12,141 research outputs found
Extensional Higher-Order Paramodulation in Leo-III
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 compatible with the TPTP/TSTP
framework for input formats, reporting results and proofs, and standardized
communication between reasoning systems, enabling e.g. proof reconstruction
from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning
in polymorphic first-order and higher-order logic, in all normal quantified
modal logics, as well as in different deontic logics. Its development had
initiated the ongoing extension of the TPTP infrastructure to reasoning within
non-classical logics.Comment: 34 pages, 7 Figures, 1 Table; submitted articl
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
Goal Translation for a Hammer for Coq (Extended Abstract)
Hammers are tools that provide general purpose automation for formal proof
assistants. Despite the gaining popularity of the more advanced versions of
type theory, there are no hammers for such systems. We present an extension of
the various hammer components to type theory: (i) a translation of a
significant part of the Coq logic into the format of automated proof systems;
(ii) a proof reconstruction mechanism based on a Ben-Yelles-type algorithm
combined with limited rewriting, congruence closure and a first-order
generalization of the left rules of Dyckhoff's system LJT.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
Towards the Integration of an Intuitionistic First-Order Prover into Coq
An efficient intuitionistic first-order prover integrated into Coq is useful
to replay proofs found by external automated theorem provers. We propose a
two-phase approach: An intuitionistic prover generates a certificate based on
the matrix characterization of intuitionistic first-order logic; the
certificate is then translated into a sequent-style proof.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
Rejection in Ĺukasiewicz's and SĹupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Ĺukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Ĺukasiewicz and developed by
his student SĹupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
Topological Models of Columnar Vagueness
This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a âtranslationâ of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfittâs recent topological reconstruction of Sainsburyâs theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzienâs account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenforsâ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamsonâs âlogic of clarityâ is explicated in terms of a generalized topology (âlocologyâ) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamsonâs logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a âslim boundaryâ are (stably) columnar. Thus, Williamsonâs logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness
Computer Science and Metaphysics: A Cross-Fertilization
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.Comment: 39 pages, 3 figure
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
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