The paper is devoted to the problem of the reduction of properties into sets of objects. It consists of three major parts. The first part deals with the conceptual framework where the notions of ‘property’ and ‘set’ are discussed. While the sets are taken straight from the mathematical set theory, properties and their relation to objects require much more complex description. The author adopts the Aristotelian approach based on an ontological relation of inherence.

The book Metaphysics by Tomasz Bigaj is intended to be an introduction to the basic issues of analytical metaphysics designed for non-philosophers. It fulfills its purpose to a large extent: the issues are competently presented in a clear and accessible way, even when referring to a very complex subjects. The unquestionable advantage of the book is also the fact that it indicates a strong relation between philosophical problems and those of other scientific disciplines, mainly physics and mathematics. It makes Bigaj’s (...) work an attractive handbook for the laymen as well as for professionals in other fields. (shrink)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We (...) provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one’s implicit commitments, and sheds light on why this is so. We argue that the theory has significant epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent analyses of implicit commitment based on truth or epistemic notions. (shrink)

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced (...) by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets. (shrink)

In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over $$ PA $$. Let $${\mathfrak {Th}}$$ denote the class of models of $$ PA $$ which admit an expansion to a model of theory $${ Th}$$. We show (combining some well known results and original ideas) that $$\begin{aligned} {{\mathfrak {PA}}}\supset {\mathfrak {TB}}\supset {{\mathfrak {RS}}}\supset {\mathfrak {UTB}}\supseteq \mathfrak {CT^-}, \end{aligned}$$ where $${\mathfrak {PA}}$$ denotes simply the class of all models of (...) $$ PA $$ and $${\mathfrak {RS}}$$ denotes the class of recursively saturated models of $$ PA $$. Our main original result is that every model of $$ PA $$ which admits an expansion to a model of $$ CT ^-$$, admits also an expansion to a model of $$ UTB $$. Moreover, as a corollary to one of the results (brought to us by Carlo Nicolai) we conclude that $$ UTB $$ is not relatively interpretable in $$ TB $$, thus answering the question from Fujimoto (Bull Symb Log 16:305–344, 2010). (shrink)

This paper is a follow-up to [4], in which a mistake in [6] (which spread also to [9]) was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT−with internal induction for total formulae${(\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}\left( {{\rm{tot}}} \right)$, denoted by PT−in [9]). We show that if to PT−the axiom of internal induction forallarithmetical formulae is added (giving${\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}$), then this theory is semantically stronger than${\rm{P}}{{\rm{T}}^ - } + (...) {\rm{INT}}\left( {{\rm{tot}}} \right)$. In particular the latter is not relatively truth definable (in the sense of [11]) in the former. Last but not least, we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in [9]. The truth theory we define is based on Weak Kleene Logic instead of the Strong one. (shrink)

James (1890) distinguished two understandings of the self, the self as “Me” and the self as “I”. This distinction has recently regained popularity in cognitive science, especially in the context of experimental studies on the underpinnings of the phenomenal self. The goal of this paper is to take a step back from cognitive science and attempt to precisely distinguish between “Me” and “I” in the context of consciousness. This distinction was originally based on the idea that the former (“Me”) corresponds (...) to the self as an object of experience (self as object), while the latter (“I”) reflects the self as a subject of experience (self as subject). I will argue that in most of the cases (arguably all) this distinction maps onto the distinction between the phenomenal self (reflecting self-related content of consciousness) and the metaphysical self (representing the problem of subjectivity of all conscious experience), and as such these two issues should be investigated separately using fundamentally different methodologies. Moreover, by referring to Metzinger’s (2018) theory of phenomenal self-models, I will argue that what is usually investigated as the phenomenal-“I” [following understanding of self-as-subject introduced by Wittgenstein (1958) ] can be interpreted as object, rather than subject of experience, and as such can be understood as an element of the hierarchical structure of the phenomenal self-model. This understanding relates to recent predictive coding and free energy theories of the self and bodily self discussed in cognitive neuroscience and philosophy. (shrink)

