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.

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.

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)

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)

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)

We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms.

For odd and for even involutive, commutative residuated chains a representation theorem is presented in this paper by means of direct systems of abelian o-groups equipped with further structure. This generalizes the corresponding result of J. M. Dunnabout finite Sugihara monoids.

We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms.

Let be the class of odd involutive even the notion of partial lex products is not sufficiently general. One more tweak is needed, a slightly even more complex construction, called partial sublex product, introduced here.

A new structure, called equality algebras, will be introduced. It has two connectives, a meet operation and an equivalence, and a constant. A closure operator will be defined in the class of equality algebras, and we call the closed algebras equivalential. We show that equivalential equality algebras are term equivalent with BCK-algebras with meet. As a by-product, we obtain a quite general result, which is analogous to a result of Kabziński and Wroński: we provide an equational characterization for the equivalential (...) fragment of BCK-algebras with meet. (shrink)

A new structure, called pseudo equality algebras, will be introduced. It has a constant and three connectives: a meet operation and two equivalences. A closure operator will be introduced in the class of pseudo equality algebras; we call the closed algebras equivalential. We show that equivalential pseudo equality algebras are term equivalent with pseudo BCK-meet-semilattices. As a by-product we obtain a general result, which is analogous to a result of Kabziński and Wroński: we provide an equational characterization for the equivalence (...) operations of pseudo BCK-meet-semilattices. Our result treats a much more general algebraic structure, namely, pseudo BCK-meet-semilattice instead of Heyting algebras, on the other hand, we also need to use the meet operation. Finally, we prove that the variety of pseudo equality algebras is a subtractive, 1-regular, arithmetical variety. (shrink)

It is shown that, under certain conditions, a subset of the graph of a commutative residuated chain is invariant under a geometric reflection. This result implies that a certain part of the graph of the monoidal operation of a commutative residuated chain determines another part of the graph via the reflection on one hand, and tells us about the structure of continuity points of the monoidal operation on the other. Finally, these results are applied for the subdomains of uniqueness problem, (...) yielding new results. (shrink)

The paper deals with involutive FLe-monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FLe-monoids over lattices are exactly involutive FLe-algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FLe-monoids, along with a new construction method, called twin-rotation. Some classes of finite involutive FLe-chains are classified by using the notion of rank of involutive FLe-chains, and a kind of duality is developed between positive and non-positive rank algebras. As a (...) side effect, it is shown that the substructural logic IUL plus t ↔ f does not have the finite model property. (shrink)

An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL$$_e$$ (...) e -chains which have finitely many positive idempotent elements. (shrink)

Classification of certain group-like FL $_e$ -chains is given: We define absorbent-continuity of FL $_e$ -algebras, along with the notion of subreal chains, and classify absorbent-continuous, group-like FL $_e$ -algebras over subreal chains: The algebra is determined by its negative cone, and the negative cone can only be chosen from a certain subclass of BL-chains, namely, one with components which are either cancellative (that is, those components are negative cones of totally ordered Abelian groups) or two-element MV-algebras, and with no (...) two consecutive cancellative components. It is shown that the classification theorem does not hold if we drop the absorbent-continuity condition. Our result is the first classification theorem in the literature on FL $_e$ -algebras that does not assume the condition of being naturally ordered (which, under certain conditions, corresponds to continuity of the monoid operation). In our classification theorem, continuity is replaced by the much weaker absorbent-continuity. (shrink)

A Correction to this paper has been published: 10.1007/s11225-020-09933-y.

