Realism’s Understanding of Negative Numbers

Author(s): Petar Bojanić,Sanja Todorović / Language(s): English Issue: 1/2016

Our topic is the understanding of the nature of negative numbers – the entities to which expressions such as ‘-1’ refer. Following Frege, we view positive whole numbers as providing the answer to the question „how many?“ In this light, how are we to view negative numbers? Both positive and negative numbers can be ordered through the relation of larger or smaller. It is then true of all negative numbers that they are entities which are (somehow) smaller than zero. For many, this has been understood as an ontological paradox: how can something be „less than nothing?“ Some propose to avoid the paradox by treating negative numbers as mere façons de parler. In this paper, we propose a more realist account, taking as our starting point the thesis that there is at least one familiar type of object, the magnitude of which can be expressed with negative numbers, namely, debt. How can the sense of an expression be ontologically paradoxical, yet the expression itself still plausibly refer to a social object such as a debt? Or, put differently, how is it possible to be, at the same time, a realist in financial theory and a nominalist in mathematical theory? The paper first shows that the paradox arises when the two distinct ways in which negative numbers are connected to real objects are run together. The first of the two refers to debt only, whereas the second could refer to debt, as well as to physical objects. Finally, we claim that a debt is at once a specifically social object and part of reality as described by physics.

Divisibility in the Stone-Čech compactification

Author(s): Boris Šobot / Language(s): English Issue: 50/2015

After defining continuous extensions of binaryrelations on the set N of natural numbers to its Stone-ˇCech ompactification βN, we establish some results about one of suchextensions. This provides us with one possible divisibility relationon βN,, and we introduce a few more, defined in a naturalway. For some of them we find equivalent conditions for divisibility.Finally, we mention a few facts about prime and irreducibleelements of (βN, ·). The motivation behind all this is to try totranslate problems in elementary number theory into βN.

Peritopological Spaces and Bisimulations

Author(s): Ahmet Hamal,Mehmet Terziler / Language(s): English Issue: 50/2015

Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We brie y describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topo- logical models, a spacial case of neighborhood semantics. A new cladding for bisimulation is presented. The concept of Alexandro peritopology is used in order to determine the logic of all peritopo- logical spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal denable, and that D is the logic of all proper peritopological spaces. Finally, among our conclusions, we show that the question whether T0, T1 peritopological spaces are modal denable in H(@) remains open.

On everywhere strongly logifiable algebras

Author(s): Tommaso Moraschini / Language(s): English Issue: 50/2015

We introduce the notion of an everywhere strongly logifiable algebra: a finite non-trivial algebra A such that for every F 2 P(A) r f;;Ag the logic determined by the matrix hA; Fi is a strongly algebraizable logic with equivalent algebraic semantics the variety generated by A. Then we show that everywhere strongly logifiable algebras belong to the field of universal algebra as well as to the one of logic by characterizing them as the finite non-trivial simple algebras that are constantive and generate a congruence distributive and n-permutable variety for some n > 2. This result sets everywhere strongly logifiable algebras surprisingly close to primal algebras. Nevertheless we shall provide examples everywhere strongly logifiable algebras that are not primal. Finally, some conclusion on the problem of determining whether the equivalent algebraic semantics of an algebraizable logic is a variety is obtained.

A Note on Wansing's expansion of Nelson's logic

Author(s): Hitoshi Omori / Language(s): English Issue: 51/2016

The present note corrects an error made by the au- thor in answering an open problem of axiomatizing an expansion of Nelson's logic introduced by Heinrich Wansing. It also gives a correct axiomatization that answers the problem by importing some results on subintuitionistic logics presented by Greg Restall.

WSZECHMOGĄCY, KTÓREMU WIERZYMY

Author(s): Miłosz Hołda / Language(s): Polish Issue: 12/2013

Concept the God’s almightiness is both one from most everyday, as well as one of the most complicated notions. The article is demonstrating how this concept is understood in Catechism of the Catholic Church and is showing why this proposal of understanding almightiness is unique. The Catechism is describing the almightiness with three expressions: universal, loving, mysterious. The universality is connected with a fact that God is a creator of the world and is supporting world in being. Almighty love is manifesting itself in the Father’s concern and forgiving sins. The mystery is becoming clearer in the perspective of Jesus - His rising from the dead, by which the evil is overcome. The faith in the God’s almightiness this way understood lets experience the God’s power and is a key to the faith in other mysteries included in the Catholic Credo.