Idea słowiańska” jest terminem ogólnym, odnoszącym się do wielu politycznych, historiozoficznych i kulturowych wątków, dla których inspiracją było językowe i kulturowe pokrewieństwo między narodami słowiańskimi. Takie ruchy jak słowianofilstwo i panslawizm miały na celu kulturalne i polityczne zjednoczenie Słowian. Idea ta wywarła silny wpływ na muzykę krajów środkowej i wschodniej Europy w XIX i XX wieku. Pod jej wpływem powstała znaczna liczba utworów muzycznych. Można utwory te podzielić na kilka grup. W jednych „słowiańskość” podkreślana była poprzez tytuł. Kompozytorzy używali również (...) typowo słowiańskich gatunków tanecznych i pieśniowych, jak polski mazurek i ukraińska dumka. Innym sposobem realizacji idei słowiańskiej były muzyczne kontakty między twórcami. Idea muzyki słowiańskiej pojawia się również w komentarzach muzykologicznych: słowiański „charakter” muzyczny bywał najczęściej łączony ze specyficznymi kategoriami emocjonalnymi, takimi jak smutek, tęsknota, szczególnie zaś żal, które to określenie stosowano do opisu przemożnego nastroju panującego w dziełach Chopina. (shrink)

This text reviews David Megger’s 2021 book entitled _Justice in Welfare Economics. __Libertarianism and the Austrian School_ (in Polish: _Sprawiedliwość w ekonomii dobrobytu, liberatarianizm i szkoła austriacka_). The review takes a critical approach, highlighting the most significant aspects of the presented considerations and emphasising their uniqueness and complexity. I intend to extensively discuss the author’s theses concerning the modification of the fundamental claims of Austrian school representatives about justice and welfare, highlighting both their strengths and weaknesses.

In this review-paper, I focus on biopsychological foundations of geometric cognition. Starting from the Kant’s views on mathematics, I attempt to show that contemporary cognitive scientists, alike the famous philosopher, recognize mutual relationships of visuospatial processing and geometric cognition. What I defend is a claim that Tinbergen’s explanatory questions are the most fruitful tool for explaining our “hardwired,” and thus shared with other animals, Euclidean intuitions, which manifest themselves in spatial navigation and shape recognition. I claim, however, that these “hardwired (...) intuitions” cannot capture full-blooded Euclidean geometry, which demands practice with cultural artifacts in various time-scales. (shrink)

The main purpose of the paper is to develop a concept of _model uncertainty_ as opposed to the existing and well-established concept of model risk. Up to date the broad literature on probability not only developed complete probability systems, but also correctly noticed limitations of probability calculus. Despite the acknowledgement of such probability restrictions, drawbacks of modeling are often related to model risk. We present an argument here to distinguish a feature limiting models even further: model uncertainty. The tenets of (...) it already exist in the literature on probability, but were not properly emphasized while the idea of model risk was developed. Our plan it to start with a broad overview of the existing knowledge about probability in order to start with fundamental principles. From them we are deriving a new concept of model uncertainty. (shrink)

Tomasz z Akwinu nie kojarzy się w pierwszym odruchu z kimś, kto kładłby mocny akcent na teologię Boga uniżonego we wcieleniu i na krzyżu. Tak przynajmniej wydaje się wielu współczesnym teologom, zwłaszcza tym, którzy są przedstawicielami szeroko rozumianego nurtu kenotycznego. Według nich Akwinata jest przykładem myśliciela, w którego teologii absolutność Boga, Jego transcendencja, niezmienność, suwerenność i potęga zostały tak mocno podkreślone, że ma to wręcz uniemożliwiać adekwatne uchwycenie zdumiewającej prawdy objawionej w Chrystusie: Bóg jest Bogiem uniżającym siebie samego. Zatem, zdaniem (...) wielu współczesnych teologów, proponowana przez Tomasza wizja Boga ma więcej wspólnego ze skonstruowaną za pomocą metafizyki filozoficzną doktryną Absolutu niż z Bogiem, który stał się człowiekiem i umarł na krzyżu dla naszego zbawienia. Czy jednak zarzut ten jest słuszny? Czy Akwinata rzeczywiście nie dostrzega uniżenia Boga? Właśnie z próby odpowiedzenia na te pytania zrodziła się niniejsza książka. Ze Wstępu. (shrink)

This article contributes to the discussion regarding the relationship between idealisation, de-idealisation and cognitive scientific progress. In this, I raise the question of the significance of the gradual de-idealisation procedure for constructing political science theories. I show that conceptions that assume the reversibility of the idealisation process can be an extremely useful theoretical perspective in reconstructions of political science modelling and analyses of scientific progress in political science. I base my position on the results of the methodological reconstruction of Richard (...) Jankowski’s theory of voting. My reconstruction and results of empirical studies show that by gradually removing simplifying assumptions, models can emerge that more accurately identify the determinants of the voting decision and the corresponding relationships. In the case I analysed, the transition from coarse-grained to fine-grained models likely demarcates the line of scientific progress. (shrink)

