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 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)

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)

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)

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)

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)

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)

In his 1887's Mitteilungen zur Lehre von Transfiniten, Cantor seeks to prove inconsistency of infinitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

This comment is analysing the last section of a paper by Piotr Blaszczyk, Mikhail G. Katz, and David Sherry on alleged misconceptions committed by historians of mathematics regarding the history of analysis, published in this journal in the first issue of 2013. Since this section abounds of wrong attributions and denouncing statements regarding my research and a key publication, the comment serves to rectify them and to recall some minimal methodological requirements for historical research.

The paper deals with reconstruction of the unique reductivecounterpart of the deductive logic. The procedure results in the deductivereductive form of logic. This extension is illustrated on the base of intuitionistic logics: Heyting’s, Brouwerian and Heyting-Brouwer’s ones.

ABSTRACT The following text is a large fragment of the lectures on heresy that Leszek Kołakowski (1927–2009) gave between November 1982 and February 1983 on the Polish radio station Radio Free Europe. These lectures have never been published in English. They were only published under the title ‘Herezja’ in Poland after the author’s death in 2010 by the publishing company Znak. Kołakowski raises the universal and timeless issues of tolerance, ideological struggles, protection of doctrine by religious institutions and the changing (...) attitude of the Catholic Church towards heretical challenges (tackling them and attempting to annex and colonize certain ideas). The concept of heresy is an opportunity to describe the history of ideological conflicts, show how they arise and how they can be resolved. The story of struggle between dogmatists and heretics is an endless story of changing social order and giving new shapes to the reality created by people and their ideas. Kołakowski points out that any doctrinal, political or ideological system must allow for criticism and otherness. Nearly 40 years after these lectures were given, they retain a special freshness in the context of the political, ideological and social crisis that Europe is experiencing in the early twenty-first century. (shrink)

The mainstream concept of death—the biological one—identifies death with the cessation of an organism. In this article, I challenge the mainstream position, showing that there is no single well-established concept of an organism and no universal concept of death in biological terms. Moreover, some of the biological views on death, if applied in the context of bedside decisions, might imply unacceptable consequences. I argue the moral concept of death—one similar to that of Robert Veatch—overcomes such difficulties. The moral view identifies (...) death with the irreversible cessation of a patient’s moral status, that is, a state when she can no longer be harmed or wronged. The death of a patient takes place when she is no longer capable of regaining her consciousness. In this regard, the proposal elaborated herein resembles that of Veatch yet differs from Veatch’s original project since it is universal. In essence, it is applicable in the case of other living beings such as animals and plants, provided that they have some moral status. (shrink)

In their insightful article, Brent Kious and Margaret Battin (2019) correctly identify an inconsistency between an involuntary psychiatric commitment for suicide prevention and physician aid in dying (PAD). They declare that it may be possible to resolve the problem by articulating “objective standards for evaluating the severity of others’ suffering,” but ultimately they admit that this task is beyond the scope of their article since the solution depends on “a deep and difficult” question about comparing the worseness of two possible (...) scenarios: letting someone die (who could have been helped) with not letting someone die (whose suffering could only be alleviated by death). In our commentary, we argue that creating such standards is more difficult than the authors assume because of the many types of deep uncertainties we have to deal with: (1) diagnostic, (2) motivational, and (3) existential. (shrink)

Singer claims that there are two ways of challenging the fact that brain-dead patients, from whom organs are usually retrieved, are in fact biologically alive. By means of the first, the so called dead donor rule may be abandoned, opening the way to lethal organ donation. In the second, it might be posited that terms such as “life” and “death” do not have any primary biological meaning and are applicable to persons instead of organisms. This second possibility permits one to (...) acknowledge that brain-dead patients are deceased because they are irreversibly unconscious. In the commentary which follows, I will argue that Singer’s second option is preferable since it provides a higher amount of organs available for transplant, and is better suited to the meaning of “death” which occurs in ordinary language. I will also defend such a concept of death against the objections raised by Michael Nair-Collins in the article Can the brain-dead be harmed or wronged? On the moral status of brain death and its implications for organ transplantation. (shrink)

