This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

Natural dualities are developed for varieties ofn-valued ukasiewicz algebras with and without negation. These dualities are based on hom-functors, and parallel Stone duality for Boolean algebras. A translation is described which relates the natural dualities to the corresponding restricted Priestley dualities. This enables a unified approach to free algebras to be presented, whence R. Cignoli's characterisations of the finitely generated free algebras are elucidated and new descriptions of arbitrary free algebras obtained. Finally it is shown how dualities for subvarieties (...) encode equational bases. An alternative approach is thereby provided to results of M.E. Adams and R. Cignoli. (shrink)

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build (...) a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation. Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras. (shrink)

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.

This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence lead on (...) to topological dualities in which the topological and correspondence components are quite separate. It is shown that, for DQAs, generalized canonicity is sufficient to yield, in a uniform way, topological dualities in the same style as those for canonical varieties. However topology and correspondence are no longer separable in the same way. (shrink)

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show (...) that topological and discrete dualities for $${\mathcal {A}}$$ exist in partnership. (shrink)

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form (...) of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra $\underline{M}$ generates a congruence-distributive variety and all subalgebras of $\underline{M}$ are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. (shrink)

The paper discusses regularisation of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of constants (...) within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Joseph Priestley's Criticisms of David Hume's Philosophy RICHARD H. POPKIN ONE OF HUME'S MOST FAMOUS CRITICS, the great scientist Joseph Priestley (1733-1804), is scarcely mentioned or studied in the Hume literature.' Perhaps because of the course philosophy followed after Hume, the Scottish Common Sense critics and the German ones connected with Kant are given almost all of the attention. In this paper 1 shall try to correct (...) this oversight, and examine the background of Priestley's critique of Hume, both in terms of Priestley's scientific and religious views and in terms of Hume's influence at the time. I shall then evaluate the merits of Priestley's critique and attempt to determine whether it deserves further study or should be returned to the obscurity it has been in for almost two centuries. 2 Joseph Priestley was a nonconforming minister whose progressively radical views caused him to lose various posts. Eventually he became a millenarian, convinced that crucial signs of the end of the world were at hand: the events of the French Revolution; the purported discovery of the Lost Tribes in Afghanistan by Sir William Jones; plans or hopes for the Jews to return to Palestine and join forces with the Christians. After years of religious wanderings Priestley ended up a millenarian Unitarian, much in the tradition of Sir Isaac Newton, whose religious writing he used extensively, and of Newton's successor, William Whiston? Priestley taught science in various schools and was immensely influenced by David Hartley's Observations on Man. He found in Hartley's theory of association of ideas a naturalistic way of explaining human mental life. And in Hartley's millenarian views he located the sort of theology he was seeking. Priestly later reedited Hartley, including introductory essays about some of the more difficult aspects of his mechanistic psychology.' (Coleridge, who was deeply impressed by ' T. E. Jessop, A Bibliography of David Hume and of the Scottish Philosophy from l-)'ancis Hutcheson to Lord Balfour (New York, 1966), lists the titles of Priestley's works relevant to Hume. However, there appear to be no discussions of Priestley on Hume listed in this work or in Roland Hill, A Bibliography of David Hume from 1930 (York, 1971). E. C. Mossner, The Life of David Hume (Edinburgh, 1954), mentions Priestley's criticisms of Hume, but only as these were made in connection with criticisms of Beattie. Joseph Priestley, Memoirs of Dr. Joseph Priestley to the Year 1795, written by Himself, with a Continuation to the time of his decease by his son, Joseph Priestley, ed. T. Cooper (London, 1806). It is also of some interest that Priestley attacked Reid, Beattie, and Oswald a few years before challenging Hume. He seemed to think that he had disposed of the Scottish Philosophy for all time. Thus Priestley was certainly no part of the mainstream of Hume's opponents in Britain. See Memoirs, pp. 78-79n, and n. 16 below. ' On Priestley's career, see his Memoirs. Modern biographies deal principally with his scientific achievements and ignore or play down his theological concerns. For an extended discussion of Priestley's millenarian views, see Clarke Garrett, Respectable Folly (Baltimore, 1975), pp. 126-143. " Priestley, Hartley's Theory of the Human Mind, on the Principle of the Association of Ideas... (London, 1775). [437] 438 HISTORY OF PHILOSOPHY Hartley, probably read Priestley's edition of his works.) For the most part, Priestley wrote long and controversial works on theology. His scientific achievements stem from his friendship with Benjamin Franklin and Richard Price, who encouraged him to write a history of discoveries in electricity. Franklin provided the resources for making the equipment. On the basis of the new material in his work, Priestley was nominated to the Royal Society by Franklin and Price. s Priestley's most famous scientific works came thereafter--his discovery of oxygen and his work on how the human body used oxygen. Later still, he became entangled in a controversy with Lavoisier over the phlogiston theory, Priestley upholding it, the French chemist opposing it. 6 Priestley was also active in politics, supporting radical causes. He supported John Wilkes, of whom Hume thoroughly disapproved, and was an early... (shrink)