We present here some Boolean connexive logics (BCLs) that are intended to be connexive counterparts of selected Epstein’s content relationship logics (CRLs). The main motivation for analyzing such logics is to explain the notion of connexivity by means of the notion of content relationship. The article consists of two parts. In the first one, we focus on the syntactic analysis by means of axiomatic systems. The starting point for our syntactic considerations will be the smallest BCL and the smallest CRL. (...) In the first part, we also identify axioms of Epstein’s logics that, together with the connexive principles, lead to contradiction. Moreover, we present some principles that will be equivalent to the connexive theses, but not to the content connexive theses we will propose. In the second part, we focus on the semantic analysis provided by relating- and set-assignment models. We define sound and complete relating semantics for all tested systems. We also indicate alternative relating models for the smallest BCL, which are not alternative models of the connexive counterparts of the considered CRLs. We provide a set-assignment semantics for some BCLs, giving thus a natural formalization of the content relationship understood either as content sharing or as content inclusion. (shrink)

The aim of the paper is to discuss the idea of dialogue with nature. Even though the idea of dialogue with animals, plants – even objects of inanimate nature – is well known, it has usually been treated as an expression of a naive or folk view. Yet, it has recently gained in importance as an idea that is used to describe an ecological approach to natural environment and tends to be treated as a foundation for an ecological culture. A (...) noteworthy contribution to the reconsideration of the concept of human-nature dialogue has also been given by a number of contemporary philosophers who have offered an environmental reinterpretation of phenomenology and hermeneutics based on, among other things, Martin Buber’s concept of the I-You relationship. The paper will focus on two aspects of the concept of dialogue, namely on its epistemological and ethical implications. Following Erazim Kohák’s theory, which will be presented as a key figure of a particular manner of speaking about nature, one that leads to ecological personalism. (shrink)

In consequence of significant social, political, economic, and demographic changes several wildlife species are currently growing in numbers and recolonizing Europe. While this is rightly hailed as a success of the environmental movement, the return of wildlife brings its own issues. As the animals arrive in the places we inhabit, we are learning anew that life with wild nature is not easy, especially when the accumulated cultural knowledge and experience pertaining to such coexistence have been all but lost. This book (...) provides a hermeneutic study of the ways we come to understand the troubling impacts of wildlife by exploring and critically discussing the meanings of 'ecological discomforts'. Thus, it begins the work of rebuilding the culture of coexistence. The cases presented in this book range from crocodile attacks to mice infestations, and their analysis consequently builds up an ethics that sees wildlife as active participants in the shaping of human moral and existential reality. This book is of interest not only to environmental philosophers, who will find here an original contribution to the established ethical discussions, but also to wildlife managers, and even to those members of the public who themselves struggle to make sense of encounters with their new wild neighbors. (shrink)

The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. -/- Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In the book, (...) Hohol argues that Euclidean geometry would not be possible without the human capacity to create and use abstract concepts, demonstrating how language and diagrams provide cognitive scaffolding for abstract geometric thinking, within a context of a Euclidean system of thought. -/- Taking an interdisciplinary approach and drawing on research from diverse fields including psychology, cognitive science, and mathematics, this book is a must-read for cognitive psychologists and cognitive scientists of mathematics, alongside anyone interested in mathematical education or the philosophical and historical aspects of geometry. (shrink)

We introduce a principle of local collection for compositional truth predicates and show that it is arithmetically conservative over the classically compositional theory of truth. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the (...) case of the induction scheme. We analyse various further results concerning end-extensions of models of compositional truth and the collection scheme for the compositional truth predicate. (shrink)

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ($\mathrm {CT}_0$). Furthermore, we (...) extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of$\mathrm {CT}_0$. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of$\mathrm {CT}_0$. (shrink)

The aim of this article is to suggest that one should think of gardens in terms of performances and not necessarily in terms of architecture, painting, or poetry, for it is possible to show that, strangely enough, gardens seem to share certain features with performance arts. Such an approach seems fruitful since it allows one both to grasp the fact that gardens combine culture and nature and to underline the role of the latter, which cannot be reduced to a sheer (...) medium as is traditionally done. The contention is that gardens should be treated more like a continuous, dynamic, partly planned process in which people can participate in different ways on a par with other non-human ‘actors’. Moreover, the category of performance seems to offer a useful framework helping to solve certain problems inherent in traditional ways of thinking about gardens. (shrink)

One of the logic defined by Richard Epstein in a context of an analysis of subject matter relationship is Symmetric Relatedness Logic S. In the monograph [2] we can find some open problems concerning relatedness logic, a Post-style completeness theorem for logic S is one of them. Our paper introduces a solution of this metalogical issue.

