Abstract. 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-e↵ect in Gödel's resp. Scott's original work.
The computer-mechanization of an ambitious explicit ethical theory, Gewirth's Principle of Generic Consistency, is used to showcase an approach for representing and reasoning with ethical theories exhibiting complex logical features like alethic and deontic modalities, indexicals, higher-order quantification, among others. Harnessing the high expressive power of Church's type theory as a meta-logic to semantically embed a combination of quantified non-classical logics, our work pushes existing boundaries in knowledge representation and reasoning. We demonstrate that intuitive encodings of complex ethical theories and their automation on the computer are no longer antipodes.
This paper presents the special issue on Formal Approaches to the Ontological Argument and briefly introduces the ontological argument from the standpoint of logic and philosophy of religion (more specifically the debate on the rationality of theistic belief). Arguments for and against the existence of God have been proposed and subjected to logical analysis in different periods of the history of philosophy. In an important sense, they all deal with the rationality of theist belief. Providing a good argument for the conclusion that God does exist, or that it is highly probable that he exists, might be a pretty strong case for the thesis that belief in his existence is rational. Similarly, a good argument for the conclusion that God does not exist could be said to support the thesis that theistic belief is irrational. A more basic approach than that would be to analyze the very concept of God. Can God create a stone so heavy that he cannot lift? If we say yes, then there is something God cannot do, namely to create such a stone; if we say no, there is also something he cannot do, namely to lift the stone. In either case he is not omnipotent. If really unsolvable, paradoxes like this (this is the paradox of the stone) show that the concept of God (who is, besides other things, omnipotent 1) is incoherent or contradictory. Like the concept of a squared circle, it could never be 1 For more on the concept of omnipotence and the paradox of the stone see [6].
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 dataset (Isabelle/HOL theory files), formal encodings of multiple deontic logics, logic combinations, deontic paradoxes and normative theories in the higher-order proof assistant system Isabelle/HOL. The data were acquired through application of the LogiKEy methodology, which supports experimentation with different normative theories, in different application scenarios, and which is not tied to specific logics or logic combinations. Our workbench consolidates related research contributions of the authors and it may serve as a starting point for further studies and experiments in flexible and expressive ethical and legal reasoning. It may also support hands-on teaching of non-trivial logic formalisms in lecture courses and tutorials.
The LogiKEy methodology and framework is discussed in more detail in the companion research article titled “Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support”
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