960 research outputs found
An Entailment Relation for Reasoning on the Web
Reasoning on the Web is receiving an increasing attention because of emerging fields such as Web adaption and Semantic Web. Indeed, the advanced functionalities striven for in these fields call for reasoning capabilities. Reasoning on the Web, however, is usually done using existing techniques rarely fitting the Web. As a consequence, additional data processing like data conversion from Web formats (e.g. XML or HTML) into some other formats (e.g. classical logic terms and formulas) is often needed and aspects of the Web (e.g. its inherent inconsistency) are neglected. This article first gives requirements for an entailment tuned to reasoning on the Web. Then, it describes how classical logic’s entailment can be modified so as to enforce these requirements. Finally, it discusses how the proposed entailment can be used in applying logic programming to reasoning on the Web
The modal logic of arithmetic potentialism and the universal algorithm
I investigate the modal commitments of various conceptions of the philosophy
of arithmetic potentialism. Specifically, I consider the natural potentialist
systems arising from the models of arithmetic under their natural extension
concepts, such as end-extensions, arbitrary extensions, conservative extensions
and more. In these potentialist systems, I show, the propositional modal
assertions that are valid with respect to all arithmetic assertions with
parameters are exactly the assertions of S4. With respect to sentences,
however, the validities of a model lie between S4 and S5, and these bounds are
sharp in that there are models realizing both endpoints. For a model of
arithmetic to validate S5 is precisely to fulfill the arithmetic maximality
principle, which asserts that every possibly necessary statement is already
true, and these models are equivalently characterized as those satisfying a
maximal theory. The main S4 analysis makes fundamental use of the
universal algorithm, of which this article provides a simplified,
self-contained account. The paper concludes with a discussion of how the
philosophical differences of several fundamentally different potentialist
attitudes---linear inevitability, convergent potentialism and radical branching
possibility---are expressed by their corresponding potentialist modal
validities.Comment: 38 pages. Inquiries and commentary can be made at
http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm.
Version v3 has further minor revisions, including additional reference
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Dimensions of Neural-symbolic Integration - A Structured Survey
Research on integrated neural-symbolic systems has made significant progress
in the recent past. In particular the understanding of ways to deal with
symbolic knowledge within connectionist systems (also called artificial neural
networks) has reached a critical mass which enables the community to strive for
applicable implementations and use cases. Recent work has covered a great
variety of logics used in artificial intelligence and provides a multitude of
techniques for dealing with them within the context of artificial neural
networks. We present a comprehensive survey of the field of neural-symbolic
integration, including a new classification of system according to their
architectures and abilities.Comment: 28 page
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