In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hájek generalized this result to the first-order versions of Łukasiewicz, Gödel and Product logics, w.r.t. their standard algebras. In this paper we extend the analysis to the first-order versions of axiomatic extensions of MTL. Our main result is the following. Let \ be a class of MTL-chains. Then the set of all first-order tautologies associated to the finite (...) models over chains in \, fTAUT\, is \ -hard. Let TAUT\ be the set of propositional tautologies of \. If TAUT\ is decidable, we have that fTAUT\ is in \. We have similar results also if we expand the language with the Δ operator. (shrink)

Around 1950, B.A. Trakhtenbrot proved an important undecidability result (known, by a pure accident, as \Trakhtenbrot's theorem"): there is no algorithm to decide, given a rst-order sentence, whether the sentence is satis able in some nite model. The result is in fact true even if we restrict ourselves to languages that has only one binary relation Tra63]. It is hardly conceivable that at that time Prof. Trakhtenbrot expected his result to in uence the development of the theory (...) of relational databases query languages, but it did. This perhaps is not surprising in view of the following two facts: 1) The theory of relational databases is strongly rooted in logic and can easily be abstracted to make it a branch of mathematical logic. 2) The main interest in the theory of relational databases is in nite relations. In the rst section we explain those two points in more detail. Then, in section 2 of the present paper, we discuss the question: \What constitutes a reasonable query to a database?" The discussion naturally leads to a certain class of queries, known as the \domain independent" queries, as an ideal query language. At this point Trakhtenbrot's theorem appears as a major obstacle, since it easily implies that this ideal class is undecidable. Real query languages cannot be allowed to have this property. Section 3 outlines then several possible ways to circumvent this di culty. The following section explains the.. (shrink)

The late Jacob Klein’s important book is, remarkably, a lucid presentation of esoteric argument. Dealing with the famed Platonic triad, Theaetetus, Sophist, and Statesman, Klein settles the dispute about the missing dialogue, "The Philosopher," by first denying that it is missing and second showing that it is unnecessary. He argues, in short, that the triad is a dyad. That argument is reinforced by the distinction Klein strongly implies between the Socratic Theaetetus and the Eleatic Sophist and Statesman. "We can now (...) understand why the conversations of the ‘trilogy’ occur in two, not three, days and why the Sophist, the Statesman, and the Philosopher are dealt with in two, not three, dialogues". (shrink)

Bridging the gap between the world of science and the realm of the spiritual, B. Alan Wallace introduces a natural theory of human consciousness that has its roots in contemporary physics and Buddhism. Wallace's "special theory of ontological relativity" suggests that mental phenomena are _conditioned_ by the brain, but do not _emerge_ from it. Rather, the entire natural world of mind and matter, subjects and objects, arises from a unitary dimension of reality that is more fundamental than these dualities, as (...) proposed by Wolfgang Pauli and Carl Jung. To test his hypothesis, Wallace employs the Buddhist meditative practice of _samatha_, refining one's attention and metacognition, to create a kind of telescope to examine the space of the mind. Drawing on the work of the physicist John Wheeler, he then proposes a more general theory in which the participatory nature of reality is envisioned as a self-excited circuit. In comparing these ideas to the Buddhist theory known as the Middle Way philosophy, Wallace explores further aspects of his "general theory of ontological relativity," which can be investigated by means of _vipasyana_, or insight, meditation. Wallace then focuses on the theme of symmetry in reference to quantum cosmology and the "problem of frozen time," relating these issues to the theory and practices of the Great Perfection school of Tibetan Buddhism. He concludes with a discussion of the general theme of complementarity as it relates to science and religion. The theories of relativity and quantum mechanics were major achievements in the physical sciences, and the theory of evolution has had an equally deep impact on the life sciences. However, rigorous scientific methods do not yet exist to observe mental phenomena, and naturalism has its limits for shedding light on the workings of the mind. A pioneer of modern consciousness research, Wallace offers a practical and revolutionary method for exploring the mind that combines the keenest insights of contemporary physicists and philosophers with the time-honored meditative traditions of Buddhism. (shrink)

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)

Georgians' acquaintance with Islam was with the first Arab raids. From the first Muslim Arab domination, Georgians started to become Muslims with cultural interaction. Islam spread especially in Eastern Georgia during the time of Muslim Arabs, Seljuks and Mongols. The spread of Islam in Western Georgia started with the Ottomans. The Ottoman Empire's contact with Georgia begins with the conquest of Trabzon by Fatih Sultan Mehmed. When the Ottomans contacted the region, the geography of Georgia was divided into small kingdoms. (...) With the conquest of Trabzon, the Georgian border was reached and in 1479, Acaristan, Borcka and Maçahel surroundings joined the Ottoman lands. On the one hand, Sukhumi was captured from the north. Then, continuing to advance in the region, Yavuz Sultan Selim paid tribute to the Kingdom of Güryel and Imeret (Açıkbaş) in 1508 during his governorship of Trabzon. Thus, the Ottoman borders had reached beyond the Black Sea coast into Western Georgia. As a result of the increasing relations around Adjaria, which came under Ottoman rule, Islamization began to spread rapidly among the people. Contrary to what is claimed, the spread of Islam in the region did not spread from the ruling strata to the lower, but from the popular strata to the ruling strata. This process was surprisingly fast. In the process of Islamization, many mosques, masjids and schools were built in Adjara, especially with the help of the people. The foundations operating in the region and the mosques and masjids serving under them show that, contrary to the claims of Russian and Georgian historians, Islamization took place rapidly at the beginning of the 17th century in Adjara. In this process, many mosques, masjids and schools were built in the region. In this research, based on the numerous Hurufat records belonging to Adjara, Batumi, Gönye, Çürüksu and Maçahel, the Islamization process will be evaluated by making a social reading with the data obtained from the religious structures established in the rural settlements of the Adjara geography and their officials. Evaluations were made on the Islamization process in Adjara by making a social reading. (shrink)

Bridging the gap between the world of science and the realm of the spiritual, B. Alan Wallace introduces a natural theory of human consciousness that has its roots in contemporary physics and Buddhism. Wallace's "special theory of ontological relativity" suggests that mental phenomena are _conditioned_ by the brain, but do not _emerge_ from it. Rather, the entire natural world of mind and matter, subjects and objects, arises from a unitary dimension of reality that is more fundamental than these dualities, as (...) proposed by Wolfgang Pauli and Carl Jung. To test his hypothesis, Wallace employs the Buddhist meditative practice of _samatha_, refining one's attention and metacognition, to create a kind of telescope to examine the space of the mind. Drawing on the work of the physicist John Wheeler, he then proposes a more general theory in which the participatory nature of reality is envisioned as a self-excited circuit. In comparing these ideas to the Buddhist theory known as the Middle Way philosophy, Wallace explores further aspects of his "general theory of ontological relativity," which can be investigated by means of _vipasyana_, or insight, meditation. Wallace then focuses on the theme of symmetry in reference to quantum cosmology and the "problem of frozen time," relating these issues to the theory and practices of the Great Perfection school of Tibetan Buddhism. He concludes with a discussion of the general theme of complementarity as it relates to science and religion. The theories of relativity and quantum mechanics were major achievements in the physical sciences, and the theory of evolution has had an equally deep impact on the life sciences. However, rigorous scientific methods do not yet exist to observe mental phenomena, and naturalism has its limits for shedding light on the workings of the mind. A pioneer of modern consciousness research, Wallace offers a practical and revolutionary method for exploring the mind that combines the keenest insights of contemporary physicists and philosophers with the time-honored meditative traditions of Buddhism. (shrink)