Diogenes Laertius ascribes the first concept of spherical Earth to both Pythagoras and Parmenides. Indeed, a major shift in cosmologies—emergence of the spherical conception of the Earth and the surrounding heaven—took place between the sixth and fifth centuries BCE. Given the poor state of preservation of early Pythagorean tradition, it is argued that primacy in formulating the notion of spherical Earth should be ascribed to Parmenides.

The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a (...) new way of thinking---thinking in terms of degrees rather than absolutes. For example, it leads to the definition of a new kind of sets, referred to as fuzzy sets, in which membership is a matter of degree. The book examines the genesis and development of fuzzy logic. It surveys the prehistory of fuzzy logic and inspects circumstances that eventually lead to the emergence of fuzzy logic. The book explores in detail the development of propositional, predicate, and other calculi that admit degrees of truth, which are known as fuzzy logic in the narrow sense. Fuzzy logic in the broad sense, whose primary aim is to utilize degrees of truth for emulating common-sense human reasoning in natural language, is scrutinized as well. The book also examines principles for developing mathematics based on fuzzy logic and provides overviews of areas in which this has been done most effectively. It also presents a detailed survey of established and prospective applications of fuzzy logic in various areas of human affairs, and provides an assessment of the significance of fuzzy logic as a new paradigm. (shrink)

The theory of concept lattices is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions “object (...) x has attribute y”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better. (shrink)

Journal Name: Apeiron Issue: Ahead of print.

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established.

We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

This paper focuses on three theories of personal identity that incorporate the idea that personal identity is the result of a person’s adopting certain attitudes towards certain mental states and actions. I call these theories subjective theories of personal identity. I argue that it is not clear what the proponents of these theories mean by “personal identity”. On standard theories, such as animalism or psychological theories, the term “personal identity” refers to the numerical identity of persons and its analysis provides (...) the persistence conditions for persons. I argue that if the subjective theories purport to provide a criterion of numerical personal identity, they fail. A different interpretation may suggest that they purport to provide a non-numerical type of identity for the purpose of providing plausible analyses of certain identity-related practical concerns. I argue that the criteria the subjective theories provide fail to capture several of the identity-related concerns. As a result, this interpretation must be rejected as well. (shrink)

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by Galois connections is provided.

We present a logic for reasoning about attribute dependencies in data involving degrees such as a degree to which an object is red or a degree to which two objects are similar. The dependencies are of the form A ⇒ B and can be interpreted in two ways: first, in data tables with entries representing degrees to which objects have attributes ; second, in database tables where each domain is equipped with a similarity relation. We assume that the degrees form (...) a scale equipped with operations representing many-valued logical connectives. If 0 and 1 are the only degrees, the algebra of degrees becomes the two-element Boolean algebra and the two interpretations become well-known dependencies in Boolean data and functional dependencies of relational databases. In a setting with general scales, we obtain a new kind of dependencies with naturally arising degrees of validity, degrees of entailment, and related logical concepts. The deduction rules of the proposed logic are inspired by Armstrong rules and make it possible to infer dependencies to degrees—the degrees of provability. We provide a soundness and completeness theorem for such a setting asserting that degrees of entailment coincide with degrees of provability, prove the independence of deduction rules, and present further observations. (shrink)

V minulém čísle časopisu Pro-Fil uveřejnil Tomáš Sobek zajímavou stať (Sobek 2015) mapující různé odpovědi na otázku, co konstituuje lidský blahobyt neboli co je pro člověka dobré. Zatímco většina kapitol má spíše deskriptivní povahu – autor představuje jednotlivé konkurenční teorie blahobytu a zdržuje se jejich hodnocení – v poslední části předkládá tezi, podle níž existuje úzký vztah mezi lidským blahobytem a autenticitou. Mým cílem v tomto krátkém příspěvku je tuto domněnku podrobněji interpretovat a poněkud zproblematizovat. Nechci tím naznačovat, že spojení (...) blahobytu a autenticity není plodné. Spíš mne zajímá, jaký je přesně vztah mezi těmito pojmy. Sobek totiž na obecné rovině říká jen tolik, že se jednotlivé teorie blahobytu musí potýkat se stejnou otázkou, totiž s otázkou autenticity, že teorie blahobytu ztělesňují různé koncepce autenticity a že diskuze mezi teoriemi blahobytu jsou zároveň spory o to, co je to autentický život. (shrink)

The article deals with a recent attack by Sam Harris on two famous arguments that purport to establish a gap between factual and evaluative statements—Hume’s Is-Ought Problem and Moore’s Open Question Argument. I present the arguments, analyze the relationship between them and critically assess Harris’ attempt to refute them. I conclude that Harris’ attempt fails.

