It is proved that the following conditions are equivalent: (a) there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore, the existence of such algebras is undecidable in ZFC. In "forcing language" condition (a) says that there exists a non-trivial σ-centered forcing not adding Cohen reals.

An analogue of Mathias forcing is studied in connection of free Boolean algebras and nowhere dense ultrafilters. Some applications to rigid Boolean algebras are given.

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...) the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. (shrink)

Galileo's paradox of infinity involves comparing the set of natural numbers, N, and the set of squares, {n2 : n ∈ N}. Galileo sets up a one-to-one correspondence between these sets; on this basis, the number of the elements of N is considered to be equal to the number of the elements of {n2 : n ∈ N}. It also characterizes the set of squares as smaller than the set of natural numbers, since ``there are many more numbers than squares". (...) As a result, it concludes that infinities cannot be compared in terms of greater--lesser and the law of trichotomy does not apply to them. Cantor's cardinal numbers provide a measure for sets. Cantor gives a definition of the relation greater–lesser between cardinal numbers and establishes the law of trichotomy for these numbers. Yet, when Cantor's theory is applied to subsets of N, it gives that any set can be either finite or of the power ℵ0. Thus, although the set of squares is the subset of N, they are of the same cardinality. Benci, Di Nasso introduces specific numbers to measure sets called numerosities. With numerosities, the following claim is true: numerosity of A < numerosity of B, whenever A ⊈ B. In this paper, we present a simplified version of the theory of numerosities that applies to subsets of N. This theory complies with Galileo's presupposition that when A ⊈ B, then the number of elements in A is smaller than the number of elements in B. Specifically, we show that as the numerosity of N is the number α, the numerosity of the set of squares is the integer part of the number √α, that is ⌊√α⌋, and the inequality ⌊√α⌋ < α holds. (shrink)

a This is an outline of a coherence theory of law. Its basic ideas are: reasonable support and weighing of reasons. All the rest is commentary.a (TM) These words at the beginning of the preface of this book perfectly indicate what On Law and Reason is about. It is a theory about the nature of the law which emphasises the role of reason in the law and which refuses to limit the role of reason to the application of deductive logic. (...) In 1989, when the first edition of On Law and Reason appeared, this book was ground breaking for several reasons. It provided a rationalistic theory of the law in the language of analytic philosophy and based on a thorough understanding of the results, including technical ones, of analytic philosophy. That was not an obvious combination at the time of the booka (TM)s first appearance and still is not. The result is an analytical rigor that is usually associated with positivist theories of the law, combined with a philosophical position that is not natural law in a strict sense, but which shares with it the emphasis on the role of reason in determining what the law is. If only for this rare combination, On Law and Reason still deserves careful study. On Law and Reason also foreshadowed and influenced a development in the field of Legal Logic that would take place in the nineties of the 20th century, namely the development of non-monotonic (a defeasiblea (TM)) logics for the analysis of legal reasoning. In the new Introduction to this second edition, this aspect is explored in some more detail. (shrink)

In this paper, we present a reinterpretation of Roman Ingarden's theory of intentional objects. There are four types of intentional objects in Ingarden's ontology, we offer a detailed analyses of an intentional object that is a correlate of a text. Such an object is characterised by Ingarden as a two-sided and schematised formation. We focus on the notion of schematism. We classify different interpretations of schematism and propose our own definition of schematism of a purely intentional object.

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

The phenomenon of weakness of will – not doing what we perceive as the best action – is not recognized by neoclassical economics due to the axiomatic assumptions of the revealed preference theory (RPT) that people do what is best for them. However, present bias shows that people have different preferences over time. As they cannot be compared by the utility measurements, economists need to normatively decide between selves (short- versus long-term preferences). A problem is that neoclassical economists perceive RPT (...) as value-free and incorporate present bias within the economic framework. The axiomatic assumption that people do what is best for them leads to theoretical and practical dilemmas. This work examines weakness of will to resolve some shortcomings of RPT. The concept of intention is used to provide multiple self conception with the framework to decide between selves, which had not been done before. The paper concludes that individuals should not always follow their revealed preferences (desires) but the intentions (reason) because the latter indicates what people really want. (shrink)

We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. We concentrate on unbounded continuous actions on metric spaces. For example, we consider the properties non‐OB, non‐FH and non‐FR.

