Most accounts of cognitive scaffolding focus on ways that external structure can support or augment an agent’s cognitive capacities. We call cases where the interests of the user are served benign scaffolding and argue for the possibility and reality of hostile scaffolding. This is scaffolding which depends on the same capacities of an agent to make cognitive use of external structure as in benign cases, but that undermines or exploits the user while serving the interests of another agent. We develop (...) criteria for scaffolding being hostile and show by reference to examples including the design features of electronic gambling machines and casino management systems that hostile scaffolding exists and can be highly effective. In cases where the scaffolding is deep and permits the offloading of significant cognitive work, hostile scaffolding exploitatively manipulates cognitive processing itself. Given the extent of human reliance on scaffolding this is an important and neglected vulnerability. [This is an updated version of the paper, added in February 2023.]. (shrink)

The random nature of immunoglobulin gene segment rearrangement inevitably leads to the generation of self‐reactive B cells. Avoidance of destructive autoimmune reactions is necessary in order to maintain physiological homeostasis. However, current central and peripheral tolerance concepts fail to explain the massive number of autoantibody‐borne autoimmune diseases. Moreover, recent studies have shown that in physiological mouse models autoreactive B cells were neither clonally deleted nor kept in an anergic state, but were instead able to mount autoantibody responses. We propose that (...) activation of autoreactive B cells is induced by polyvalent autoantigen complexes that can occur under physiological conditions. Repeated encounter of autoantigen complexes leads to the production of affinity‐matured autoreactive IgM that protects its respective self‐targets from degradation. We refer to this novel mechanism as adaptive tolerance. This article discusses the discovery of adaptive tolerance and the unexpected role of high affinity IgM autoantibodies. (shrink)

I argue that in “Empiricism and the Philosophy of Mind,” Wilfrid Sellars significantly misconstrued the early twentieth-century empiricists he was criticizing (empiricists such as Bertrand Russell, H.H. Price and C.I. Lewis). Because these philosophers and their theories were becoming passé (partly due to EPM itself but also due to broader trends), this misconstrual was not noted. As a result, Sellars’s most influential claim – that the given is a myth – did not receive the critical scrutiny that it should have (...) received prior to its widespread acceptance. (shrink)

A striking empirical finding about color perception is that normal perceivers disagree about which hues are pure. (Pure hues contain no perceived admixture of any other color.) This finding poses a prima facie problem for color objectivism and representationalist accounts of perceptual experience. Michael Tye attempts to resolve this problem by arguing that pure hues do exist as objective properties of ordinary objects, but that human color detection mechanisms did not evolve with sufficient refinement to allow us to determine which (...) hues are the pure ones. I argue that Tye’s attempted solution fails, and that the most reasonable conclusion consistent with color objectivism is that pure hues do not exist. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Book Reviews 137 The Meanings of Landscape: Essays on Place, Space, Environment and Justice BY KENNETH R. OLWIG London: Routledge, 2019 REVIEWED BY TIMM SCHÖNFELDER Landscape is more than spatial scenery that meets the eye: it is an anthropogenic artefact, an intellectual construct, a mirror of culture; it even has its own language.1 This broadness is reflected in the compilation of nine authoritative essays by the geographer and (...) professor of landscape architecture Kenneth R. Olwig that were published during the past quarter of a century and are now molded into a more coherent whole. In this regard, the present volume can be considered the pinnacle of Olwig’s intellectual oeuvre. In a fervent foreword, the anthropologist Tim Ingold sets the tone when he urges the reader to shed the contemporary notion of what he terms “topophobia”—­ a lack of bond betweenpeopleandplaces—­andtoreinstatephilologyasacentralfield of interest after it was “relegated by techno-­ science to the margins of scholarship” (xv). To achieve this, the essays deal with an eclectic set of themes ranging from Ptolemy’s notion of “choros” as something similar to today’s understanding of landscape, to the humanistic geographer Yi-­Fu Tuan’s concept of “topophilia,” which is hailed as a guiding ideal for much of Olwig’s work. The articles are bound together by an introduction that quickly transcendsthegeometricallimitationsoflandscape,ontologicallyquestioning whether these can be representative of the world itself at all (3). The first chapter thus delves right into the etymologies of “landscape” and its historical implications in the Nordic countries and England with detours through Italy and the German lands. In a cursory, highly associative manner Olwig fathoms the respective cultural implications. Central is the idea that the concept of landscape is more than the mere scenicrepresentationofspaceitcametobeknownasinRenaissance art and culture.2 Rather, the Nordic variants of the word, like the Swedish landskap, are used to denote natural regions with their own polity that were then gradually subjected to a monarchic state, not least through the surveying and mapping of their territory (see also chapter two). ENVIRONMENT, SPACE, PLACE / VOLUME 13 / ISSUE 2 / 2021 138 At times, Olwig’s philological musings appear rather stretched, as when he associates the German word Grundgesetz (the “basic/­ fundamental law” of 1949, which until today serves as Germany’s constitution ) with a biopolitical dimension of “ground” as in “soil” (41).3 In cases like this, his use of literature is selective. This culminates in a problematic assessment of Nazi ideology as being motivated by a specific understanding of landscape. Statements like “there can be no doubt that the modern German conception of Landschaft was implicated in the promotion of blood-­ and-­ soil fascist ideology” (42) neglect important literature on the topic.4 Such generalizations are owed to the author’s sweeping tour de force approach. The reviewer would have preferred more guidance through the theoretical thickets by way of a clear, chronologically ordered argument. This is not helped by the occasional seeming inconsistency, as in the claim that the U.S. “inherited the British scenic idea of landscape” as a former colony while also being “very much the inheritor of German romantic ideas concerning the relation of culture to nature as expressed in the physical landscape” (45), which, in turn, showed parallels to “fascist landscapes” in the “attempt to cleanse geography of its environmentalist taint by making it a science of space” (42). This truly is a lot of ground to cover for one essay. The ensuing chapters, however, try to disentangle this inspired jumble. The second article, “Landscape, Place, and the State of Progress,” identifies a switch since the Renaissance from the idea of circuitous development to linear progress towards utopia. For Olwig, there is a “darker, Faustian side” to it that he identifies as the “dialectic of modernity,” which “opposes cosmic dreams of a utopian future to the topian exigencies of present and past”: to advance, the path had to be cleared of traditional elements (52). This materialized in courteous theatre as well as in Shakespeare’s popular plays, and in the new architecture of parks where landscape was symbolic for cultural and social progress (64). “Choros, place, and the spatialization of landscape” traces important theoretical threads within geography... (shrink)

