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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 series of theory simplification experiments conducted in interaction with a modern proof assistant system. The starting point for these experiments was the computer encoding of Gödel's argument, and then automated reasoning techniques were systematically applied to arrive at the simplified variant presented. The presented work thus exemplifies a fruitful human-computer interaction in computational metaphysics. Whether the presented result increases or decreases the attractiveness and persuasiveness of the ontological argument is a question I would like to pass on to philosophy and theology.
(Article to appear in Sophia)

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 existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem proving technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argument’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In computational hermeneutics, the utilization of automated reasoning tools effectively boosts our capacity to expose the assumptions we indirectly commit ourselves to every time we engage in rational argumentation and it fosters the explicitation and revision of our concepts and commitments.

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 simplifications of Gödel's original theory is the exploitation of a link to the notions of filter and ultrafilter in topology.
The paper illustrates how modern knowledge representation and reasoning technology for quantified non-classical logics can contribute new knowledge to other disciplines. The contributed material is also well suited to support teaching of non-trivial logic formalisms in classroom.

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 reading, they are in fact closely related, as our computer-supported formal analysis (conducted in the proof assistant system Isabelle/HOL) reveals. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.

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 simplifications of Gödel's original theory is the exploitation of a link to the notions of filter and ultrafilter from topology. The paper illustrates how modern knowledge representation and reasoning technology for quantified non-classical logics can contribute new knowledge to other disciplines. The contributed material is also well suited to support teaching of non-trivial logic formalisms in classroom.

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 reading they are in fact closely related. This has been revealed in the computer-supported formal analysis presented in this article. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.

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 reading, they are in fact closely related, as our computer-supported formal analysis (conducted in the proof assistant system Isabelle/HOL) reveals. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.

Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.

Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.

An ambitious explicit ethical theory, Gewirth's Principle of Generic Consistency, is mechanized on the computer. Utilizing Church's type theory as meta-logic to semantically embed a rich combination of expressive non-classical logics as required for the task, our work pushes existing boundaries in knowledge representation and reasoning. We demonstrate that intuitive encodings of ambitious ethical theories and their mechanization, resp. automation, on the computer are no longer antipodes.

Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.

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 generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category theory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments.

Mathematical proofs should be paired with formal proofs, whenever feasible.

The authors universal (meta-)logical reasoning approach is demonstrated and discussed with a challenge puzzle in epistemic reasoning: the wise men puzzle. The presented solution puts a particular emphasis on the adequate modeling of common knowledge.
(For further explanations see the related article at http://dx.doi.org/10.1016/j.scico.2018.10.008)

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 logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.

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

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

A semantical embedding of input/output logic in classical higher-order logic is presented. This embedding enables the mechanisation and automation of reasoning tasks in input/output logic with off-the-shelf higher-order theorem provers and proof assistants. The key idea for the solution presented here results from the analysis of an inaccurate previous embedding attempt, which we will discuss as well.

A flexible infrastructure for normative reasoning is outlined. A small-scale demonstrator version of the envisioned system has been implemented in the proof assistant Isabelle/HOL by utilising the first authors universal logical reasoning approach based on shallow semantical embeddings in meta-logic HOL. The need for such a flexible reasoning infrastructure is motivated and illustrated with a contrary-to-duty example scenario selected from the General Data Protection Regulation.

An ambitious ethical theory ---Alan Gewirth's "Principle of Generic Consistency"--- is encoded and analysed in Isabelle/HOL. Gewirth's theory has stirred much attention in philosophy and ethics and has been proposed as a potential means to bound the impact of artificial general intelligence.

A flexible infrastructure for the automation of deontic and normative reasoning is presented. Our motivation is the development, study and provision of legal and moral reasoning competencies in future intelligent machines. Since there is no consensus on the “best” deontic logic formalisms and since the answer may be application specific, a flexible infrastructure is proposed in which candidate logic formalisms can be varied, assessed and compared in experimental ethics application studies. Our work thus links the historically rich research areas of classical higher-order logic, deontic logics, normative reasoning and formal ethics.

