively poor proof...

This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization of the notion of elementary topoi. Additionally, we formalize symmetrical monoidal closed categories expressing...

We report some results regarding the mechanization of normative (preference-based) conditional reasoning. Our focus is on Aqvist's system E for conditional obligation (and its extensions). Our mechanization is achieved via a shallow semantical embedding in Isabelle/HOL. We consider two possible uses of the framework. The first one is as a tool for...

This paper describes an evaluation of Automated Theorem Proving (ATP) systems on problems taken from the QMLTP library of first-order modal logic problems. Principally, the problems are translated to higher-order logic in the TPTP language using an embedding approach, and solved using higher-order logic ATP systems. Additionally, the results from n...

Written reflective practice is a regular exercise pre-service teachers perform during their higher education. Usually, their lecturers are expected to provide individual feedback, which can be a challenging task to perform on a regular basis. In this paper, we present the first open-source automated feedback tool based on didactic theory and implem...

This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization of the notion of elementary topoi. Additionally, we formalize symmetrical monoidal closed categories expressing...

Higher-order logic HOL offers a very simple syntax and semantics for representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a rich type system, but has rather substantial conceptual differences to HOL, as well as comparatively poor proof...

In the Russo-Ukrainian war, propaganda is produced by Russian state-run news outlets for both international and domestic audiences. Its content and form evolve and change with time as the war continues. This constitutes a challenge to content moderation tools based on machine learning when the data used for training and the current news start to di...

The full-scale conflict between the Russian Federation and Ukraine generated an unprecedented amount of news articles and social media data reflecting opposing ideologies and narratives. These polarized campaigns have led to mutual accusations of misinformation and fake news, shaping an atmosphere of confusion and mistrust for readers worldwide. Th...

The full-scale conflict between the Russian Federation and Ukraine generated an unprecedented amount of news articles and social media data reflecting opposing ideologies and narratives. These polarized campaigns have led to mutual accusations of misinformation and fake news, shaping an atmosphere of confusion and mistrust for readers worldwide. Th...

This paper reports on an exploration of Boolos' Curious Inference, using higher-order automated theorem provers (ATPs). Surprisingly, only suitable shorthand notations had to be provided by hand for ATPs to find a short proof. The higher-order lemmas required for constructing a short proof are automatically discovered by the ATPs. Given the observa...

We report our first results regarding the automated verification of deontic correspondences (broadly conceived) and related matters in Isabelle/HOL, analogous to what has been achieved for the modal logic cube.

This paper describes an evaluation of Automated Theorem Proving (ATP) systems on problems taken from the QMLTP library of first-order modal logic problems. Principally, the problems are translated to higher-order logic in the TPTP languages using an embedding approach, and solved using higher-order logic ATP systems. Additionally, the results from...

Boolos' Curious Inference is automated in Isabelle/HOL after interactive speculation of a suitable shorthand notation (one or two definitions).

Over the past decade, variants of Gödel's ontological arguments have been critically examined using modern symbolic AI technology. Computers have unearthed new insights about them and even contributed to the exploration of new, simplified variants of the argument, which now need to be further investigated by theologians and philosophers. In this ar...

This paper reports on an exploration of variants of Boolos' curious inference, using higher-order automated theorem provers (ATPs). Surprisingly, only a single shorthand notation had to be provided by hand. All higher-order lemmas required for obtaining short proof are automatically discovered by the ATPs. Given the observations and suggestions in...

Participatory Design means recognizing that those who will be affected by a future technology should have an active say in its creation. Yet, despite continuous interest in involving people as future users and consumers into designing novel and innovative future technology, participatory approaches in technology design remain relatively underdevelo...

In this talk I give an overview of HOL, Universal Meta-Logical Reasoning in HOL, and the LogiKEy knowledge representation and reasoning methodology.
New side result:
Automation* of Boolos Curious Inference in Isabelle/HOL with the Prover E.
(Disclaimer *: Automation here means that only some key definitions still need to be given by hand. All...

Non-classical logics are used in a wide spectrum of disciplines, including artificial intelligence, computer science, mathematics, and philosophy. The de-facto standard infrastructure for automated theorem proving, the TPTP World, currently supports only classical logics. This paper describes the latest extension of the TPTP World, providing langua...

A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e. precision and intuition, through the advent...

A shallow semantical embedding for public announcement logic (PAL) with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way and (ii) how non-trivial...

The logico-pluralist LOGIKEY knowledge engineering methodology and framework is applied to the modelling of a theory of legal balancing in which legal knowledge (cases and laws) is encoded by utilising context-dependent value preferences. The theory obtained is then used to formalise, automatically evaluate, and reconstruct illustrative property la...

Non-classical logics are used in a wide spectrum of disciplines, including artificial intelligence, computer science, mathematics, and philosophy. The de-facto standard infrastructure for automated theorem proving, the TPTP World, currently supports only classical logics. Similar standards for non-classical logic reasoning do not exist (yet). This...

A simplified variant of G\"odel's ontological argument is presented. The simplified argument is valid already in basic modal logics K or KT, it does not suffer from modal collapse, and it avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by G\"odel. The variant presented has been obtained as a side result o...

