In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.

Left-continuous t-norms are much more complicated than the continuous ones, and obtaining a complete classification of them seems to be a very hard task. In this paper we investigate some aspects of left-continuous t-norms, with emphasis on their continuity points. In particular, we are interested in left-continuous t-norms which are isomorphic to t-norms which are continuous in the rationals. We characterize such a class, and we prove that it contains the class of all weakly cancellative left-continuous t-norms.

We introduce the notion of a dice model as a framework for describing a class of probabilistic relations. We investigate the transitivity of the probabilistic relation generated by a dice model and prove that it is a special type of cycle-transitivity that is situated between moderate stochastic transitivity or product-transitivity on the one side, and Lukasiewicz-transitivity on the other side. Finally, it is shown that any probabilistic relation with rational elements on a three-dimensional space of alternatives which possesses this particular (...) type of cycle-transitivity, can be represented by a dice model. The same does not hold in higher dimensions. (shrink)

Hahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains (...) which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented. (shrink)

The authors of the article are trying to identify the signs of a crisis of our time as a consequence of the refusal from the philosophical justification of the goals and objectives of scientific pedagogy, which as a chain reaction entail ignoring the results of sciences related to pedagogy: neurobiology and psychology. The reproductive approach in education selectively takes into account the results of the empirical sciences, as they serve to justify it. The result is a shift in emphasis: being (...) driven by the hypothetical imperatives of competitiveness in the labor market, teachers and other adults feel the need for the achievements of the pupils, for their effectiveness. As a consequence, the child's psyche does not develop properly which manifests itself in the inability to see the subjects of interaction in the vis-a-vis, in an effort to objectify what is not subject to objectification - a person. The authors of the article see the solution to the problem in the appeal in the philosophical and pedagogical system of I. Kant, that may be reconstructed from his works. The authors of the article conclude that I. Kant may rightfully be considered the founder of a productive approach to education. (shrink)

This short novel by the writer S´ndor M´rai, regarded by many as the finest Hungarian writer of our time, was published in 1942 but became known in the West only many years later and was finally translated into English in 2002. Its central subject is the importance and value of friendship between men and love between men and women, involving the three central characters, Henrik, Konrad and Krisztina. The men, who had been friends since their early youth, meet again 41 (...) years later. Although Krisztina, Henrik's wife and Konrad's lover, is long dead, she is fully present in the memory of both. Like Proust's A La Recherche du Temps Perdu, Embers is remembrance of things past, but with a difference. M´rai's novel plays on the contrast between their past friendship and their present broken friendship, and the contrast between Henrik's love for his wife, and her love for Konrad. But whereas Proust's novel is a recapture of the past, M´rai's novel is a recapture of the past through the present. What is left are only embers of friendship and love. (shrink)

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic (...) interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational (...) completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational (...) completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

Papers delivered at the joint meeting of the Hegel Society of America and the Hegel Society of Great Britain held at Merton College, Oxford, Sept. 1-4, 1981, to mark the 150th anniversary of Hegel's death. Includes bibliographical references and index.

Hegel’s Doctrine of the Concept advances a theory of conceptual determinacy. As I will demonstrate, Hegel’s theory of conceptual determinacy leads him to endorse self-predication and existential implication as features endemic to conceptual content. I first demonstrate some features of this logic, and some of its entailments. Following the reconstruction of Hegel’s logic of self-predication, I apply this logic to illuminate Hegel’s critique of formal logic. Finally, the self-predicative feature of Hegel’s logics offers a measure by which to determine the (...) potential and the limitations of any formal account of Hegel’s logic. (shrink)

On the life and works of Śaṅkarācārya, religious reformer and exponent of the Advaita Vedanta school in Hindu philosophy; includes an account of the institutions founded by him.

Saiva commentary, called Śrīkaṇṭhabhāṣyā, with text of the Brahmasūtra of Bādarāyaṇa, aphoristic work on the Vedanta; with super commentaries.

Commentary, with supercommentary, on the first four aphorisms of Brahmasūtra by Bādarāyaṇa, explaining Śaktiviśiṣṭādvaita approach to Vedanta philosophy of the Hindus; includes text of Brahmasūtra.