ABSTRACT A new algebraic construction -called rotation- is introduced in this paper which from any left-continuous triangular norm which has no zero divisors produces a left-continuous but not continuous triangular norm with strong induced negation. An infinite number of new families of such triangular norms can be constructed in this way which provides a huge spectrum of choice for e.g. logical and set theoretical connectives in non-classical logic and in fuzzy theory. On the other hand, the introduced construction brings us (...) closer to the understanding the structure of these connectives and the corresponding logics. From the application point of view, results of this paper can be especially useful in the field of non-classical logic, fuzzy sets, and fuzzy preference modeling. (shrink)

This paper is the continuation of [11] where the rotation construction of left-continuous triangular norms was presented. Here the class of triangular subnorms and a second construction, called rotation-annihilation, are introduced: Let T1 be a left-continuous triangular norm. If T1 has no zero divisors then let T2 be a left-continuous rotation invariant t-subnorm. If T1 has zero divisors then let T2 be a left-continuous rotation invariant triangular norm. From each such pair the rotation-annihilation construction produces a left-continuous triangular norm with (...) strong induced negation. An infinite number of new families of such triangular norms can be constructed in this way, and this further extends our spectrum of choice for the proper triangular norm e.g. in probabilistic metric spaces, or for logical and set theoretical connectives in non-classical logic, or e.g. in fuzzy sets theory and its applications. On the other hand, the introduced construction brings us closer to the understanding of the structure of these operations. (shrink)

This volume gathers together for the first time are all the key texts in a crucial debate in modern philosophy, centered on Leibniz's famous 1695 essay, the "New System of the Nature of Substances and their Communication," in which he introduced his strikingly original theory of metaphysics. His "system" became increasingly famous and drew him into discussion and development of these ideas, both in public and in private, with a variety of thinkers, most notably the great French philosopher Pierre Bayle. (...) Woolhouse's and Francks's new English edition gives the only full representation of this debate, and will therefore be essential reading for anyone who wishes to gain a proper understanding of Leibniz's philosophy and its intelletual context. All the texts are newly translated and extensively annotated; many appear in English for the first time. (shrink)

Plato's dialogues frequently criticize traditional Greek myth, yet Plato also integrates myth with his writing. Daniel S. Werner confronts this paradox through an in-depth analysis of the Phaedrus, Plato's most mythical dialogue. Werner argues that the myths of the Phaedrus serve several complex functions: they bring nonphilosophers into the philosophical life; they offer a starting point for philosophical inquiry; they unify the dialogue as a literary and dramatic whole; they draw attention to the limits of language and the limits of (...) knowledge; and they allow Plato to co-opt cultural authority as a way of defining and legitimating the practice of philosophy. Platonic myth, as a species of traditional tale, is thus both distinct from philosophical dialectic and similar to it. Ultimately, the most powerful effect of Platonic myth is the way in which it leads readers to participate in Plato's dialogues and to engage in a process of self-examination. (shrink)

t. 1. Apo ton prōto skeptomeno anthrōpo mechri ton 6. aiōna M.Ch. -- t. 2. Apo ton 7. aiōna M.Ch. mechri to sēmera.

This article is the guest editors’ introduction to the special issue in Business & Society on “SMEs and CSR in Developing Countries.” The special issue includes four original research articles by Hamann, Smith, Tashman, and Marshall; Allet; Egels-Zandén; and Puppim de Oliveira and Jabbour on various aspects of the relationship of small and medium enterprises to corporate social responsibility in developing countries.

Since the launch of applications such as DALL-E, Midjourney, and Stable Diffusion, generative artificial intelligence has been controversial as a tool for creating artwork. While some have presented longtermist worries about these technologies as harbingers of fully automated futures to come, more pressing is the impact of generative AI on creative labour in the present. Already, business leaders have begun replacing human artistic labour with AI-generated images. In response, the artistic community has launched a protest movement, which argues that AI (...) image generation is a kind of theft. This paper analyzes, substantiates, and critiques these arguments, concluding that AI image generators involve an unethical kind of labour theft. If correct, many other AI applications also rely upon theft. (shrink)

