The book is addressed to all readers interested in cognitive science, and especially in research combining a logical analysis with psychological, linguistic and neurobiological approaches. The publication is the result of a collaboration between the Department of Cognitive Science at University of Lodz and the Department of Cognitive Science at Lund University. It is intended to provide a comprehensive presentation of the key research issues undertaken in both Departments, including considerations on meaning, natural language and reasoning, linguistic as well as (...) numerical competence, the theory of decision making, modelling of conceptual representations, cognitive and game theoretic approach to social interactions. (shrink)
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.
This article presents an empirical examination of the consequences of the virtual entailment principle proposed by Jean Buridan to resolve the Liar paradox. This principle states that every sentence in natural language implicitly asserts its own truth. Adopting this principle means that the Liar sentence is not paradoxical but false, because its content is contradictory to what is virtually implied. As a result, humans should perceive the Liar sentence the same way as any other false sentence. This solution to the (...) Liar paradox received criticism for making ad hoc claims about the natural language. However, thanks to modern advancements in psychophysiology, it became possible to empirically investigate if the human brain really perceives the Liar sentence like a false sentence. We designed and conducted an experiment to examine brain activity in response to true sentences, false sentences and self-referential sentences. Our results provide support for the Buridan’s hypothesis and show that the Liar sentence is processed by the human brain identically to false sentences, while the Truthteller sentence is perceived identically to true sentences. This agrees with predictions derived from the virtual entailment principle and supports the idea that humans think with the logic of truth—a logic for which the truth is a designated value of its adequate semantics. (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 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.
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  (...) obtained system S5 on the base of the propositional calculus with identity. Next, Cresswell ,  reconstructed S4 and S5 in a similar way. Systematic investigations of this problem were undertaken by Suszko within the broader frame of SCI programme , , . 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)
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)
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)
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)
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 . I present them below an abbreviated version of them. (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)
This paper is a proposal of continuation of the work of C. Rauszer. The logic of falsehood created by her may constitute the starting point for construction of logic formalising reductive reasonings. The extension of Heyting-Brouwer logic to its deductive-reductive form sheds new light upon those classical tautologies which are rejected in intuitionism. It turns out that among HBtautologies there can be found all the classical ones. Some of them are characteristic for deductive reasoning and they are accepted by intuitionism. (...) Others formulate the laws of reductive reasoning. Many of them, including the law of excluded middle has been rejected in Heyting’s intuitionism. Intuitionism only permits for those reductive tautologies, which at the same time bear deductive character. Thus, the complete HB intuitionism does not reject any of the classical tautologies. Every classical tautology appears in HB logic in its deductive or reductive part. Some are even present in both parts. At the end it is shown that the law of excluded middle α∨¬α is a law of the reductive part of the traditional, Heyting’s intuitionistic propositional calculus. (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.