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

The epistemicist theory aims to explain ignorance due to vagueness by semantic plasticity: the shiftiness of intensions across close possible worlds resulting from shiftiness in usage. This explanation is challenged by the Omniscient Speaker Puzzle (Sennet in Philos Stud 161(2):273–285, 2012). Suppose that an omniscient speaker, Barney, who knows all the facts about usage and how these facts determine the intensions of expressions, cooks up a scheme to stabilise the intension of a normally semantically plastic term like ‘rich’. It seems (...) that ‘rich’ would display all the phenomena associated with vagueness without being semantically plastic, thus making the epistemicist explanation of ignorance due to vagueness insufficient. In this paper, I present a few choice points for epistemicism that arise as a result of the puzzle. (shrink)

Impossible Minds: My Neurons, My Consciousness has been written to satisfy the curiosity each and every one of us has about our own consciousness. It takes the view that the neurons in our heads are the source of consciousness and attempts to explain how this happens. Although it talks of neural networks, it explains what they are and what they do in such a way that anyone may understand. While the topic is partly philosophical, the text makes no assumptions of (...) prior knowledge of philosophy; and so contains easy excursions into the important ideas of philosophy that may be missing in the education of a computer scientist. The approach is pragmatic throughout; there are many references to material on experiments that were done in our laboratories. The first edition of the book was written to introduce curious readers to the way that the consciousness we all enjoy might depend on the networks of neurons that make up the brain. In this second edition, it is recognized that these arguments still stand, but that they have been taken much further by an increasing number of researchers. A post-script has now been written for each chapter to inform the reader of these developments and provide an up-to-date bibliography. A new epilogue has been written to summarize the state-of-the art of the search for consciousness in neural automata, for researchers in computation, students of philosophy, and anyone who is fascinated by what is one of the most engaging scientific endeavours of the day. This book also tells a story. A story of a land where people think that they are automata without much in the way of consciousness, a story of cormorants and cliffs by the sea, a story of what it might be like to be a conscious machine... (shrink)

Organizational deviance represents a costly behavior to many organizations. While some precursors to deviance have been identified, we hope to add to our predictive capabilities. Utilizing social cognitive theory and psychological contract theory as explanatory concepts, we explore the role of moral disengagement and turnover intentions, testing our hypotheses using two samples: a sample of 44 nurses from a hospital system in the Southwestern United States (Study 1), and a sample of 52 working adults collected from an online survey system (...) (Study 2). Results strongly supported our hypotheses in both samples, indicating that the self-regulatory deactivation inherent in moral disengagement led to increased organizational deviance; effects that were much more pronounced when turnover intentions were high. Our findings support the increased role of cognition in determining behavior when environmental pressures stemming from the psychological contract have been altered, leading to a number of theoretical and practical implications, particularly in industries with high turnover rates. (shrink)

The goal of this article is to show that instrumental rationality and utility that have been used in economics for many years does not work well. What is presented in the article is how significant the influence of utilitarianism has been on economics and why the economists get rid of humans’ goals and motivations. It is shown in the article that the human who decides in present is absolutely different from the human who decides over time. Many economists neglected this (...) problem because they wanted to have an effective and simple model. Becker’s economic method is presented as a dead end to which economics has been brought to. It is impossible to connect different selves of one human being by using the utility measure. The works of Schelling and J.S. Mill are used to explain this impossibility. The conclusion of this article is that instrumental rationality and utility have affected economics significantly. But, this simplified view on human nature is no longer valid. Hence, economics needs to think not only about the means but also about the human goals. Economics needs to rebuff relativism and show people how to achieve well-being. If we want to help people with their self-governance, we will have to choose reason over emotions. (shrink)

Ifeel that Iam apartof, but separatefrom an 'out there' world. 2. Ifeel that my perception of the world mingles with feelings of past experience. 3. My experienceof the world is selective and purposeful. 4. I am thinking ahead allthe timeintrying ...