We propose a formal framework for interpretable machine learning. Combining elements from statistical learning, causal interventionism, and decision theory, we design an idealised explanation game in which players collaborate to find the best explanation for a given algorithmic prediction. Through an iterative procedure of questions and answers, the players establish a three-dimensional Pareto frontier that describes the optimal trade-offs between explanatory accuracy, simplicity, and relevance. Multiple rounds are played at different levels of abstraction, allowing the players to explore overlapping causal (...) patterns of variable granularity and scope. We characterise the conditions under which such a game is almost surely guaranteed to converge on a optimal explanation surface in polynomial time, and highlight obstacles that will tend to prevent the players from advancing beyond certain explanatory thresholds. The game serves a descriptive and a normative function, establishing a conceptual space in which to analyse and compare existing proposals, as well as design new and improved solutions. (shrink)

In Seeing Things as They Are (Searle 2015), Searle developed a direct realist’s theory of perception. According to direct realism, physical objects are directly and immediately perceived. Searle claims that Berkeley’s theory of perception goes against direct realism. For Searle, Berkeley’s theory suggests that only subjective experiences (ideas) are directly and immediately perceived, not physical objects. Contrary to Searle, G. S. Pappas claims that Berkeley’s theory of perception is consistent with the view that physical objects are immediately perceivable (Pappas 1982; (...) Pappas 1987). Given the conflicting interpretations of Berkeley’s theory of perception, this paper attempts to defend Searle’s position on Berkeley’s theory of perception. It refutes Pappas’ interpretation by arguing that he is inconsistent with Berkeley’s theses of ontological phenomenalism and heterogeneity thesis. (shrink)

Capital punishment—the legally authorized killing of a criminal offender by an agent of the state for the commission of a crime—stands in special need of moral justification. This is because execution is a particularly severe punishment. Execution is different in kind from monetary and custodial penalties in an obvious way: execution causes the death of an offender. While fines and incarceration set back some of one’s interests, death eliminates the possibility of setting and pursuing ends. While fines and incarceration narrow (...) one’s routes to happiness, death eliminates its possibility. Not surprisingly, there is much debate about the moral permissibility of capital punishment. This entry maps the terrain of this debate. The first section discusses justifications of the death penalty as they appear in major theories of punishment. The second section surveys moral objections to execution that apply to most justifications. The third addresses procedural criticisms, which do not target the morality of execution so much as the justice of its implementation. (shrink)

Many people are worried about the harmful effects of climate change but nevertheless enjoy some activities that contribute to the emission of greenhouse gas (driving, flying, eating meat, etc.), the main cause of climate change. How should such people make choices between engaging in and refraining from enjoyable greenhouse-gas-emitting activities? In this chapter, we look at the answer provided by decision theory. Some scholars think that the right answer is given by interactive decision theory, or game theory; and moreover think (...) that since private climate decisions are instances of the prisoner’s dilemma, one rationally should engage in these activities provided that one enjoys them. Others think that the right answer is given by expected utility theory, the best-known version of individual decision theory under risk and uncertainty. In this chapter, we review these different answers, with a special focus on the latter answer and the debate it has generated. (shrink)

Critics of the computational connectionism of the last decade suggest that it shares undesirable features with earlier empiricist or associationist approaches, and with behaviourist theories of learning. To assess the accuracy of this charge the works of earlier writers are examined for the presence of such features, and brief accounts of those found are given for Herbert Spencer, William James and the learning theorists Thorndike, Pavlov and Hull. The idea that cognition depends on associative connections among large networks of neurons (...) is indeed one with precedents, although the implications of this for psychological issues have been interpreted variously — not all versions of connectionism are alike. (shrink)

The practical goal of preventing premature death seems uncontroversial. But the term ‘premature death’ is vague with several, sometimes conflicting definitions. This ambiguity results in several conceptions with which not all will agree. Moreover, the normative rationale behind the goal of preventing premature deaths is masked by the operational definition of existing measures. In this article, we argue that ‘premature death’ should be recognized as a normative concept. We propose that normative theories should be used to justify measures of premature (...) death to provide them with normative validity and public legitimacy. (shrink)