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reaso...

This special issue of the Journal of Logic and Computation presents 11 carefully selected articles in the field of Logic and Argumentation. Authors of the strongest contributions to 4th International Conference on Logic and Argumentation (CLAR 2021) were invited to submit extended versions of their papers for inclusion in this special issue. All ar...

A shallow semantical embedding of a dyadic deontic logic by Carmo and Jones in classical higher-order logic is presented. The embedding is proven sound and complete, that is, faithful. This result provides the theoretical foundation for the implementation and automation of dyadic deontic logic within off-the-shelf higher-order theorem provers and p...

Invited keynote speech at the Inauguration Ceremony and Kick-off Meeting of ZLAIRE, the Zhejiang University—University of Luxembourg Joint Lab on Advanced Intelligent Systems and REasoning

This book constitutes the refereed proceedings of the 4th International Conference on Logic and Argumentation, CLAR 2021, held in Hangzhou, China, in October 2021. The 20 full and 10 short papers presented together with 5 invited papers were carefully reviewed and selected from 58 submissions. The topics of accepted papers cover the focus of the CL...

Literature in AI & Law contemplates argumentation in legal cases as an instance of theory construction. The task of a lawyer in a legal case is to construct a theory containing: (a) relevant generic facts about the world, (b) relevant legal rules such as precedents and statutes, and (c) contingent facts describing or interpreting the situation at h...

Presentation at the AAAI Spring Symposium on Implementing AI Ethics

Reductio ad absurdum can easily be avoided in the proof of the surjective Cantor theorem.

Literature in AI & Law contemplates argumentation in legal cases as an instance of theory construction. The task of a lawyer in a legal case is to construct a theory containing: (a) relevant generic facts about the world, (b) relevant legal rules such as precedents and statutes, and (c) contingent facts describing or interpreting the situation at h...

A shallow semantical embedding of a dyadic deontic logic by Carmo and Jones in classical higher-order logic is presented. The embedding is proven sound and complete, that is, faithful. This result provides the theoretical foundation for the implementation and automation of dyadic deontic logic within o�-the-shelf higher- order theorem provers and p...

Introduction to the Special Issue on Logic Rules and Reasoning: Selected Papers from the 2nd International Joint Conference on Rules and Reasoning (RuleML+RR 2018) - CHRISTOPH BENZMÜLLER, XAVIER PARENT, FRANCESCO RICCA

Die Autorinnen und Autoren präsentieren in diesem Buch Argumente, die die Unmöglichkeit des Reduktionismus aus philosophischer, naturwissenschaftlicher bzw. mathematisch-logischer Perspektive zu begründen suchen. Der Reduktionismus behauptet, dass Eigenschaften auch von komplexen Systemen (bis hin zu Lebensvorgängen und menschlichem Bewusstsein) vo...

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reaso...

Der zunehmende Einsatz von sogenannter „Künstlicher Intelligenz“ (KI) verspricht viele Verbesserungen, beispielsweise durch Bilderkennung in der Medizindiagnose. Er birgt aber auch das Risiko, dass Menschen durch irrtümliche Vorhersagen von KI-Systemen zu Schaden kommen können. In solchen Fällen wird es immer schwieriger zu bestimmen, wer die Veran...

We present a case example on how to conduct computer aided reasoning on legal texts. The basis is an anecdote of Kurt Gödel's citizenship hearing in which he claimed that the US Constitution allowed for the erection of a dictatorship. We shall model relevant parts of the US Constitution and conduct reasoning on them. This is done using the language...

To appear in: VerantwortungKI – Künstliche Intelligenz und gesellschaftliche Folgen (Schriftenreihe der Berlin-Brandenburgische Akademie der Wissenschaften), Interdisziplinäre Arbeitsgruppe Verantwortung: Maschinelles Lernen und Künstliche Intelligenz der Berlin-Brandenburgischen Akademie der Wissenschaften, volume 3, 2020.

The LogiKEy workbench and dataset for ethical and legal reasoning is presented. This workbench simultaneously supports development, experimentation, assessment and deployment of formal logics and ethical and legal theories at different conceptual layers. More concretely, it comprises, in form of a data set (Isabelle/HOL theory files), formal encodi...

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reaso...

A shallow semantical embedding of a dyadic deontic logic by Carmo and Jones in classical higher-order logic is presented. The embedding is proven sound and complete, that is, faithful. This result provides the theoretical foundation for the implementation and automation of dyadic deontic logic within o-the-shelf higher-order theorem provers and pro...

Future intelligent autonomous systems (IAS) are inevitably deciding on moral and legal questions, e.g. in self-driving cars, health care or human-machine collaboration. As decision processes in most modern sub-symbolic IAS are hidden, the simple political plea for transparency, accountability and governance falls short. A sound ecosystem of trust r...

Free logics are a family of logics that are free of any existential assumptions. Unlike traditional classical and non-classical logics, they support an elegant modeling of nonexistent objects and partial functions as relevant for a wide range of applications in computer science, philosophy, mathematics, and natural language semantics. While free fi...

Computers may help us to better understand (not just verify) arguments. In this chapter we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language argument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the ex...