It seems that nonmonotonicity of our reasoning is an obvious truth. Almost every logician not even believes, but simply knows very well that a human being thinks in a nonmonotonic way. Moreover, a nonmonotonicity of thinking seems to be a phenomenon parallel to the existence of human beings. Examples allegedly illustrating this phenomenon are not even analyzed today. They are simply quoted. Nowadays, this is a standard approach to nonmonotonicity. However, even simple analysis of those “obvious” examples shows that they (...) illustrate various problems of our thinking, among which none concerns nonmonotonicity. (shrink)

The paper is a continuation of A deductive-reductive form oflogic: general theory and intuitionistic case and considers the problem of definability of modal operators on the intuitionistic base. Contrary tothe classical case, it seems that the fact whether the connective is Heyting’sor Brouwerian is essential for the intuitionistic logic. The connective of possibility has the classical interpretation, i.e. w |= ✸α iff ∃t, if it is defined on the base of the logicwith Brouwerian connective of coimplication.

As far as existence of enduring subjects of change is concerned, Roman Ingarden's substance ontology stands in direct opposition to Whitehead's process metaphysics. However, with regard to domain of necessary relations, both systems are deeply platonic. Thus in this paper I pursue towards revealing some parallels and differences between both systems regarding views on ideal entities, necessary relations and pure possibilities. I examine whether having assumed contrary concepts of reality (substantial/procesual) both philosophers are forced to accept diverse conclusions in respect (...) to ideal entities domain. I also point out some crucial problems that are to be found in both systems: what is the ground of necessary relations – is it a nature of so called ideal qualities, or is it content of ideas (Ingarden); is it an individual or a relational essence of eternal objects, so called "primordial nature of God", or evolving actuality (Whitehead). Finally I investigate whether points of convergence between both concepts could serve as a basis for some modifications to overcome relevant problems. (shrink)

On the base of the classical logic the connectives of necessity and possibility have the equivalent positions in this sense that each of them is definable by the other one. The consequence of this fact is the possibility to define of the both modalities using the connective of identity. Thus, the connective of propositional identity defining the congruence of the propositional language has become the base of the reconstruction of necessity operator in some modal systems. Already in 1957 Greniewski [9] (...) obtained system S5 on the base of the propositional calculus with identity. Next, Cresswell [3], [4] reconstructed S4 and S5 in a similar way. Systematic investigations of this problem were undertaken by Suszko within the broader frame of SCI programme [16], [17], [18]. The connections between modal connectives and the connective of identity settle some philosophical sense of both modalities. Something is necessary, if it is identical with a logical truth. If something is different from the negation of some logical truth, it is possible. It is however, the “ontological” characteristics typical for modalities defined on the base of the classical logic. The settling of similar connections between intuitionistic identity and intuitionistic modalities uncover the philosophical sense of the “epistemic” necessity and possibility. Unfortunately, the symmetry between the classical and intuitionistic case is not entire. The acceptance of both equalities: = ¬ ¬ and = ¬ ¬, in the intuitionistic logic yields the classical versions of both modalities . Thus, there is a problem with defining the connective of intuitionistic possibility by the connective of identity. In Sections 2 and 3 two “partial” solutions of the mentioned problem are presented. They are “partial” because the connective of possibility has undesirable properties. The second solution yields the strengthening of the necessity connective, too. The main aim of this paper is obtained in Section 5, where a definition of the intuitionistic possibility is given. This definition uses the new connective of nonidentity, presented in Section 4. The new approach bases on the idea that some connectives like implication, negation, identity, necessity are “Heyting's” ones. The case of possibility is different. Together with coimplication, weak negation, and nonidentity, possibility belongs to the so-called “Brouwerian” connectives. The connectives of the first kind have such property that in Kripke's style semantics their interpretations in each point of this semantics are related to the interpretations in next points. In the case of the “Brouwerian” connectives the interpretations in each point depend on the interpretations in previous points. In this sense, “Heyting's” connectives are interpreted from the point of view of the future, while the interpretations of the “Brouwerian” connectives are from the point of view of the past. The relations between nonidentity and possibility connectives allow to say that in temporal semantics something is possible in the moment a, if before a there was such moment b in which it was different from the contradiction. Thus, “something is possible” does not mean that it held once in the past. A broader philosophical remarks about the nature of intuitionistic possibility are contained in two Comments in the last Section. (shrink)