Introduction. PHilosophy of Language and Linguistics: The Formal Turn Piotr Stalmaszczyk Gottlob Frege, Philosophy of Language, and Predication Piotr Stalmaszczyk Philosophy, Linguistics and Semantic Interpretation Christian Bassac An Unresolved Issue: Nonsense in Natural Language and Non-Classical Logical and Semantic Systems Elzbieta Chrzanowska-Kluczewska Varieties of Context-Dependence Tadeusz Ciecierski The Logos of Semantic Structure Marie Du í, Bjørn Jespersen and Pavel Materna The Good Samaritan and the Hygienic Cook: A Cautionary Tale About Linguistic Data Chris Fox The Meaning of Multiple (...) Quantified Sentences: Where Formal Semantics Meets Psycholinguistics Justyna Grudzinska The Hybrid Theory of Reference for Proper Names Filip Kawczynski On the Nature of Statistical Language Laws Agnieszka Kulacka Vagueness and Contextualism Joanna Odrowaz-Sypniewska The Myth of Semantic Structure Jaroslav Peregrin Scalar Implicatures, Communication and Language Evolution Salvatore Pistoia Reda Semantics and Contextuality: The Case of Pia s Leaves Stefano Predelli Is Logico-Semantical Analysis of Natural Language Expressions a Translation? Jirí Raclavský Beyond the Fregean Myth: The Value of Logical Values Fabien Schang Modal Calculus of Illocutionary Logic Andrew Schumann Subjectivity in Philosophy and Linguistics Barbara Sonnenhauser Asymmetrical Semantics Mieszko Talasiewicz Truth: An Anti-realist Adequacy Condition Luca Tranchini Belief Reports: Defaults, Intentions and Scorekeeping Giacomo Turbanti On Truth in Time Bartosz Wieckowski Order William J. SUllivan Indexes. (shrink)
Topic modeling—a text‐mining technique often used to uncover thematic structures in large collections of texts—has been increasingly frequently used in the context of the analysis of scholarly output. In this study, we construct a corpus of 19,488 texts published since 1971 in seven leading journals in the field of bioethics and philosophy of medicine, and we use a machine learning algorithm to identify almost 100 topics representing distinct themes of interest in the field. On the basis of intertopic correlations, we (...) group the content‐based topics into eight clusters, thus providing a novel, fine‐grained intellectual map of bioethics and philosophy of medicine. Moreover, we conduct a number of diachronic analyses, examining how the “prominence” of different topics has changed across time. In this way, we are able to observe the distinct patterns in which bioethics and philosophy of medicine have evolved and changed their focus over the past half a century. (shrink)
W świadomości społecznej religie politeistyczne uchodzą za „mniej dojrzałe” niż inne typy religijności. Również w starszych opracowaniach naukowych można napotkać przyjmowaną wprost gradację religii, która sytuuje wielobóstwo niżej niż monoteizm. Uznawanie istnienia wielu bogów jest jednak nieracjonalne tylko z perspektywy teologii monoteistycznej i wizji świata przyjętej pod jej wpływem. W rzeczywistości z wierzeń politeistycznych można wyabstrahować ciekawe założenia metafizyczne, są one zdolne również do wytworzenia bogatej, racjonalnej teologii. W artykule postaram się obronić politeizm przed zarzutami dotyczącymi jego filozoficznej jałowości. Najpierw (...) spróbuję zdiagnozować błędy w powszechnej interpretacji wielobóstwa, następnie przedstawię różnice w mono- i politeistycznym rozumieniu pojęcia boga, by ostatecznie przedstawić możliwe strategie uzasadnienia wiary w osobowy i intencjonalny charakter bogów. (shrink)
The government debt portfolio is usually the largest financial portfolio in the country. It often contains complex and risky financial structures and can generate significant risk to the state budget and the country’s financial stability. Therefore, governments are required to have sound risk management and sound public debt structures to limit exposure to market risk, debt financing or rolling risk, liquidity risk, credit, settlement and operational risk. In recent years, the debt market crises have highlighted the importance of sound public (...) debt management practices and related risks, and the need for an effective and well-developed domestic capital market. This may reduce the vulnerability of the economy to adverse economic and financial shocks. However, it is also important for the government to maintain a macroeconomic policy that ensures sound fiscal and monetary management. The aim of the research is to present the theoretical and practical aspects of extremely important issues such as public debt management and to indicate the most important implications for the financial stability of the country on the example of the Polish economy. The study uses a research method based on literature studies in the field of macroeconomics, economic policy and finance, as well as statistical analysis of the studied phenomenon. Results of research indicate that effective public debt management can reduce the economy’s vulnerability to financial threats, contribute to the financial stability of the country, maintain debt stability and protect the government’s reputation among investors. (shrink)