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 logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.
DOI: 10.1016/j.scico.2018.10.008

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 generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed.
The purpose of this article is not to claim any novel results in category the- ory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments.
(Preprint of an article in JAR; see doi:10.1007/s10817-018-09507-7 )

In the past decades, several emendations of Gödel’s (resp.Scott’s) modal ontological argument have been proposed, many of which preserve the intended conclusion (the necessary existence of God), while avoiding a controversial side result of the Gödel/Scott variant called the “modal collapse”, which expresses that there are no contingent truths (everything is determined, there is no free will). In this paper we provide a summary on recent computer-supported veri- fication studies on some of these modern variants of the ontological argument. The purpose is to provide further evidence that the interaction with the computer technology, which we have developed together with colleagues over the past years, can not only enable the formal assessment of ontological arguments, but can, in fact, help to sharpen our conceptual understanding of the notions and concepts involved. From a more abstract perspective, we claim that interreligious understanding may be fostered by means of formal argumentation and, in particular, formal logical analysis supported by modern interactive and automated theorem proving technology.

We motivate and illustrate the utilization of (higher-order) automated deduction technologies for natural language understanding and, in particular, for tackling the problem of finding the most adequate logical formalisation of a natural language argument. Our approach, which we have called computational hermeneutics, is grounded on recent progress in the area of automated theorem proving for classical and non-classical higher-order logics, and it integrates techniques from argumentation theory. It has been inspired by ideas in the philosophy of language , especially semantic inferentialism and Donald Davidson's radical interpretation; a systematic approach to interpretation that does justice to the inherent circularity of understanding: the whole is understood (compositionally) on the basis of its parts, while each part is understood only in the context of the whole (hermeneutic circle). Computational hermeneutics is a holistic, iterative, trial-and-error approach, where we evaluate the adequacy of some candidate formalisation of a sentence by computing the logical validity of (i) the whole argument it appears in, and (ii) the dialectic role the argument plays in some piece of discourse.

In the past decades, several emendations of Gödel's (resp. Scott's) modal ontological argument have been proposed, many of which preserve the intended conclusion (the necessary existence of God), while avoiding a controversial side result of the Gödel/Scott variant called the "modal collapse", which expresses that there are no contingent truths (everything is determined, there is no free will). In this paper we provide a summary on recent computer-supported verification studies on some of these modern variants of the ontological argument. The purpose is to provide further evidence that the interaction with the computer technology, which we have developed together with colleagues over the past years, can not only enable the formal assessment of ontological arguments, but can, in fact, help to sharpen our conceptual understanding of the notions and concepts involved. From a more abstract perspective, we claim that interreligious understanding may be fostered by means of formal argumentation and, in particular, formal logical analysis supported by modern interactive and automated theorem proving technology.

The quest for a most general framework supporting universal reasoning is very prominently represented in the works of Leibniz. He envisioned a scientia generalis founded on a characteristica universalis, that is, a most universal formal language in which all knowledge about the world and the sciences can be encoded. A quick study of the survey literature on logical formalisms suggests that quite the opposite to Leibniz’ dream has become reality. Instead of a characteristica universalis, we are today facing a very rich and heterogenous zoo of different logical systems, and instead of converging towards a single superior logic, this logic zoo is further expanding, eventually even at accelerated pace. As a consequence, the unified vision of Leibniz seems farther away than ever before. However, there are also some promising initiatives to counteract these diverging developments. Attempts at unifying approaches to logic include categorial logic algebraic logic and coalgebraic logic.

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 recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is simply adjoined to AOT’s specially formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics .
Our results were made technically possible by a suitable adaptation of Benzmüller’s metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables one to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics.

A shallow semantic embedding of an intensional higher-order modal logic (IHOML) in Isabelle/HOL is presented. IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Gödel's God in order to discuss his emendation of Gödel's ontological argument for the existence of God. Utilizing IHOML, the most interesting parts of Fitting's textbook are formalized, automated and verified in the Isabelle/HOL proof assistant. A particular focus thereby is on three variants of the ontological argument which avoid the modal collapse, which is a strongly criticized side-effect in Gödel's resp. Scott's original work.

A computer-formalisation of the essential parts of Fitting’s text- book Types, Tableaus and Gödel’s God in Isabelle/HOL is presented. In particular, Fitting’s (and Anderson’s) variant of the ontological argument is verified and confirmed. This variant avoids the modal collapse, which has been criticised as an undesirable side-effect of Kurt Gödel’s (and Dana Scott’s) versions of the ontological argument. Fitting’s work is employing an intensional higher-order modal logic, which we shallowly embed here in classical higher-order logic. We then utilize the embedded logic for the formalisation of Fitting’s argument.

Reasoning with embedded formulas is relevant for the SUMO ontology but there is limited automation support so far. We investigate whether higher-order automated theorem provers are applicable for the task. Moreover, we point to a challenge that we have revealed as part of our experiments: modal operators in SUMO are in conflict with Boolean extensionality. A solution is proposed.

Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is well suited for realising a universal logic reasoning approach. Universal logic reasoning in turn, as envisioned already by Leibniz, may support the rigorous formalisation and deep logical analysis of rational arguments within machines. A respective universal logic reasoning framework is described and a range of exemplary applications are discussed. In the future, universal logic reasoning in combination with appropriate, controlled forms of rational argumentation may serve as a communication layer between humans and intelligent machines.

Simple type theory is suited as framework for combining classical and
non-classical logics. This claim is based on the observation that various
prominent logics, including (quantified) multimodal logics and intuitionistic
logics, can be elegantly embedded in simple type theory. Furthermore, simple
type theory is sufficiently expressive to model combinations of embedded logics
and it has a well understood semantics. Off-the-shelf reasoning systems for
simple type theory exist that can be uniformly employed for reasoning within
and about combinations of logics.

We report on the application of higher-order automated theorem proving in ontology reasoning. Concretely, we have integrated the Sigma knowledge engineering environment and the Suggested Upper-Level Ontology (SUMO) with the higher-order theorem prover LEO-II. The basis for this integration is a translation from SUMO’s SUO-KIF representations into the new typed higher-order form representation language TPTP THF. We illustrate the benefits of our integration with examples, report on experiments and analyze open challenges. 1

Numerous classical and non-classical logics can be elegantly embedded in Church’s simple type theory, also known as classical
higher-order logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security,
and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded
logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly
employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic
logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example
problems and report on experiments with off-the-shelf higher-order automated theorem provers.
KeywordsClassical and non-classical logics–Quantified multimodal logics–Logic combinations–Classical higher-order logic–Semantic embeddings–Knowledge representation–Higher-order automated theorem proving

The modal collapse that afflicts Gödel’s modal ontological argument for God’s existence is discussed from the perspective of the modal square of opposition.

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.

A notion of quantified conditional logics is provided that includes quantification over individual and propositional variables. The former is supported with respect to constant and variable domain semantics. In addition, a sound and complete embedding of this framework in classical higher-order logic is presented. Using prominent examples from the literature it is demonstrated how this embedding enables effective automation of reasoning within (object-level) and about (meta-level) quantified conditional logics with off-the-shelf higher-order theorem provers and model finders.

The mechanization and automation of combination of logics, expressive ontologies and notions of context are prominent current challenge problems. I propose to approach these challenge topics from the perspective of classical higher-order logic. From this perspective these topics are closely related and a common, uniform solution appears in reach.

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited.
Our embedding supports the application of off-the-shelf higher- order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

Goedel's ontological proof has been analysed for the first-time with an
unprecedent degree of detail and formality with the help of higher-order
theorem provers. The following has been done (and in this order): A detailed
natural deduction proof. A formalization of the axioms, definitions and
theorems in the TPTP THF syntax. Automatic verification of the consistency of
the axioms and definitions with Nitpick. Automatic demonstration of the
theorems with the provers LEO-II and Satallax. A step-by-step formalization
using the Coq proof assistant. A formalization using the Isabelle proof
assistant, where the theorems (and some additional lemmata) have been automated
with Sledgehammer and Metis.

A semantic embedding of quantified conditional logic in classical higher-order logic is utilized for reducing cut-elimination in the former logic to existing results for the latter logic. The presented embedding approach is adaptable to a wide range of other logics, for many of which cut-elimination is still open. However, special attention has to be payed to cut-simulation, which may render cut-elimination as a pointless criterion.

We present an interactive and automated theorem prover for free higher-order logic. Our implementation on top of the Isabelle/HOL framework utilizes a semantic embedding of free logic in classical higher-order logic. The capabilities of our tool are demonstrated with first experiments in category theory.

This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott's manuscript about Gödel's argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic KB instead of S5 and by avoiding the equality relation. The second proof is also technically shorter than the first one, because it eliminates unnecessary detours and uses Axiom 1 for the positivity of properties only once. The third and fourth proofs formalize, respectively, Anderson's and Bjørdal's variants of the ontological argument, which are known to be immune to modal collapse.

This paper discusses the inconsistency in Gödel's ontological argument. Despite the popularity of Gödel's argument, this inconsistency remained unnoticed until 2013, when it was detected automatically by the higher-order theorem prover Leo-II. Complementing the meta-logic explanation for the inconsistency available in our IJCAI 2016 paper [6], we present here a new purely object-logic explanation that does not rely on semantic argumentation.