Studie předkládá novou interpretaci zpráv o oběhu nebeských těles podle Anaximena z Mílétu. Nebeská tělesa se v jeho pojetí nemají pohybovat pod Zemí, ale naopak kolem Země. Anaximenés měl danou koncepci přiblížit motivem plstěného klobouku, který se otáčí kolem naší hlavy. Případné vysvětlení oběhů nebeských těles pomocí nakloněné Země podél její horizontální osy však patrně mylně vychází až z koncepce kulové Země a sklonu ekliptiky. Je možné, že dochované texty ve skutečnosti navazují na problém sklonu nebeské osy a popisují původní (...) situaci oběhů nebeských těles předtím, než se „nebe naklonilo“. (shrink)

This study focuses on the origin of life as presented in the thought of Anaximander of Miletus but also points to some parallel motifs found in much later conceptions of both the pre-Darwinian German romantic science and post-Darwinian biology. According to Anaximander, life originated in the moisture associated with earth (mud). This moist environment hosted the first living creatures that later populated the dry land. In these descriptions, one can trace the earliest hints of the notion of environmental adaptation. The (...) origin of humans was seen as connected in some way with fish: ancient humans were supposed to have developed inside fish-like animals. Anaximander took into account changes in the development of living creatures (adaptations) and speculated on the origins of humans. Similar ideas are found also in the writings of much later, eighteenth and nineteenth century authors who were close to the tradition of German romantic science. We do not argue that these later concepts are in any way directly linked with those of the pre-Socratics, but they show surprising parallels in, e.g., the hypothesis that life originated in a moist environment or the supposition that human developed from fish-like ancestors. These transformations are seen as a consequence of timeless logic rather than as evolution in historical terms. Despite the accent on the origin of living things, both Anaximander and the later Naturphilosophen lack in their notions the element most characteristic of Darwin’s thought, that is, the emphasis on historicity and uniqueness of all that comes into being. (shrink)

An algebra with fuzzy equality is a set with operations on it that is equipped with similarity ≈, i.e. a fuzzy equivalence relation, such that each operation f is compatible with ≈. Described verbally, compatibility says that each f yields similar results if applied to pairwise similar arguments. On the one hand, algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic. On the other hand, they are the formal counterpart to the intuitive idea of having functions (...) that are not allowed to map similar objects to dissimilar ones. In this paper, we present a generalization of the well-known Birkhoff’s variety theorem: a class of algebras with fuzzy equality is the class of all models of a fuzzy set of identities iff it is closed under suitably defined morphisms, substructures, and direct products. (shrink)

The paper studies closure properties of classes of fuzzy structures defined by fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and Horn theories by means of closedness under suitable algebraic constructions.

The paper presents generalizations of results on so-called Horn logic, well-known in universal algebra, to the setting of fuzzy logic. The theories we consider consist of formulas which are implications between identities (equations) with premises weighted by truth degrees. We adopt Pavelka style: theories are fuzzy sets of formulas and we consider degrees of provability of formulas from theories. Our basic structure of truth degrees is a complete residuated lattice. We derive a Pavelka-style completeness theorem (degree of provability equals degree (...) of truth) from which we get some particular cases by imposing restrictions on the formulas under consideration. As a particular case, we obtain completeness of fuzzy equational logic. (shrink)

Although the theory of memetics appeared highly promising at the beginning, it is no longer considered a scientific theory among contemporary evolutionary scholars. This study aims to compare the genealogy of memetics with the historically more successful gene-culture coevolution theory. This comparison is made in order to determine the constraints that emerged during the internal development of the memetics theory that could bias memeticists to work on the ontology of meme units as opposed to hypotheses testing, which was adopted by (...) the gene-culture scholars. I trace this problem back to the diachronic development of memetics to its origin in the gene-centered anti-group-selectionist argument of George C. Williams and Richard Dawkins. The strict adoption of this argument predisposed memeticists with the a priori idea that there is no evolution without discrete units of selection, which in turn, made them dependent on the principal separation of biological and memetic fitness. This separation thus prevented memeticists from accepting an adaptationist view of culture which, on the contrary, allowed gene-culture theorists to attract more scientists to test the hypotheses, creating the historical success of the gene-culture coevolution theory. (shrink)

[The neo-Meinongian conception of existence].

