55,901 research outputs found
Fundamental Properties of Fuzzy Implications
In the article we continue in the Mizar system [8], [2] the formalization of fuzzy implications according to the monograph of BaczyĆski and Jayaram âFuzzy Implicationsâ [1]. We develop a framework of Mizar attributes allowing us for a smooth proving of basic properties of these fuzzy connectives [9]. We also give a set of theorems about the ordering of nine fundamental implications: Ćukasiewicz (ILK), Gödel (IGD), Reichenbach (IRC), Kleene-Dienes (IKD), Goguen (IGG), Rescher (IRS), Yager (IYG), Weber (IWB), and Fodor (IFD).This work is a continuation of the development of fuzzy sets in Mizar [6]; it could be used to give a variety of more general operations on fuzzy sets [13]. The formalization follows [10], [5], and [4].Institute of Informatics, University of BiaĆystok, PolandMichaĆ BaczyĆski and Balasubramaniam Jayaram. Fuzzy Implications. Springer Publishing Company, Incorporated, 2008. doi:10.1007/978-3-540-69082-5.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, and Karol PÄ
k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9â32, 2018. doi:10.1007/s10817-017-9440-6.Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.Adam Grabowski. Formal introduction to fuzzy implications. Formalized Mathematics, 25(3):241â248, 2017. doi:10.1515/forma-2017-0023.Adam Grabowski. Basic formal properties of triangular norms and conorms. Formalized Mathematics, 25(2):93â100, 2017. doi:10.1515/forma-2017-0009.Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51â54, 2013.Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing - 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160â171. Springer, 2015. doi:10.1007/978-3-319-19324-3_15.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191â198, 2015. doi:10.1007/s10817-015-9345-1.Petr HĂĄjek. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer, 1998.Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics, 9(2):351â356, 2001.Yasunari Shidama. The Taylor expansions. Formalized Mathematics, 12(2):195â200, 2004.Philippe Smets and Paul Magrez. Implication in fuzzy logic. International Journal of Approximate Reasoning, 1(4):327â347, 1987. doi:10.1016/0888-613X(87)90023-5.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338â353, 1965. doi:10.1016/S0019-9958(65)90241-X.26427127
The Pondicherry interpretation of quantum mechanics: An overview
An overview of the Pondicherry interpretation of quantum mechanics is
presented. This interpretation proceeds from the recognition that the
fundamental theoretical framework of physics is a probability algorithm, which
serves to describe an objective fuzziness (the literal meaning of Heisenberg's
term "Unschaerfe," usually mistranslated as "uncertainty") by assigning
objective probabilities to the possible outcomes of unperformed measurements.
Although it rejects attempts to construe quantum states as evolving ontological
states, it arrives at an objective description of the quantum world that owes
nothing to observers or the goings-on in physics laboratories. In fact, unless
such attempts are rejected, quantum theory's true ontological implications
cannot be seen. Among these are the radically relational nature of space, the
numerical identity of the corresponding relata, the incomplete spatiotemporal
differentiation of the physical world, and the consequent top-down structure of
reality, which defies attempts to model it from the bottom up, whether on the
basis of an intrinsically differentiated spacetime manifold or out of a
multitude of individual building blocks.Comment: 18 pages, 1 eps figure, v3: with corrections made in proo
This elusive objective existence
Zurek's existential interpretation of quantum mechanics suffers from three
classical prejudices, including the belief that space and time are
intrinsically and infinitely differentiated. They compel him to relativize the
concept of objective existence in two ways. The elimination of these prejudices
makes it possible to recognize the quantum formalism's ontological implications
- the relative and contingent reality of spatiotemporal distinctions and the
extrinsic and finite spatiotemporal differentiation of the physical world -
which in turn makes it possible to arrive at an unqualified objective
existence. Contrary to a widespread misconception, viewing the quantum
formalism as being fundamentally a probability algorithm does not imply that
quantum mechanics is concerned with states of knowledge rather than states of
Nature. On the contrary, it makes possible a complete and strongly objective
description of the physical world that requires no reference to observers. What
objectively exists, in a sense that requires no qualification, is the
trajectories of macroscopic objects, whose fuzziness is empirically irrelevant,
the properties and values of whose possession these trajectories provide
indelible records, and the fuzzy and temporally undifferentiated states of
affairs that obtain between measurements and are described by counterfactual
probability assignments.Comment: To appear in IJQI; 21 pages, LaTe
Are we at the dawn of quantum-gravity phenomenology?
A handful of recent papers has been devoted to proposals of experiments
capable of testing some candidate quantum-gravity phenomena. These lecture
notes emphasize those aspects that are most relevant to the questions that come
to mind when one is exposed for the first time to these research developments:
How come theory and experiments are finally meeting in spite of all the gloomy
forecasts that pervade traditional reviews? Is this a case of theorists having
put forward more and more speculative ideas until a point was reached at which
conventional experiments could rule out the proposed phenomena? Or has there
been such a remarkable improvement in experimental techniques and ideas that we
are now capable of testing plausible candidate quantum-gravity phenomena? These
questions are analysed rather carefully for the recent proposals of
interferometry-based tests and tests using observations of gamma rays of
astrophysical origin. I also briefly discuss other proposed experiments
(including tests of quantum-gravity-induced decoherence using the neutral-kaon
system and accelerator tests of models with large extra dimensions). The
emerging picture suggests that we are finally starting the exploration of a
large class of plausible quantum-gravity effects. However, our chances to
obtain positive (discovery) experimental results depend crucially on the
magnitude of these effects. In most cases the level of sensitivity that the
relevant experiments should achieve within a few years corresponds to effects
suppressed only linearly by the Planck length.Comment: 47 pages, Latex. Based on lectures given at the XXXV Karpacz Winter
School of Theoretical Physics "From Cosmology to Quantum Gravity", Polanica,
Poland, 2-12 February, 1999. To appear in the proceeding
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