713 research outputs found
Embedding and Automating Conditional Logics in Classical Higher-Order Logic
A sound and complete embedding of conditional logics into classical
higher-order logic is presented. This embedding enables the application of
off-the-shelf higher-order automated theorem provers and model finders for
reasoning within and about conditional logics.Comment: 15 pages, 1 Figure, 1 Tabl
Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support
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 deontic logics, logic
combinations and ethico-legal domain theories in expressive classic
higher-order logic (HOL). This meta-logical approach enables the provision of
powerful tool support in LogiKEy: off-the-shelf theorem provers and model
finders for HOL are assisting the LogiKEy designer of ethical intelligent
agents to flexibly experiment with underlying logics and their combinations,
with ethico-legal domain theories, and with concrete examples---all at the same
time. Continuous improvements of these off-the-shelf provers, without further
ado, leverage the reasoning performance in LogiKEy. Case studies, in which the
LogiKEy framework and methodology has been applied and tested, give evidence
that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure
Systematic Verification of the Modal Logic Cube in Isabelle/HOL
We present an automated verification of the well-known modal logic cube in
Isabelle/HOL, in which we prove the inclusion relations between the cube's
logics using automated reasoning tools. Prior work addresses this problem but
without restriction to the modal logic cube, and using encodings in first-order
logic in combination with first-order automated theorem provers. In contrast,
our solution is more elegant, transparent and effective. It employs an
embedding of quantified modal logic in classical higher-order logic. Automated
reasoning tools, such as Sledgehammer with LEO-II, Satallax and CVC4, Metis and
Nitpick, are employed to achieve full automation. Though successful, the
experiments also motivate some technical improvements in the Isabelle/HOL tool.Comment: In Proceedings PxTP 2015, arXiv:1507.0837
Harnessing Higher-Order (Meta-)Logic to Represent and Reason with Complex Ethical Theories
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.Comment: 14 page
The Higher-Order Prover Leo-II.
Leo-II is an automated theorem prover for classical higher-order logic. The prover has pioneered cooperative higher-order-first-order proof automation, it has influenced the development of the TPTP THF infrastructure for higher-order logic, and it has been applied in a wide array of problems. Leo-II may also be called in proof assistants as an external aid tool to save user effort. For this it is crucial that Leo-II returns proof information in a standardised syntax, so that these proofs can eventually be transformed and verified within proof assistants. Recent progress in this direction is reported for the Isabelle/HOL system.The Leo-II project has been supported by the following grants: EPSRC grant EP/D070511/1 and DFG grants BE/2501 6-1, 8-1 and 9-1.This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s10817-015-9348-y
Universal (Meta-)Logical Reasoning: Recent Successes
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
Semantical Investigations on Non-classical Logics with Recovery Operators: Negation
We investigate mathematical structures that provide a natural semantics for
families of (quantified) non-classical logics featuring special unary
connectives, called recovery operators, that allow us to 'recover' the
properties of classical logic in a controlled fashion. These structures are
called topological Boolean algebras. They are Boolean algebras extended with
additional unary operations, called operators, such that they satisfy
particular conditions of a topological nature. In the present work we focus on
the paradigmatic case of negation. We show how these algebras are well-suited
to provide a semantics for some families of paraconsistent Logics of Formal
Inconsistency and paracomplete Logics of Formal Undeterminedness, which feature
recovery operators used to earmark propositions that behave 'classically' in
interaction with non-classical negations. In contrast to traditional semantical
investigations, carried out in natural language (extended with mathematical
shorthand), our formal meta-language is a system of higher-order logic (HOL)
for which automated reasoning tools exist. In our approach, topological Boolean
algebras become encoded as algebras of sets via their Stone-type
representation. We employ our higher-order meta-logic to define and interrelate
several transformations on unary set operations (operators), which naturally
give rise to a topological cube of opposition. Furthermore, our approach allows
for a uniform characterization of propositional, first-order and higher-order
quantification (also restricted to constant and varying domains). With this
work we want to make a case for the utilization of automated theorem proving
technology for doing computer-supported research in non-classical logics. All
presented results have been formally verified (and in many cases obtained)
using the Isabelle/HOL proof assistant
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