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لقد خلقت المثنوية تأرجحاً ملازماً في المطارحات الفكرية والنظائر العلمية، أي صراعاً بين النازع الروحي، والعوز المادي. وتوضح لنا أن التذبذب، يميل في أغلب الأحيان إلى صدام بين الروحانية والعلمية. وهذا ما شهدناه بين الدين وفلسفة المنطق، وبين الدين ومكتشفات العلوم الطبيعية، ما جعل الفكر ينشط في توليد قضايا مختلفة في ألوانها وأجناسها وقيمها وأهدافها وطبيعة منجزاتها. وهكذا ينتهي بنا القول إلى أن المثنوية، هي فلسفة المتضاد المثنوي في منظومة القيم المُحدثة بفعل تأملات العقل النظري، والمحور الناظم لبنية النظائر المفترضة (...) في جدلية التغريب التي ساهمت في مأسسة مواريث ثقافة العماء في سائر المطارحات التي تخلّقها العقل الإبداعي. (shrink)

Translation and interpretation of selected Hadith on Islamic ethics and coduct.

Study of the religious and philosophical views of Sir Muḥammad Iqbal, 1877-1938.

Religion; Gods; Sufism; philosophical aspects.

كل ما أعنيه في مؤلفي، هو التعبير عن وجهات نظر تجاه آراء، طرحها بعض الفلاسفة والمفكرين والمنظرين والمبدعين الذين تناولوا جانباً حساساً في ما يتعلق بدور الوعي الفـني والجمالي في عملية إستلهام الحقائق الجمالية، والإفصاح عنها بواسطة الفن، والكشف عن البِنى القاعدية والتاريخية والمعرفية والفنية والتراثية والروحية والفكرية والجمالية.. سواء في تطابق وجهات النظر، أم دحض وتفنيد الأفكار التي رأت الفن والجمال في الوجود رؤية ســـطحية، وتوضيح الآراء المتباينة وتعددية المناهج المختلفة، التي حددت وظيفة الفن في بنائية الحياة. ويجدر الانتباه (...) إلى أن ما أعنيه من جمالي الفن، يشمل كافة الإشراقات الإبداعية العظيمة (أسطورية، أدبية، طقسية، تصوير، رسوم، لغة كتابية، عقائد دينية، أيديولوجية، دراما، نحت، عمارة، موسيقا، إعلام، رياضة، تكنولوجيا..إلخ) في حياتنا الأناسية. فوددت أن أستعرض بلمحة موجزة بدايات نشوء الفن كمقدمة منطقية، تفرضها ضرورات البحث، وذلك لما للفن من أثـر عميق في نشوء الحضارة والتاريخ، ولما له من قدرة نفاذية إلى الأثر الإلهي المتماهي في كينونة الوجود، ومبيناً أيضاً دور الدين في إبراز جلال الفن عـبر التاريخ، والإفصاح عن الإشكالية التي تتعلق به، بوصفه وعياً جمالياً ما وراء الواقع العياني. (shrink)

On Sikh doctrines inunciated by Guru Nanak.