The goal of this paper is to assess three arguments that have been proposed to rebut the idea that the notion of indeterminate identity is incoherent. In the first part, the author presents Gareth Evans’ argument purporting to show the incoherence of indeterminate identity. Next, the author assesses a rebuttal proposed by E. J. Lowe. Although the rebuttal seems sound, Harold Noonan has shown that its scope is limited. After that, a rebuttal by Peter van Inwagen is analysed. The author (...) compares it with Lowe’s and shows that consistent application of the principles van Inwagen uses leads to objects having inconsistent properties. In the final part, it is shown that although the answer proposed by Terence Parsons seems superior to both van Inwagen’s and Lowe’s, its scope is also limited. As a result, Evans’ argument seems to stand unrefuted by these three counterarguments. (shrink)

This essay presents a comparison of modal theories with flexible and constant domain of individuals. Whatś called the simplest quantified modal logic represents the latter type. However, it has some controversial consequences: the fact that all objects of discourse exist necessarily, and the fact that to every merely possible object there corresponds an actual one. These drawbacks are eliminated by the Kripke models, which postulate possible worlds with a flexible domain of objects. Proponents of TIL return to the theory with (...) the constant domain and attempt to elucidate what mistakes lead Kripkeś followers to endorsement of models with the flexible domain, and explain-away the alleged problems of the constant domain. This work aims at clarifying the character of the attitude of TIL proponents and showing that the arguments for accepting the constant domain they provide are not completely satisfactory due to the role proper names play in their theory. (shrink)

An integral part of Odo Marquard’s texts is both humour and reflections of humour, which he connects with human reason. That is why the theme of this contribution to commemorate Odo Marquard is the potential connection between humour, laughter, and reason. Marquard’s reflections on the connections between humour and reason are preceded in the text by an interpretation of evolutionary biological research on the emergence and role of laughter, which is considered to be important in understanding the development of man, (...) and, as a result, man’s reason. From biological perspective, laughter is considered to be a form of social behaviour. Due to social aspects of laughter, one can trace a clear line between humour, which causes laughter, and reason, which is capable of producing humour and laughter. I continue with Marquard’s thoughts on potential connections between reason and humour in the framework of his philosophical anthropology. In the final part of my micro-etymological contribution I outline why humour is one the philosophically supreme activities of the human brain and show where Marquard’s ideas can supplement the evolutionary biological research of laughter. (shrink)

Článek nabízí kritickou reflexi způsobů, jimiž se filozofie snaží zajistit a zprostředkovat vědění o sobě samé, svých dějinách a klíčových problémech. Podrobně se věnuje procesům redukce a explikace, které jsou nedílnou součástí tohoto úsilí. S pomocí mnoha konkrétních příkladů ukazuje jejich úskalí a vzájemnou provázanost.

The article is divided into five parts that take readers through a historical and sociological analysis of the birth of European nationalism and concludes by emphasizing the need to overcome nationalism. In the first three parts, the author provides readers with detailed arguments on the historical background of nationalism. These show that the ideas of nationalism provided modern society with an important type of social bond. However, the article also focuses on why this type of social bond became the source (...) of serious trouble in central Europe. It has produced many disastrous events, such as wars, the holocaust, and ethnic-cleansing. Moreover, it has brought about a style of thinking that is too rigid and out of date in our globalizing world. The fourth part shows that nationalism operates through nationalist identity-myths. In the final part of the article, the author demonstrates how this new type of nationalism type has influenced the philosophy of education, teaching and national curricula and suggests a few maxims that should govern the transition from nationalist to trans-nationalist perspectives (not only) in education. (shrink)

In this work - both psychologists working on concepts and mathematicians working on fuzzy logic - reassess the usefulness of fuzzy logic for the psychology of concepts.

This essay examines theoretical arguments surrounding the use of post-colonial theory as a way to fill in the epistemological lacuna in the studies of post-socialism. It reviews the various streams of this theoretical development and employs Edward Said’s notion of “traveling theory” to demonstrate that theoretical claims made by proponents and opponents of this particular comparative perspective are historically, socially, and geographically situated, although not fixed. Disciplinary, national, and institutional affiliations, instead of theoretical justifications, are identified as important factors in (...) the propensity to accept or resist the introduction of post-colonial perspective on Central and Eastern Europe and the former Soviet Union. The essay concludes by acknowledging the potential usefulness as well as the limits of post-colonialism in the conceptualization of the post- socialist space. (shrink)

Aristotle ascribes to Thales of Miletus the notion of the Earth resting on water. However, in archaic Ionian cosmologies the Earth was not viewed as a cosmic body resting on physical support. The very motif is likely to have originated in the Near East. If Aristotle claims that this was the oldest conception of stability of the Earth it is possible that Thales may have been associated with it anachronistically.

The goal of the essayistic speculation (intended also as a brief laudatio) is to illustrate on an example the way that Sloterdijk could transform literary impulses into themes that he subsequently processes philosophically and develops in his works. I will present several of such potential transformations by means of parts and themes from Bonaventura’s Nachtwachen (1804, especially vigils 9 and 14), which Sloterdijk commented on in his Vienna lectures in 1998. The aim is not to seek parallels between Bonaventura’s and (...) Sloterdijk’s works, but to illustrate the possibilities of a philosophical transformation, the exploitation of literary themes by philosophical imagination and subsequently a detailed elaboration of philosophical concepts. (shrink)