This article identifies problems with regard to providing criteria that regulate the matching of logical formulae and natural language. We then take on to solve these problems by defining a necessary and sufficient criterion of adequate formalization. On the basis of this criterion we argue that logic should not be seen as an ars iudicandi capable of evaluating the validity or invalidity of informal arguments, but as an ars explicandi that renders transparent the formal structure of informal reasoning.

Examines how the modern city is portrayed in art and literature, discusses modernism, futurism, and expressionism, and looks at the work of Rilke, Eliot, Pound, Joyce, and Brecht.

One of the central logical ideas in Wittgenstein’sTractatus logico-philosophicusis the elimination of the identity sign in favor of the so-called “exclusive interpretation” of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier’s account, the first concerning the treatment of individual constants, the second concerning so-called “pseudo-propositions” (Scheinsätze) of (...) classical logic such as$a=a$or$a=b \wedge b=c \rightarrow a=c$. We argue that overcoming these problems requires two fairly drastic departures from Wehmeier’s account: (1) Not every formula of classical first-order logic will be translatable into asingleformula of Wittgenstein’s exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original “inclusive” formulas to beambiguous. (2) Certain formulas of first-order logic such as$a=a$will not be translatable into Wittgenstein’s notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a “correct” conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations. (shrink)

In order to provide an alternative to the Cartesian myth that knowledge of our thoughts and sensations is "given," Sellars posits a community of "Rylean ancestors" - humans at an early stage of conceptual development who possess a language containing sophisticated concepts about the physical world and about their own language and behavior, but who lack any concepts of thoughts or sensations. Sellars's presentation of this thought experiment leaves many important details sketchy. In the following dialogue, we offer our differing (...) assessments of how well those details could be filled out. TT questions how the Rylean hypothesis could provide a plausible account of human thought and sensory experience at any stage of human conceptual development. WdeV responds to TT's challenge by filling out the picture of what the Ryleans' conceptual world would look like. TT and WdeV debate the merits and the plausibility of the resulting picture. (shrink)

In this engaging and accessible dialogue, four students offer contrasting arguments on the nature and scope of morality. While specific social policy issues, such as animal rights and racism, come into play, the discussions focus on more general--and fundamental--questions, including: Does morality limit personal freedom? Is morality relative to culture, or is it universal? What is the motivation to be moral? Is religion in tension with secular moral principles? Does science undermine morality? Can a common morality emerge out of the (...) diversity of human interests? A glossary of important terms and suggestions for further reading are included. (shrink)