In this paper, we investigate abstract model-theoretic properties which hold for models in which a truth or satisfaction predicate for a sublanguage of the signature is definable. We analyze in which cases those properties in fact ensure the definability of the respective truth predicate. In some cases, we formulate different axiomatic theories which are indispensable for such properties to hold and we analyze the mutual definability relations between those theories.

The book presents a critical overview of Peirce's views on modern nominalism and offers a novel approach to the social-anthropological underpinnings of his realism, especially Pragmatic Realism vis à vis the individualist tendencies in modern thought.

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional connectives of Łukasiewicz’s (...) logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete. (shrink)

-/- Medytacje filozoficzne, Fundacja na Rzecz Myślenia im. Barbary Skargi, Warszawa 2015. -/- Publikacja współfinansowana przez Muzeum Łazienki Królewskie w Warszawie. -/- Zbiór tekstów powstałych na podstawie wykładów, które zostały wygłoszone w ramach cyklu "Medytacje filozoficzne w Łazienkach" w latach 2013-2015. -/- -/- Spis treści: -/- Cezary Wodziński, Premedytacja Krzysztof Pomian, O wyjątkowości człowieka Zygmunt Bauman, O miłości i nienawiści… Tropami Barbary Skargi Tadeusz Sławek, Ciemne liturgie. Język, historia i gest błogosławieństwa Lech Witkowski, Rozprawa z autorytetem: w strone autorytu przejścia (...) Władysław Stróżewski, Doświadczenie czasu, doświadczenie istnienia Juliusz Domański, „Pradawna waśń filozofów z poetami” a kłopoty z dziedzictwem antycznym Serhij Żadan, Wolność jako odpowiedzialność Marcin Poręba, Co to jest rzeczywistość? Szymon Wróbel, Co to jest historia filozofii? Magdalena Środa, Moralność życia publicznego Kostiantyn Tyszczenko, Po co językoznawstwu metateoria? (shrink)

This paper discusses main issues concerning Toulmin’s theory of evolutionary development of scientific knowledge. The reconstruction of Toulmin’s assumptions, which underlie the theory, and the presentation of the theory itself, is followed by the analysis of the difficulties of the conception in general. The author tries to show that these difficulties consist primarily in the discrepancy between what Toulmin aimed at and what he actually achieved. The problem is that although incorporating the means of Darwin’s theory may itself be — (...) in the context of describing the growth of scientifi c knowledge — justifi able, it is not consistent with Toulmin’s demand for the rationality of science. The purpose of creating the evolutionary theory was, as Toulmin claimed, to give an account of the development of scientific knowledge which would be non-relativistic and rationalistic. However, this purpose is, as the author argues, not achievable on the grounds of Toulmin’s accounts of rationality. Key words S. TOULMIN, SCIENCE, EVOLUTION. (shrink)

The paper presents a partial evaluation of employment and factors related to the labour markets in European countries in 2007-2016. The interconnectedness of these determinants in the context of GDP dynamics per capita for each country was examined. The quoted partial subject literature and empirical research allowed to formulate the most important conclusions, among others: in the context of GDP dynamics per capita, at least four groups of countries can be distinguished in Europe, each of them has completely different characteristics (...) having an influence (in the Granger causality sense) on change in GDP per capita of these countries for various time steps. (shrink)

Over the past decade work engagement has gained both business and academia attention. With growing number of studies and meta-analyses the concept of work engagement is one of the pillars of positive work and organizational psychology. This systematic review presents the current state of research on work engagement in Poland. Results confirmed that work-engagement studies have not yet reached the threshold to conduct meta-analysis. The review of measurement methods and synthesis of findings allows to identify strengths and gaps in Polish (...) studies. Discussion of limitations and biases in current research is accompanied with urge to overcome them and develop thriving stream of research on work engagement. (shrink)