This paper explores Nicholas of Cusa’s framing of the De pace fidei as a dialogue taking place incaelo rationis. On the one hand, this framing allows Nicholas of Cusa to argue that all religious rites presuppose the truth of a single, unified faith and so temporally manifest divine logos in a way accommodated to the historically unique conventions of different political communities. On the other hand, at the end of the De pace fidei, the interlocutors in the heavenly dialogue are (...) enjoined to return to earth and lead their countrymen in a gradual conversion to the acceptance of rites which would explicitly acknowledge the metaphysically presupposed transcendent unity of all true faiths. In light of these two aspects of the literary framing of the De pace fidei, the question that motivates this paper concerns the extent to which the understanding of history subtending Cusanus’ temporal political aims is consistent with the understanding of history grounded in his metaphysical presupposition that there is una religio in omni diversitate rituum. In addressing this question, I shall argue that the literary strategy of the De pace fidei sacrifices Nicholas of Cusa’s apologetic doctrinal aims insofar as the text creates an allegorical space in which the tension between its literal and figurative dimensions assigns to its readers the task of choosing their own orientations to the significance of history as a foundation for future action. (shrink)

Artykuł stanowi krytyczne omówienie książki Kevina Duttona zatytułowanej Mądrość psychopatów. Lekcja życia pobrana od świętych, szpiegów i seryjnych morderców. Celem pracy jest przybliżenie zasadniczych tez stawianych przez autora oraz zaproszenie do namysłu nad psychopatią – zarówno jako fenomenem naukowym, jak i egzystencjalnym.

Lev Isakovic Sestov e annoverato fra gli eccezionali filosofi del XX secolo. Questo pensatore di origine giudea, e un continuatore della corrente esistenziale del pensiero religioso di Pascal e Kierkegaard. Viene, pero annoverato pure - del resto insieme ai filosofi menzionati - alla corrente del pensiero tragico. Questo non e contradditorio perche ambedue le trame sono presenti alla riflessione di Sestov, anzi esse sono Ii, cosi inseparabilmente intrecciate, che fino ad oggi la riflessione del pensatore russo essenzialmente sfugge da ogni (...) interpretazione univoca. Nel frammento seguente del teste sto dimostrando che per Sestov l'esperienza tragica non e soltanto un inizio deU'esperienza religiosa, ma anche la sua condizione indispensabile, come un'IIniziazione. Sestov cerca l'esemplificazione di questa coincidenza particolare - l'esperienza tragica come un'iniziazione religiosa - sugli esempi dei vari pensatori di cui ha scrutato non soltanto le opere ma anche le biografie, partendo dalla premessa che esse sono delle realta collegate inseparabilmente. Sestov cerca l'affermazione di questa verita studiando la vita e le opere dei pensatori non soltanto tragici come Shakespeare, Ibsen, Nietzsche o Dostoevskij, ma sopratutto dei pensatori religiosi come Pascal o Lutero e Kierkegaard. In questi ultimi quel collegamento del tragico e della reiigiosita si e fatto evidente, mentre negli altri esso era soltanto parziale e embrionale. (shrink)