Piotr Cyciura Human Freedom and MetaphysicsHuman freedom is in a sense the topic of metaphysics. Freedom is usually regarded as what is the most perfect in the reality. For man being perfect is the same as acting in the way which is proper for him only, viz., acting in a rational manner. However, what man desires is more perfect than what he is. The human existence is participated whereas what man aims at in the most rational manner is the (...) Existence itself. Therefore God (as loved and known) guarantees the human freedom. The value is everything that brings us closer to Him – everything save God and persons – provided we offer it to Him. It is our choice whether we make whatever is ours obey to God or not. However, what is eventually achieved is not what is ours but we ourselves – living in the Life itself. There are no borders for the human freedom in the physical world; whatever bears upon us, i.e. what is natural, could become an offering, whereas what is offered to Him in the way He cannot reject it is the only absolute value. Keywords: freedom, values, metaphysics, God. (shrink)
The article presents the question of understanding divine causality and its analogical character in the context of Thomas Aquinas’s teaching on Divine Providence. Analyzing Aquinas’s texts concerning the relation of God’s action towards nature and its activities it is necessary to emphasize the proper understanding of mutual relations between secondary causes and the primary cause which are not on the same level. Influenced by the reflection of M. Dodds and I Silva, the author of the article refers to Aquinas’s biblical (...) commentaries which have not been discussed so far from the perspective of the character of God’s action. In the final part of the article, metaphors used by Thomas in reference to the relation of God towards the world will be presented. (shrink)
Was Heidegger a 'realist' or an 'idealist'? The issue has been and continues to be hotly debated in Heidegger scholarship. Here it is argued that the much more desirable realistic interpretation of Heidegger can be sustained, provided his theory of moods is given its due. Moods, I argue, are not only 'equiprimordial' with Dasein's understanding of being, but are also irreducible to the latter. It is often held - correctly, as it seems to the author - that Heidegger's idealism is (...) all but inevitable if Dasein's awareness of entities is grounded only in Dasein's understanding of being. But in Being and Time Heidegger speaks also of how what there is is 'disclosed moodwise'. The essay closely analyzes this specifically moody mode of disclosure, and shows both its autonomy vis-à-vis the understanding of being and its function of securing, for Dasein, an access to a truly independent reality. (shrink)
Piotr Kropotkin był najważniejszym przedstawicielem tzw. darwinizmu rosyjskiego oraz teorii pomocy wzajemnej, która podkreślała znaczenie współpracy wewnątrzgatunkowej w procesie doboru naturalnego oraz walki o byt. Na jej kanwie rosyjski badacz wysunął tezę o ewolucyjnych źródłach moralności, będącej problem badawczym w niniejszym artykule. Stanowi to jednocześnie powód dlaczego poza przedstawieniem wspomnianej tezy i jej założeń, w publikacji zilustrowana została także teoria pomocy wzajemnej. Zaproponowana przez Kropotkina wizja moralności suponowała, iż stanowi ona „przyrodzoną” właściwość ludzkiej natury. Podstawę dla niej stanowić ma (...) instynkt społeczny oraz jego przejawy, takie jak: równość, solidarność, sprawiedliwość, itd., a także pomoc wzajemna. Stawiało to propozycję rosyjskiego darwinisty w opozycji wobec dominujących trendów w myśleniu o ludzkiej moralności na przełomie XIX i XX wieku, zwłaszcza wobec tzw. darwinizmu społecznego i charakterystycznego dlań dualizmu moralnego. Ponadto zrywała z antropocentryzmem zakładając, iż poza gatunkiem ludzkim, także inne gatunki cechują się uczuciami moralnymi. (shrink)
The article presents a thought experiment aimed at indicating a possibility for thinking education beyond the logic of progress. In its first part, the argument reconstructs the entanglement of the modern idea of progress (as found in Francis Bacon and Comenius) and education, while tracking down the specific coupling of obedience and conquest at work. Through such an analysis a link between the ideas of progress and of emancipation is determined, which leads to the acknowledgement of the difficulty of the (...) task of imagining education outside of the logic of progress. In the second part an attempt is made to match this task by suggesting that when studying with a teacher, the logic of progress is deactivated. A phenomenological analysis of the practices involved in studying with a teacher points to a specific way of living together that such a collective study involves. This way of living together is formed by continual exercises in the humble equality realized through shared attention to something worthy of study: this stems from attentiveness to what exists, and it develops an attitude of care and respect for being as such. (shrink)