Celem tego artykułu jest analiza powiązań jakie zachodzą pomiędzy fenomenami metodyczności i etyczności we wczesnej filozofii Martina Heideggera. Nie pojmuje się tu ani metody jako procedury badawczej prowadzącej do określonych rezultatów, ani etyki jako odrębnej dyscypliny filozoficznej posiadająca własne pole przedmiotowe. Chodzi o dwa wymiary bycia, które wzajemnie się implikują. Na początku badana jest fenomenologia jako sposób pytania o bycie, a następnie zostaje wydobyty metodyczny sens postępowania, które Heidegger określa mianem formalnego wskazywania. W rezultacie metodyczność zostaje odsłonięta jako droga prowadząca (...) do bycia jako źródła sensu, droga, którą zapewnia samo to bycie. W dalszej kolejności analizowany jest fenomen filozofowania, który Heidegger interpretuje jako wyraźne transcendowanie. Transcendowanie to ostatecznie okazuje się wolnością, której fundamentalny charakter stanowi pozwolić-być, kładące jednocześnie fundament pod odpowiedzialność. Odpowiedzialność zaś jest odpowiedzią udzielaną na wezwanie bycia i przyjmuje postać (zobo)wiązania Dasein wobec bytu, któremu pozwoliło ono być. Etyczność odsłania się tym samym jako domaganie się takiej odpowiedzi na wezwanie bycia, która sprosta ciężarowi jego przesłania. W tym miejscu na jaw wychodzi spotkanie etyczności z metodycznością – ta ostatnia jest drogą do otwartości bycia zapewnioną przez samą tę otwartość, by postulat etyczności mógł zostać spełniony. Oznacza to, że etyczność domaga się metodyczności, a metodyczność (a wraz z nią filozofowanie) niesie ze sobą konsekwencje etyczne dla rozumienia bycia i bytu. Słowa kluczowe: Heidegger, fenomenologia, metoda, etyka, wolność, odpowiedzialność. (shrink)

We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with \Delta 0 -induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

Author: Bonecki Mateusz Title: HUMANISTIC INTERPRETATION AND THEORETICAL RECONSTRUCTION OF CULTURE (Interpretacja humanistyczna a teoretyczna rekonstrukcja kultury) Source: Filo-Sofija year: 2011, vol:.12, number: 2011/1, pages: 189-211 Keywords: CULTURE, SOCIO-REGULATORY THEORY OF CULTURE, KMITA, INTERPRETATION, HUMANITIES, FUNCTIONALISM, CULTURAL STUDIES, KULTURWISSENSCHAFTEN Discipline: PHILOSOPHY Language: POLISH Document type: ARTICLE Publication order reference (Primary author’s office address): E-mail: www:In this paper the author presents crucial aspects of Jerzy Kmita’s socio-regulatory conception of culture in order to define it as a theoretical background of “humanistic (...) interpretation”. When the latter is to be understood as a method of explanation of human behavior with regard to its cultural meaning, the theoretical background is provided by variety of research activities which the author defines as “theoretical reconstruction of culture”. They consist in e.g. socio-functional explanation, linguistic and semantic analysis, or even in epistemological considerations, and guarantee theoretically independent justification of the interpretive hypotheses applied in the course of “humanistic interpretation”. Such “reconstruction” specifies further conditions of intersubjective acceptability of the statements within the discourse of humanities and therefore it also limits the subjective and individual character of the attribution of cultural beliefs to the members of a certain cultural community. (shrink)

In my paper I undertake an attempt at an archaeological analysis of the increasing interest in emotions and affects in the humanities. Simultaneously, it is to some extent an attempt to continue Foucault's reflection from The Order of Things. In doing so, I reconstruct (a) previous ways – both philosophical and literary (starting from the seventeenth century) – of defining the relationship between “reason” and “heart” and (b) the origins, transformations and disintegration of the subject of desire. “Affectology” itself turns (...) out to be the result of (i) the decay of the traditional humanities (ii) the influence of structuralism, (iii) the emergence of “culture” which has replaced “human” as the primary reference point of the humanities. (shrink)

The article concerns two axiom systems of Słupecki for the functionally complete three-valued propositional logic: W1–W6 and A1–A9. The article proves that both of them are inadequate—W1–W6 is semantically incomplete, on the other hand, A1–A9 governs a functionally incomplete calculus, and thus, it cannot be a semantically complete axiom system for the functionally complete three-valued logic.

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)