Definitions and the Growth of Knowledge: The Main Ideas

Author(s): Robert Kublikowski / Language(s): English Issue: 65/2016

Are definitions useful in an empirical knowledge-gaining process? What roles do definitions play in the process of the growth of empirical knowledge? Two attitudes towards definitions can be distinguished in the history of the theory of definitions. According to the first and positive one, definitions have been useful in science. The second attitude has been a critical one. I try to defend the view about the usefulness of definitions, on the one hand, by application of Hilary Putnam’s theory of reference of natural kind terms. On the other hand, Karl Popper’s fallibilism is implemented to the theory of definitions, especially to the theory of real definitions.The structure of this text is as follows: (I) the origin and the development of the theory of definitions, (II) Popperian antidefinitionism, (III) the theory of definitions and the Putnamian theory of meaning and (IV) the theory of stipulative, lexical and persuasive definitions.

Aniccam - Budistička teorija prolaznosti (Pristup sa stanovišta moderne filosofije)

Author(s): Čedomil Veljačić / Language(s): Serbian Issue: 261/1980

— Da li su oko... oblik ... opazajna svest, stalni ili prolazni? — Prolazni, poštovani gospodaru. — A ono što je prolazno, da li je to bol ili radost? — Bol, poštovani gospodaru. — Da li je ispravno smatrati ono što je prolazno, bolno i podlezno promeni kao »ovo je moje, to sam ja, to je moje sopstvo«? — Ne, poštovani gospodaru. Saznanja i otkrića do kojih je ljudski um došao pre dve i po hiljade godlima, u vreme Buddhe (ili čak nekoliko vekova pre tog vremena), verovatno su imali duboke i revolucionarne posledice u evolucdji postojedih pogleda na svet, i to posledica ne manjle značajne nego što su to bila oktrića Galileja i Kopemika za konačni raspad pogleda na svet srednjovekovne hrišćanske civilizacije. Ova druga otrikća, koja označavalju počeltak modeme civilizacije, toliko su postala deo uobiičajene ili opšte informisanosti da se mogu prelneti deci, koja ih normalno usvajaju, bez teškoča, i u najnižim razredima osnovtnog obralzovanja.

Jak przygodne i jak a priori są przygodne prawdy a priori?

Author(s): Jacek Wawer / Language(s): Polish Issue: 1/2015

W artykule poddaję analizie słynne twierdzenie Saula Kripkego, że niektóre prawdy a priori są przygodne. Pokazuję, że wbrew tezie Kripkego, przy historycznym rozumieniu przygodności, pojęcia przygodności i aprioryczności stoją ze sobą w głębokim konflikcie. Przy tym rozumieniu przygodności przeszłość, którą można poznać a priori, nie jest przygodna, a o przyszłości, która jest przygodna, trudno zdobyć wiedzę a priori. Doprecyzowawszy tezę Kripkego, proponuję trzy sposoby jej obrony w kontekście historycznego rozumienia możliwości: (a) przez wprowadzenie pojęcia “faktycznej” przyszłości, (b) przez zastąpienie pojęcia aprioryczności pojęciem aprioryczności-w-przyszłości; (c) przez zastąpienie pojęcia aprioryczności pojęciem historycznej aprioryczności, a pojęcia przygodności pojęciem niegdyś-przygodności. W aneksie do artykułu przedstawiam formalną analizę postawionego przeze mnie problemu oraz trzech zaproponowanych przeze mnie rozwiązań w języku temporalno-modalnej logiki predykatów dla modeli indeterministycznego czasu. ===== In the presented article, I have analyzed the famous Saul Kripke statement that some a priori truths are contingent. I show, that despite Kripke’s thesis, in the historical understanding of contingency, the notions of contingency and apriority are in deep conflict with each other. In this understanding of contingency, the past, which can be known a priori, is not contingent, and the future, which is contingent, has difficulty acquiring a priori knowledge. Having stated Kripke’s thesis more precisely, I propose three means in order to defend it in the historical understanding of possibility: (a) by introducing the notion of “factual” future, (b) by replacing the notion of apriority with the notion of apriority-in-the-future; (c) by replacing the notion of apriority with the notion of historical apriority, and the notion of contingency with the notion of once-apriority. In the annex of the article, I present the formal analysis of the problem that I have introduced and three solutions which I have proposed in the language of temporal-modal logic of predicates for models of indeterministic time.