In this text I take into account Roman Ingarden's theory of general objects. Polish phenomenologist avoided most of paradoxes implied by traditional theories of universals pointing out the difference between two kinds of form inherent in general object, namely form of subject of properties and form of whole and parts. Every idea has twofold construction: as an idea (qua idea) it is a subject of peculiar properties and on the other hand it has a content consisted of so called constants (...) and variables. The content of the idea has a whole-parts form. This however is inconsistent with other analyses of whole-parts form and of the character of constants and variables given by Roman Ingarden. In the consequence I claim that a third form, unique for general object, is needed. I also point out difficulty of indicating positive characteristic of general object (i.e. other than negation of properties of individuals). (shrink)

ABSTRACTThe author traces the impact of Abramowski's ideas on the recent history of Poland. His concepts were not only popular in the Polish Socialist Party and the syndicalist movement in the interwar period, but they also exerted a profound influence on the cooperative movement and democratic left-wing opposition in the 1970s and 1980s. The leaders of the Workers’ Defence Committee were much influenced by Abramowski's ideas and, according to some researchers, the Solidarity movement from 1980 to 1981 in Poland was (...) the culmination of his concepts. Today's anti-systemic movements in Poland are also inspired by Abramowski. The author also draws attention to certain similarities between Abramowski's ideas, Kropotkin's idea, Gramsci's concept of civil society and the thought of the young Marx. The author also outlines Abramowski's social ideas in the context of ideas promoted by the main theoreticians of the Polish left in the late nineteenth and early twentieth centuries. (shrink)

Volume 17, Issue 5, November 2020, Page 566-588.

ABSTRACT The article analyses the language used by the Polish nationalist right in relation to LGBT communities and the right to abortion. The authors show links between the language of Church hierarchs and right-wing columnists as the ideological backbone of the governing right-wing populist right. According to the authors, the attack on gender is the same method of political mobilisation and power management as the campaign against refugees and the anti-immigrant hysteria. On the one hand, the anti-gender discourse may strengthen (...) the narrative against the ‘liberal EU’ and, on the other hand, it is used to show substitute ‘scapegoats’ in Poland. Moral panic makes it possible to release class tensions in the manner that is safe for the system through the artificially induced cultural and ideological conflict. However, the dispersed anti-gender discourse has a real impact on social attitudes – on the one hand, it polarises social sympathies and, on the other hand, it strengthens conservative attitudes. The analysis is based on right-wing press articles, Church hierarchs’ statements, videos on YouTube and a parliamentary debate about the right to abortion. (shrink)

Interpretations of logics with only truth-functional connectives create a number of problems regarding the understanding of interpreted sentences. A particular problem is caused by the understanding of a sentence that is the negation of another. What is the meaning of sentence ¬p, for a particular sentence p? Even when we know what the semantic correlate of the sentence p is, we still do not know how to understand the semantic correlate of the sentence ¬p. The standard algebraic approach does not (...) explain much. The problem is still open, because it is still unclear how to understand the element of algebra −v(p). The propositional logic with the connective of the content implication, as it is a non-Fregean logic, proposes a simple solution to this problem. The key to understanding the content of the sentence ¬p is to understand the content of the sentence p. Because, the content of the natural language sentence usually has a partial understanding, so it can be understood differently. Therefore, the negation ¬p is here understood adequately to the current and partial understanding of the sentence p. It seems that the proposed approach is consistent with our daily thinking. (shrink)

Epistemicism seems to be the most dominating approach to vagueness in the recent twenty years. In the logical and philosophical tradition, e.g. Peirce, vagueness does not depend on human knowledge. Epistemicists deny this fact and contend that vagueness is merely the result of our imperfect mind, our dearth of knowledge, sort of phantom, finally, that it simply does not exist. In my opinion, such a stance not only excludes vagueness comprehended in terms of human knowledge, but which is worse, stems (...) from spurious logical arguments. The part of arguments called Sorensen’s Arguments or even Proofs were the subject of my analysis in the book Paradoksy and in the paper “Epistemicism and Roy Sorensen Arguments” published in the Bulletin of the Section of Logic. Here I shall only briefly refer to these works and focus mainly on the arguments launched by Tymothy Williamson. One of them is to uncover why we are not able to recognize the alleged sharp boundary between positive and negative extensions of any vague predicates. Williamson’s reasoning is based on his margin for error principle. Another argumentation of Williamson aims at the refutation of the principle I know that I know. It should be emphasized that all the aforementioned arguments are fundamental for epistemicism and all of them are fallacious because of either formal or false-premise fallacy. There is the circumstance that we cannot deem epistemicism logical. Finally, we show that within the epistemic frame the following thesis is valid: if what epistemicism states is the case, then what epistemicism states is not the case. This immediately implies → ¬p’) that it is not the case what epistemicism states. So, either epistemicism or logic. (shrink)

