Abstract
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.
Supported by VolkswagenStiftung, grant Consistent, Rational Arguments in Politics (CRAP).
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Notes
- 1.
The formal content of this paper has been generated directly by Isabelle from our source files. A benefit is the prevention of typos. As a side contribution we showcase the usability of modern proof assistants for the non-initiated in order to foster their application.
- 2.
Neighbourhood semantics is a generalisation of Kripke semantics, developed independently by Dana Scott and Richard Montague. Whereas a Kripke frame features an accessibility relation \(R: W{\rightarrow }2^W\) indicating which worlds are alternatives to (or, accessible from) others, a neighborhood frame \(N: W{\rightarrow }2^{2^W}\) (or, as in our case, \(N: 2^W{\rightarrow }2^{2^W}\)) features a neighbourhood function assigning to each world (or set of worlds) a set of sets of worlds.
- 3.
Note that in addition to the ASCII name “cjboxa”, Isabelle/HOL supports graphical notation
This is essential for obtaining intuitive mathematical representations. - 4.
Note that
is not part of Kaplan’s original system. It has been added by us in order to better highlight some semantic features of our formalization of Gewirth’s theory in the next section and for enabling the use of the necessitation rule for drawing inferences. - 5.
Our work constitutes a most relevant first step for further assessment of Kornai’s claim. E.g. we plan to embody our encoding of Gewirth’s theory in virtual agents and devise and conduct respective empirical studies. The merits of the work presented here are however not tied to the validity of Kornai’s claim. We illustrate that representation and reasoning with complex ethical theories is meanwhile feasible to an extent unmatched before; and this is highly relevant for implementing explicit ethical intelligent systems. In the following, we will present some commented extracts of our formal encoding of Gewirth’s theory and of the computer-supported verification of the argument leading to the PGC.
- 6.
We were indeed able to formally verify Gewirth’s claim, on condition of committing to an alternative notion of (logical) necessity: Kaplan’s “indexical validity”.
- 7.
Definitions and axiomatized conceptual interrelations framing the inferential role of terms. We also refer to them as “explications”. Meaning postulates were introduced in Carnap (1952).
- 8.
Lemma “recognizeOtherPPA” below is indeed inferred from axiom “essentialPPA” using Isabelle’s blast tactic (a tableaux prover).
- 9.
Their higher-order and modal nature well illustrates the need for expressive knowledge representation and reasoning techniques.
- 10.
This theorem is indeed derivable directly in DDL from the definition of obligations: If
oughts to obtain then
is possible. - 11.
Below we use Isabelle’s simp tool to prove that Kant’s lemma follows from one of the DDL semantic conditions (not shown here).
This is essential for obtaining intuitive mathematical representations.
is not part of Kaplan’s original system. It has been added by us in order to better highlight some semantic features of our formalization of Gewirth’s theory in the next section and for enabling the use of the necessitation rule for drawing inferences.
oughts to obtain then
is possible.References
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Fuenmayor, D., Benzmüller, C. (2019). Harnessing Higher-Order (Meta-)Logic to Represent and Reason with Complex Ethical Theories. In: Nayak, A., Sharma, A. (eds) PRICAI 2019: Trends in Artificial Intelligence. PRICAI 2019. Lecture Notes in Computer Science(), vol 11670. Springer, Cham. https://doi.org/10.1007/978-3-030-29908-8_34
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