Teizm a twardy inkompatybilizm

Author(s): Dariusz Łukasiewicz / Language(s): Polish Issue: 3/2017

The aim of the article is to present and to compare the view on human freedom called hard incompatibilism with the contemporary Christian doctrine on human free will. Hard incompatibilism claims that human free will understood both in a libertarian and compatibilist way does not exist. One stresses in the paper that there is a similarity between hard incompatibilism and Christian wisdom rooted in the Bible and this similarity consists in the fact that humans are deeply dependent in their existence on external conditions. Hard incompatiblism identifies that conditions simply as the external or physical world and Christian wisdom points to God as an ontological and axiological foundation of human being and prospects. However, one argues in the paper that the doctrine of human freedom and responsibility for sin and moral evil is a crucial part of the Christian theology and philosophy. Thus, the Christian doctrine is incoherent with hard incompatibilism. There is a proposal, put forth in the last part of the article, how one can reconcile metaphysical indeterminism—which is coherent with hard incompatibilism—with the libertarian doctrine on the human free will, which is coherent with the Christian view on the nature of human freedom.

Acorduri sentimentale. Instantanee filosoficești și logicești cu Ion Ungureanu

Author(s): Mihai Cimpoi / Language(s): Romanian Issue: 2/2017

Kıyas Şekillerinin Geometrik Yöntemle İrcâ‘ı

Author(s): Ekrem Sefa Gül / Language(s): Turkish Issue: 2/2017

This study suggests a method for establishing a correct syllogism which regulates principles of logic and for reduction of syllogisms to first figure. This geometric method depends on the correspondence of four figures of syllogism with four straight triangles formed by a second square drawn in a main square. The middle term is on the right angled corners of the triangles. The minor term and the minor premise of syllogism are on the vertical edge, the major term and the major premise are on the horizontal edge. The conclusion has drawn diagonally from the minor term to the major term. In this method, the syllogism can be shown quickly. In addition, the reduction of syllogisms can be done easily and accurately without requiring any knowledge other than knowing the general rules of syllogism and the two specific rules of first figure and geometrical appearance of this figure’s four valid modes. Summary: There is a distinct importance for syllogism among the types of reasoning in terms of logic which it is main goal is protect the mind from falling into the wrong during thinking. The type of syllogism that most emphasized by logicians is categorical syllogism and when mentioned the term of syllogism usually meant this. A valid categorical syllogism contains three categorical propositions. Two of them premises and one conclusion. The middle term must be distributed in at least one of the premises but cannot be distributed in conclusion.In comparison with this the categorical syllogism was expressed in four figures. These figures are; the first figure which the middle term becomes attribute in minor premise and subject in major term, the second figure which the middle term becomes attribute in both premises, the third figure which the middle term becomes subject in both premises and finally the fourth figure which the middle term becomes subject in minor premise and attribute in major term. According to quantity and quality of premises and conclusion, every figures involve another sub-figures named mods. When considered this rules there is nineteen valid syllogism mods among the sixty-four total mods. Four of this mods for first figure; four ones for second figure, six ones for third figure and five ones for fourth figure.The first figure has been regarded as the perfect figure because it is the closest match to the human mind and nature. For the control of validity, the other figures have done by reduction. The process of reduction is a method used not only to show validity of a valid syllogism but also to show the invalidity of incorrect ones. Generally accepted three ways for reduction. Converting of premises, transposing of premises and indirect reduction that also called reduction per impossible. In order to do the process of reduction, it is necessary to know the name of each mode, the meaning of the mnemonic letters on it that show how the reduction process is done. This is the case when the valid syllogism mods are reduced. However, there is neither a name and nor mnemonic letters for invalid syllogisms as like as valid ones. For this reason, although there are some methods, but no certain method guided by mnemonic letters for reduction of invalid syllogism to first figure some methods.The geometric reduction method that proposed in this article removes this difficulty and makes possible to reduction of invalid syllogisms as easily as valid ones. Geometrical representation of syllogisms is a very effective and easy method for identify the terms and premise in syllogism, understanding of invalidity in a syllogism which resulted from lacking attention to general and specific rules of syllogism. In addition, by this method the process of reduction can be done easily a correctly for demonstration of syllogism’s validity. Although there have been previous studies on the syllogism figures by venn diagrams, these are mostly confined to a geometric representation of the reasoning in syllogism. The method suggested in this article is new method offering far more opportunities than a mere demonstration for checking the syllogisms and reduction of them to first figure.This geometric method depends on the correspondence of four figures of syllogism with four straight triangles formed by a second square drawn in a main square. The middle term is on the right angled corners of the triangles. The minor term and the minor premise of syllogism are on the vertical edge, the major term and the major premise are on the horizontal edge. The conclusion has drawn diagonally from the minor term to the major term. On arrows indicating propositions, there are one notch for positive, two for universal negative, and three notches for particular negative propositions. When reduced the syllogism valid or invalid to first figure’s geometrical shape, if there is an event of a contradictory for the specific and general rules of syllogism, this can be shown automatically. For example, if the three notched arrow of geometric shape when converted, this means converting of particular-negative which cannot be done and thus can be shown invalidity of syllogism. In this method, the syllogism can be shown quickly and reduction of syllogisms can be done easily and accurately without requiring any knowledge other than knowing the general rules of syllogism and the geometrical appearance of the first figure’s four valid modes. The syllogisms in other three figures when drawn according to the first figure’s geometric shape the conversions and transpositions has done spontaneously.