The idea of belief revision is strictly connected with the notion of contraction given by the set of postulates formulated by Alchourrón, Gärdenfors and Makinson. In the present paper expansion and contraction are defined by Tarski's consequence relation and Tarski-like elimination relation. The logic of falsehood (i.e. a logic dual in Wójcicki's sense to the given logic of truth) plays a key role for defining the elimination relation. A decision of adding or refusing of some sentences is arbitrary and depends (...) on our wish only. This decision cannot be logical and logic cannot justify it. In our approach logic is a tool for faultless and precise realization of extension or reducing of the set of our beliefs. Step forward extends the set of our beliefs and it is used when some new belief appears. Step backward reduces the set of our beliefs and it is used when we reject from some previously accepted belief. But why some "initial" sentences should be added or refused depends on extralogical reasons. The logic for the back-reasoning uses the class of models adequate for the logic extending the set of our beliefs. However, the class is used in a specific i.e. dual form. That is why the step forward (expansion) and the step backward (contraction) constitute the one whole. Procedure of contraction satisfies the well known AGM postulates. We limit our considerations to first six conditions for contraction. Satisfaction of almost every postulate is a good sign that our approach is reasonable. The only exception we make for the controversial fifth postulate. (shrink)

A few years ago, believing that human thinking is nonmonotonic, I tried to reconstruct a nonmonotonic reasoning by application of two monotonic procedures. I called them “step forward” and “step backward” . The first procedure is just a consequence operation responsible for an extension of the set of beliefs. The second one, defined on the base of the logic of falsehood reconstructed for the given logic of truthfulness, is responsible for a reduction of the set of beliefs. Both procedures taken (...) together were successfully verified by using so-called AGM , postulates for expansion, contraction and revision formulated by Alchourrón, Gärdenfors and Makinson . Reasoning composed of the mutual application of both procedures seemed to be quite natural for modeling our thinking. At that time, I supposed that it should be nonmonotonic but I was wrong. It turned out impossible to satisfy a definition of the nonmonotonic inference by reasoning composed both steps. To understand why this is impossible, I began to analyze how nonmonotonicity is obtainable in some well-known cases in the literature. I analyzed the problem from two points of view: non-formal examples for nonmonotonicity and formal constructions of nonmonotonic operations/relations. The result of those investigations was astonishing: none of the considered by me cases of nonmonotonicity belonging to point and almost none belonging to satisfies the definition of nonmonotonic inference. Arguments against the nonmonotonic character of well-known examples for nonmonotonicity of human thinking are more precisely presented in [6]. I present them below an abbreviated version of them. (shrink)

The ancient paradox of Protagoras had the opinion of an unsolved problem. The two solutions proposed in the 20th century by W. Lenzen and L. Aqvist are considered to be the best. In fact none of them may be treated as proper. In the paper we show that both of authors avoid contradiction solely by means of mere neglect. However quite a simple solution seems to be feasible when the paradox is approached as an amphibolic construction, thus an ambiguity.

This book, provides a critical approach to all major logical paradoxes: from ancient to contemporary ones. There are four key aims of the book: 1. Providing systematic and historical survey of different approaches – solutions of the most prominent paradoxes discussed in the logical and philosophical literature. 2. Introducing original solutions of major paradoxes like: Liar paradox, Protagoras paradox, an unexpected examination paradox, stone paradox, crocodile, Newcomb paradox. 3. Explaining the far-reaching significance of paradoxes of vagueness and change for philosophy (...) and ontology. 4. Proposing a novel, well justified and, as it seems, natural classification of paradoxes. (shrink)