Lev Isakovic Sestov e annoverato fra gli eccezionali filosofi del XX secolo. Questo pensatore di origine giudea, e un continuatore della corrente esistenziale del pensiero religioso di Pascal e Kierkegaard. Viene, pero annoverato pure - del resto insieme ai filosofi menzionati - alla corrente del pensiero tragico. Questo non e contradditorio perche ambedue le trame sono presenti alla riflessione di Sestov, anzi esse sono Ii, cosi inseparabilmente intrecciate, che fino ad oggi la riflessione del pensatore russo essenzialmente sfugge da ogni (...) interpretazione univoca. Nel frammento seguente del teste sto dimostrando che per Sestov l'esperienza tragica non e soltanto un inizio deU'esperienza religiosa, ma anche la sua condizione indispensabile, come un'IIniziazione. Sestov cerca l'esemplificazione di questa coincidenza particolare - l'esperienza tragica come un'iniziazione religiosa - sugli esempi dei vari pensatori di cui ha scrutato non soltanto le opere ma anche le biografie, partendo dalla premessa che esse sono delle realta collegate inseparabilmente. Sestov cerca l'affermazione di questa verita studiando la vita e le opere dei pensatori non soltanto tragici come Shakespeare, Ibsen, Nietzsche o Dostoevskij, ma sopratutto dei pensatori religiosi come Pascal o Lutero e Kierkegaard. In questi ultimi quel collegamento del tragico e della reiigiosita si e fatto evidente, mentre negli altri esso era soltanto parziale e embrionale. (shrink)

We assume—in contrast to many “legal realists”—that law is a part of reality. Law exists because people believe in law, but law is not identical with beliefs. Law supervenes on human beliefs, preferences, actions, dispositions and artefacts. Moreover, the morally binding personal interpretation of the law supervenes on two things together: on the individual's knowledge of legal institutions and on moral obligation. The first supervenes in its turn on mutual beliefs; the second supervenes on motivations and dispositions of the individual, (...) provided that she is morally sensitive and rational. Personal interpretation of law converts into social law, if other persons on the basis of overriding reasons do not contest it. Morally binding social law supervenes on moral motivation that is triggered by institutions that supervene on mutual beliefs. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

Epistemicism associates vagueness with ignorance produced by semantic plasticity: the shiftiness of intensions in our language resulting from small changes in usage. The recent literature (Caie 2012; Magidor 2018; Yli-Vakkuri 2016) points to a missing piece in the epistemicist theory of vagueness, namely a clear account of the semantics of the definiteness operator Δ. The fundamentals of the epistemicist theory are well understood. However, the technical work of defining the definiteness operator has proven difficult. There are several desiderata that we (...) would like Δ to satisfy. For instance, we would like the epistemicist notion of ‘definiteness’ to interact well with modal operators and validate intuitive principles like ‘necessarily, if φ is definitely true, then φ is true’. Providing an account that would meet all such desiderata has eluded the epistemicists so far. In this paper, I present a novel version of a multidimensional model inspired by the work of Robert Stalnaker and David Kaplan. Using this model, I provide an account of epistemicist definiteness that meets our desiderata. (shrink)

The aim of this paper is to show the main thesis concerning the theory of cognition of the eminent neo-Kantian Heinrich Rickert, as presented in his work “Der Gegenstand der Erkenntnis”. On the one hand, Rickert finds out that thinking is fated to “clash with nothingness”, thus creating a temptation to reject all rigours and to yield to complete discretion. On the other hand, he attributes axiological status to nothingness which subjects thinking to a particular kind of “ought”. In his (...) view, the cognizing subject is faced with an axiological choice: either discretion or truth and argues that it is worth opting for truth. His argumentation could be an interesting point of reference for contemporary culture gradually moving away from the type of thinking rooted in objectively existing principles. (shrink)

The books’ goal is to answer the question: Do the weaknesses of value-free economics imply the need for a paradigm shift? The author synthesizes criticisms from different perspectives (descriptive and methodological). Special attention is paid to choices over time, because in this area value-free economics has the most problems. In that context, the enriched concept of multiple self is proposed and investigated. However, it is not enough to present the criticisms towards value-free economics. For scientists, a bad paradigm is better (...) than no paradigm. Therefore, the author considers whether value-based economics with normative approaches such as economics of happiness, capability approach, libertarian paternalism, and the concept of multiple self can be the alternative paradigm for value-free economics. This book is essential reading to everyone interested in the current state of economics as a discipline. (shrink)

