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Whitehead's pointfree geometry and diametric posets

100%
Gerla G., Paolillo B.
Logic and Logical Philosophy
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2010
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vol. 19
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issue 4
289–308
EN
This note is motivated by Whitehead’s researches in inclusion-based point-free geometry as exposed in An Inquiry Concerning the Principles of Natural Knowledge and in The concept of Nature. More precisely, we observe that Whitehead’s definition of point, based on the notions of abstractive class and covering, is not adequate. Indeed, if we admit such a definition it is also questionable that a point exists. On the contrary our approach, in which the diameter is a further primitive, enables us to avoid such a drawback. Moreover, since such a notion enables us to define a metric in the set of points, our proposal looks to be a good starting point for a foundation of the geometry metrical in nature (as proposed, for example, by L.M. Blumenthal).
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ALGEBRAICZNE ASPEKTY MEREOLOGII NIEEKSTENSJONALNEJ

75%
Obojska L.
Roczniki Filozoficzne
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2012
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vol. 60
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issue 1
105-124
EN
An extensional mereology was subjected to analysis of many authors. It was proved that it corresponds to a Boolean algebra without a null element. A slightly modified version of this model in which the primitive relation of being a part does not fullfill the Extensional Principle, will be called: Non-extensional Mereology. There is no systematic analysis for such a model until now. Some authors present partial descriptions of it. In this work we would like to propose a detailed and systematic analysis of Non-extensional Mereology. We present a minimal set of axioms and show that this model, under certain conditions, corresponds to an implicative lattice.
PL
Mereologia klasyczna, nazywana również˙ mereologia˛ ekstensjonalna˛ została dość szczegółowo przebadana przez wielu autorów. Udowodniono, z˙e jest to model odpowiadający algebrom Boole’a bez zera. Model nieco słabszy, w którym relacja pierwotna bycia częścią nie spełnia zasady ekstensjonalności, może zostać nazwany mereologią nieekstensjonalną. Jak dotychczas nie istnieje systematyczna analiza takiego modelu. Kilka prac przedstawia jedynie pewne jej fragmentaryczne opisy. W niniejszej pracy pra- gniemy zaproponować formalna˛ i kompletna˛ analizę części tej teorii. Wprowadzając minimalny układ aksjomatów wykazujemy, że odpowiada ona algebraicznej strukturze kraty implikatywnej.
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