Влияние современных web-технологий на развитие контентного вида мышления

Author(s): Tatyana Eduardovna Sizikova,Nadezda Alexandrovna Stunzha,Alexander Fedorovich Poveshenko,Ruben Oganesovich Agavelyan,Tatyana Viktorovna Voloshina / Language(s): Russian Issue: 6/2017

Introduction. This article explores the impact of Web 3.0 and Web 4.0 contents dominant in the information environment on modern thinking. The purpose of the article is an analytical study of the changes that occur in the thinking of Internet consumers. Materials and Methods. The research is conducted on the basis of the method of theoreti-cal analysis by adopting an activity and synergistic approaches. Results. The authors have identified the following main characteristics of Web 3.0 and Web 4.0 content: semantic structure, cooperativity, clustering, wide opportunities for consumer self-expression, self-developing basic personal content, self-correcting errors system, efficient and convenient information management, accessibility, simplicity and maximum convenience, development and use of additional features, human resource management in the current time mode, crystallization, and the availability of maximum possible security of consumers. The authors emphasize that in the present context of accelerating scientific and technological progress and the development of global information networks, qualitative changes in the thinking activity of the modern person are taking place. Content has a developing effect on consumers’ thinking and thinking itself becomes like content. The authors suggest the hypothesis that there exist a combination of two mechanisms of self-development of content thinking: the external mechanism – the self-development of content, and the internal mechanism – the reflection. Conclusions. Conclusions are drawn that the main characteristics of such thinking are the following: accumulation of many different types and styles of thinking, the leading method is synthetic deduction, increasing entropy, increasing singularity, operation within the framework of the "peace" category, expanding and tapering integrity, variability and different levels of clusters of processing and packaging of information, bringing thought to the ultimate limits of philosophical categories, multi-vector focus of reality research, and creative self-development. The limitation of content thinking is a reduction of a reflection level as a consequence of a scanning method of working with information.

What are the limits of mathematical explanation?

Author(s): Piotr Urbańczyk,David Charles McCarty / Language(s): English Issue: 60/2016

Interview with Charles McCarty by Piotr Urbańczyk

Matematyka nie opisuje świata, lecz wychodzi mu naprzeciw

Author(s): Jerzy Mioduszewski / Language(s): Polish Issue: 58/2015

In everyday experience mathematics rarely appears to us as a whole, and certainly never as a system in the sense of David Hilbert’s considerations from early 20th Century. Mathematical disciplines seem to be independent and autonomous. We do not see that specific deduction goes beyond particular convention applicable in given discipline. In the late 19th Century this view was shared by Felix Klein and Richard Dedekind. The latter’s work “What are numbers and what should they be?” (Was Was sind und was sollen die Zahlen?) was the inspiration for writing this article. This essay is an attempt to see mathematics not as a building, but as a living organism seeking its explanation.