The main stream of legal theory tends to incorporate unwritten principles into the law. Weighing of principles plays a great role in legal argumentation, inter alia in statutory interpretation. A weighing and balancing of principles and other prima facie reasons is a jump. The inference is not conclusive.To deal with defeasibility and weighing, a jurist needs both the belief-revision logic and the nonmonotonic logic. The systems of nonmonotonic logic included in the present volume provide logical tools enabling one to speak (...) precisely about various kinds of rules about rules, dealing with such things as applicability of rules, what is assumed by rules, priority between rules and the burden of proof. Nonmonotonic logic is an example of an extension of the domain of logic. But the more far-reaching the extension is, the greater problems it meets. It seems impossible to make logical reconstruction of the totality of legal argumentation. (shrink)

This paper relates to a formal statement of the mechanisms that are thought minimally necessary to underpin consciousness. This is expressed in the form of axioms. We deem this to be useful if there is ever to be clarity in answering questions about whether this or the other organism is or is not conscious. As usual, axioms are ways of making formal statements of intuitive beliefs and looking, again formally, at the consequences of such beliefs. The use of this style (...) of exposition does not entail a claim to provide a mathematically rigorous formal deductive system. Conventional mathematical notation is used to achieve clarity, although this is elaborated with natural language in an attempt to reduce terseness. In our view, making the approach axiomatic is synonymous with building clear usable tests for consciousness and is therefore a central feature of the paper. The extended scope of this approach is to lay down some essential properties that should be considered when designing machines that could be said to be conscious. In the broader discussion about the nature of consciousness and its neurological mechanisms, it may seem to some that axiomatisation is premature and continues to beg many questions. However, the approach is meant to be open- ended so that others can build further axiomatic clarifications that address the very large number of questions which, in the search for a formal basis for consciousness, still remain to be answered. Of course, in discussions about consciousness many will also argue that the subject is not one that may ever be formally addressed by means of axioms. The view taken in this paper is 'let's try to do it and see how far it gets'. (shrink)

All through the XXth century it has been repeated that "there is an exact correspondence, almost coincidence between Euclid's definition of equal ratios and the modern theory of irrational numbers due to Dedekind". Since the idea was presented as early as in 1908 in Thomas Heath's translation of Euclid's Elements as a comment to Book V, def. 5, we call it in the paper Heath's thesis. Heath's thesis finds different justifications so it is accepted yet in different versions. In the (...) paper its historical and mathematical version is reconstructed. We next reconstruct Eudoxos' theory of proportions in an axiomatic fashion. Finally, we show that Heath's thesis both in the historical and mathematical version is false. To this end a counterexample is given; it is based upon a specific interpretation of the uniform distribution theorem. (shrink)

In this paper we show that the field of the real numbers is an intentional object in the sense specified by Roman Ingarden in his Das literarische Kunstwer and Der Streit um die Existenz der Welt. An ontological characteristics of a classic example of an intentional object, i.e. a literary character, is developed. There are three principal elements of such an object: the author, the text and the entity in which the literary character forms the content. In the case of (...) the real numbers the triad consists of Richard Dedekind, his work Steitigkeit und irratinale Zahlen, and the intentional object determined by this work. Showing that we are indeed faced with the inten-tional object we analyse three moments: two-sidedness of the formal structure of the intentional object, the moment of its existentional derivation, and schematism of the intentional object. Since there are many constructions of the real numbers known in mathematics, we show what relation obtains between Dedekind's and Cantor's constructions of the real numbers as these are taken as intentional objects. Moreover, we show what relation obtains between Dedekind's construction and the axiomatic characteristics of the real numbers. (shrink)

The author analyses three aspects of the correspondence between theories: physical, methodological and philosophical.