Future intelligent autonomous systems (IAS) are inevitably deciding on moral and legal questions, e.g. in self-driving cars, health care or human-machine collaboration. As decision processes in most modern sub-symbolic IAS are hidden, the simple political plea for transparency, accountability and governance falls short. A sound ecosystem of trust r...

Deontic logic is the logic of normative concepts such as obligations, permissions, and rights, and it is used in knowledge representation to formalize ethical and legal reasoning. Norms can be conflicting and subject to exceptions, they can originate from various authorities, and they can specify ideal as well as sub-ideal situations and actions. A...

Free logics are a family of logics that are free of any exis-tential assumptions. Unlike traditional classical and non-classical logics, they support an elegant modeling of nonexistent objects and partial functions as relevant for a wide range of applications in computer science, philosophy , mathematics, and natural language semantics. While free...

We present an approach towards the deep, pluralistic logical analysis of argumentative discourse that benefits from the application of state-of-the-art automated reasoning technology for classical higher-order logic. Thanks to its expressivity this logic can adopt the status of a uniform \textit{lingua franca} allowing the encoding of both formaliz...

Enabling machines to legal balancing is a non-trivial task challenged by a multitude of factors some of which are addressed and explored in this work. We propose a holistic approach to formal modeling at different abstraction layers supported by a pluralistic framework in which the encoding of an ethico-legal value and upper ontology is developed i...

An approach to universal (meta-)logical reasoning in classical higher-order logic is employed to explore and study simplifications of Kurt Gödel's modal ontological argument. Some argument premises are modified, others are dropped, modal collapse is avoided and validity is shown already in weak modal logics K and T. Key to the gained simplification...

Future intelligent autonomous systems (IAS) are inevitably deciding on moral and legal questions, e.g. in self-driving cars, health care or human-machine collaboration. As decision processes in most modern sub-symbolic IAS are hidden, the simple political plea for transparency, accountability and governance falls short. A sound ecosystem of trust r...

A framework and methodology---termed LogiKEy---for the design and engineering of ethical reasoners, normative theories and deontic logics is presented. The overall motivation is the development of suitable means for the control and governance of intelligent autonomous systems. LogiKEy's unifying formal framework is based on semantical embeddings of...

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automorphisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fraïssé games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse c...

Climate Engineering (CE) is the intentional large-scale intervention in the Earth’s climate system to counter climate change. CE is highly controversial, spurring global debates about whether and under which conditions it should be considered. We focus on the computer-supported analysis of a small subset of the arguments pro and contra CE intervent...

An approach to universal (meta-)logical reasoning in classical higher-order logic is employed to explore and study simplifications of Kurt Gödel's modal ontological argument. Some argument premises are modified, others are dropped, modal collapse is avoided and validity is shown already in weak modal logics K and T. Key to the gained simplification...

Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of Ernst Mally, formalized by Zalta) that distinguishes between ordinary and abstract objects.
This article reports on recen...

We discuss ongoing work on reusing existing (higher-order) automated reasoning infrastructure for seamlessly combining and reasoning with different non-classical logics (modal, deontic, epistemic, paraconsistent, etc.), particularly suited for normative reasoning. Our work illustrates, in particular, the utilisation of the Is-abelle/HOL proof assis...

Leo-III is an effective automated theorem prover for ex-tensional type theory with Henkin semantics. It is based on an exten-sional higher-order paramodulation calculus and supports reasoning in monomorphic and rank-1 polymorphic first-order and higher-order logics. Leo-III also automates various non-classical logics, including almost every normal...

Climate Engineering (CE) is the intentional large-scale intervention in the Earth's climate system to counter climate change. CE is highly controversial, spurring global debates about whether and under what conditions it should be considered. We focus on the computer-supported analysis of a small subset of the arguments pro and contra CE interventi...

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automor-phisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fra¨ısséFra¨ıssé games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an...

Three variants of Kurt Gödel's ontological argument, proposed by Dana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory read...

A shallow semantical embedding of free logic in classical higher-order logic is presented, which enables the off-the-shelf application of higher-order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a ge...

This book constitutes the refereed proceedings of the 13th International Conference on Intelligent Computer Mathematics, CICM 2020, held in Bertinoro, Italy, in July 2020*.
The 15 full papers, 1 invited paper and 2 abstracts of invited papers presented were carefully reviewed and selected from a total of 35 submissions. The papers focus on advances...

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automor-phisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fraisse games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse...

We present a computer-supported approach for the logical analysis and conceptual explicitation of argumentative discourse. Computational hermeneutics harnesses recent progresses in automated reasoning for higher-order logics and aims at formalizing natural-language argumentative discourse using flexible combinations of expressive non-classical logi...

Three variants of Kurt Gödel's ontological argument, as proposed byDana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument, the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory r...

Three variants of Kurt Gödel's ontological argument, as proposed byDana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument, the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory r...

A shallow semantical embedding of Input/output logic in classical higher-order logic is presented, and shown to be faithful (sound and complete). This embedding has been implemented in Isabelle/HOL, a proof assistant tool. It is applied to a well-known example in moral philosophy, the example of moral luck.