Czy logiczna możliwość implikuje metafizyczną możliwość?

Author(s): Paweł Zięba / Language(s): Polish Issue: 55/2014

According to Chalmers, the argument from the conceivability of philosophical zombies disproves materialism in the philosophy of mind. This claim depends on the assumption that conceivability (logical possibility) entails metaphysical possibility. Such entailment is incorrect, however, because a materialist might formulate an analogous argument from the conceivability of anti-zombies. A clash between two mutually excluding logical possibilities prevents one from inferring a metaphysical possibility from any of them.

Aksjomat wyboru w pracach Wacława Sierpińskiego

Author(s): Katarzyna Lewandowska / Language(s): Polish Issue: 52/2013

This paper presents Wacław Sierpiński – the first advocate of the axiom of choice. We focus on the philosophical and mathematical topics related to the axiom of choice which were considered by Sierpiński. We analyze some of his papers to show how his results effected the debate over Zermelo’s axiom. Sierpiński’s impact on this discussion is of particular importance since he was the first who tried to explore consequences of the axiom of choice thoroughly and asserted its undoubted significance to mathematics as a whole.

Czy konceptualizm jest wystarczającą podstawą dla odrzucenia niekonstruktywnych dowodów istnienia w matematyce?

Author(s): Daniel Chlastawa / Language(s): Polish Issue: 51/2012

Non-constructive existence proofs (which prove the existence of mathematical objects of a certain kind without giving any particular examples of such objects) are rejected by constructivists, who hold a conceptualist view that mathematical objects exist only if they are constructed. In the paper it is argued that this conceptualist argument against non-constructive proofs is fallacious, because those proofs establish the existence of objects belonging to certain kinds rather than the existence of those objects per se. Moreover, to engage in proving existence theorems in a given mathematical theory one has to define all of the objects of this theory at the very beginning, which can be interpreted as establishing the existence of these objects before any theorem about them is proven. It is also argued that the constructivist may escape these objections by adopting the actualistic view, according to which a mathematical sentence is true if and only if it is established as true, but this view is very implausible, as it seems unable to explain the strictness and objectiveness of mathematics and the fact that it differs so fundamentally from, for example, fictional discourse.

Pytania i odpowiedzi. Analiza krytyczna koncepcji Kazimierza Ajdukiewicza

Author(s): Anna Brożek / Language(s): Polish Issue: 42/2008

The aim of the paper is to present critical analysis of Kazimierz Ajdukiewicz's theory of questions. Generally, the author concentrates on four elements of Ajdukiewicz's theory of questions: the structure of question, the classification of questions, the concept of presuppositions of questions and the concept of answer. Ajdukiewicz's contributions in the domain of logic of questions are unquestionable. However, there are still many 'gaps' in his conception of questions that need to be filled. He proposes some improvements to this conception. They include an explication of the concepts of 'query' and 'tenor', a classification of complementation questions by use of categorial grammar, enriching the concept of 'assumptions' by the concept of 'pragmatic assumptions', and an explication of the concept of 'answer' (in general).

Topic, logical subject and sentence structure in Hungarian

Author(s): Zsuzsanna Gécseg / Language(s): English Issue: 2/2006

This paper investigates the "discourse-configurationality" hypothesis in Hungarian, based on the current assumption that Hungarian sentence structure is largely determined by information structure. It argues for the necessity of differentiating between the notion of topic, defined on a pragmatic level with respect to possible contexts, and the notion of logical subject, defined on a decontextualized logico-semantic level. On the basis of the distinction between these two levels of sentence analysis, Hungarian should be taken as a logical subject-prominent language rather than a topic-prominent one. As for the so-called contrastive topic in Hungarian, only a subclass of contrastive topic expressions meets the topicality conditions established in this paper on a pragmatic ground, and other types of contrastive topic expressions, namely those that can hardly be differentiated from ordinary topics, display the properties of logical subjects rather than topics.