This theoretical paper is offered in the spirit of advancing the debate on the socioemotional wealth construct and its impact on how family firms conceptualize and practise corporate social responsibility. The study builds on Kellermanns et al.’s :1175–1182, 2012) claim that the SEW dimensions can be positively and negatively valenced as well as makes a distinction between the selective and instrumental approach to CSR and the holistic and normative one. Drawing on these considerations, it provides a theoretical underpinning in favour (...) of the view that SEW has ambivalent nature and therefore can produce detrimental outcomes for stakeholders of family companies. In this way, the study challenges the implicit assumption prevalent in the literature that SEW is “a prosocial and positive stimulus”. Crucially, it expands on the SEW construct by arguing that, given its ambivalent nature, SEW, as such, is at odds with the “strategic, whole-business view of responsibility”. Consequently, it posits that family firms—because of their concern with SEW—may be more likely to adopt the instrumental and selective rather than strategic and normative approach. Hence, it also makes the case for regarding the latter as a reference point to investigate the family company’s attitude towards social responsibility. It concludes by summarising the argument and offering future research avenues. (shrink)
Predictive processing (PP) has been repeatedly presented as a unificatory account of perception, action, and cognition. In this paper, we argue that this is premature: As a unifying theory, PP fails to deliver general, simple, homogeneous, and systematic explanations. By examining its current trajectory of development, we conclude that PP remains only loosely connected both to its computational framework and to its hypothetical biological underpinnings, which makes its fundamentals unclear. Instead of offering explanations that refer to the same set of (...) principles, we observe systematic equivocations in PP‐based models, or outright contradictions with its avowed principles. To make matters worse, PP‐based models are seldom empirically validated, and they are frequently offered as mere just‐so stories. The large number of PP‐based models is thus not evidence of theoretical progress in unifying perception, action, and cognition. On the contrary, we maintain that the gap between theory and its biological and computational bases contributes to the arrested development of PP as a unificatory theory. Thus, we urge the defenders of PP to focus on its critical problems instead of offering mere re‐descriptions of known phenomena, and to validate their models against possible alternative explanations that stem from different theoretical assumptions. Otherwise, PP will ultimately fail as a unified theory of cognition. (shrink)
Blumenthal-Barby and her colleagues (2022) situate their discussion of philosophy and bioethics in the context of (reportedly) widely held assumption that, when compared to the early days of bioethics, the role of philosophy is now diminished across the field – the assumption we call the Disconnection Thesis. This assumption can be summarized, to use the authors’ own words, by the phrase “philosophy’s glory days in bioethics are over“. While in no place of the article they explicitly endorse the Disconnection Thesis, (...) at least some of the authors had previously endorsed it in print (Savulescu 2015). Such expressions of collective expert wisdom might be a valuable source of information on the discipline's history, but they should not be accepted uncritically. Given the explosion in the size and scope of bioethical research in recent decades, any scholar’s familiarity with the area is necessarily based on selective reading and might be biased. Hence, in this commentary, we examine what kind of more rigorous evidence could corroborate the Disconnection Thesis. In other words, if the role of philosophy in bioethics has been indeed diminishing, what kind of observable patterns should we expect to see? Drawing on our previous research (Bystranowski, Dranseika, Żuradzki 2022), here we focus on two useful perspectives: citation analysis and topic modeling. While the first approach allows us to indirectly measure the level of engagement of bioethicists with philosophical literature (by measuring the proportion of references from articles published in journals in bioethics that cite philosophy journals), the latter provides a window into the content and argumentative style of bioethical texts. (shrink)
The article reviews the book Słowa i zbawienie. Dyskurs religijny w perspektywie filozofii Hilarego Putnama [Words and Salvation: Religious Discourse in the Perspective of Hilary Putnam's Philosophy], by Piotr Sikora.
The article reviews the book Słowa i zbawienie. Dyskurs religijny w perspektywie filozofii Hilarego Putnama [Words and Salvation: Religious Discourse in the Perspective of Hilary Putnam's Philosophy], by Piotr Sikora.
The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ and (...) c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics. (shrink)