Bernard Gert’s account of morality is straightforward, clear and, in its essentials, easily grasped. As such, it offers rich pedagogical resources for teaching morality, not just in undergraduate courses but also in pre-college philosophy classes or workshops, including those offered during the elementary school years. Gert’s account, properly calibrated to the age group in question, can provide a unified framework for students to think about morality, clarify their understanding of it, and engage in discussions with each other about it. After (...) summarizing Gert’s account, I illustrate several ways in which it can be applied in the classroom, beginning with applications at the elementary school level and working up to pedagogy appropriate to high school and college. I conclude by considering and responding to some possible objections to this approach. (shrink)

Bernard Gert’s account of morality is straightforward, clear and, in its essentials, easily grasped. As such, it offers rich pedagogical resources for teaching morality, not just in undergraduate courses but also in pre-college philosophy classes or workshops, including those offered during the elementary school years. Gert’s account, properly calibrated to the age group in question, can provide a unified framework for students to think about morality, clarify their understanding of it, and engage in discussions with each other about it. After (...) summarizing Gert’s account, I illustrate several ways in which it can be applied in the classroom, beginning with applications at the elementary school level and working up to pedagogy appropriate to high school and college. I conclude by considering and responding to some possible objections to this approach. (shrink)

Much of what we understand about heterochromatin formation in mammals has been extrapolated from forward genetic screens for modifiers of position‐effect variegation (PEV) in the fruit fly Drosophila melanogaster. The recent identification of the HUSH (Human Silencing Hub) complex suggests that more recent evolutionary developments contribute to the mechanisms underlying PEV in human cells. Although HUSH‐mediated repression also involves heterochromatin spreading through the reading and writing of the repressive H3K9me3 histone modification, clear orthologues of HUSH subunits are not found in (...) Drosophila but are conserved in vertebrates. Here we compare the insights into the mechanisms of PEV derived from genetic screens in the fly, the mouse and in human cells, review what is currently known about the HUSH complex and discuss the implications of HUSH‐mediated silencing for viral latency. Future studies will provide mechanistic insight into HUSH complex function and reveal the relationship between HUSH and other epigenetic silencing complexes. (shrink)

, David Bloor suggests that logical reasoning is radically relativistic in the sense that there are incompatible ways of reasoning logically, and no culturally transcendent rules of correct logical inference exist which could allow for adjudication of these different ways of reasoning. Bloor cites an example of reasoning used by the Azande as an illustration of such logical relativism. A close analysis of this reasoning reveals that the Azande's logic is in fact impeccably Aristotelian. I argue that the conclusions Bloor (...) can legitimately draw from his case study are not controversial and do nothing to make plausible the thesis of logical relativism. (shrink)

Newton claims to have proven the heterogeneity of light through his experimentum crucis. However, Olaf Müller has worked out in detail Goethe’s idea that one could likewise prove the heterogeneity of darkness by inverting Newton’s famous experiment. Müller concludes that this invalidates Newton’s claim of proof. Yet this conclusion only holds if the heterogeneity of light and the heterogeneity of darkness is logically incompatible. This paper shows that this is not the case. Instead, in Quine’s terms, we have two logically (...) compatible theories based on mutually irreducible theoretical terms. From a Quinean point of view, this does no harm to the provability of the corresponding statements. (shrink)

Das Buch entwickelt einen neuartigen, physikalistischen Interpretationsansatz zu Wittgensteins Tractatus Logico-Philosophicus. Das traditionelle Urteil, Wittgenstein habe im Tractatus keine klare Vorstellung der Analyse gehabt, wird widerlegt. Auf der Basis der Rekonstruktion der um die Jahrhundertwende etablierten Sinnesdatenanalysen im allgemeinen und der Farbanalysen im besonderen wird nachgewiesen, daß Wittgensteins Tractatus eine physikalische Sinnesdatenanalyse voraussetzt. Auf diesem Hintergrund werden Wittgensteins allgemeine Auffassungen zur Analyse der Welt und Sprache gedeutet, begründet und exemplifiziert. Der Tractatus liefert die philosophische Klärung des mechanistischen Weltbildes von Boltzmann (...) und Hertz. Er stellt die Mittel bereit, um den Sinn der Sätze zu analysieren und die logische Zulässigkeit von Aussagen zu prüfen. Die Anwendung dieser Mittel ist Aufgabe der Philosophie. Daß sie anwendbar sind und wie sie anzuwenden sind, demonstriert dieses Buch. (shrink)

In this paper I problematize the use of appeals to the common intuitions people have about the morality of our society’s current treatment of animals in order to defend that treatment. I do so by looking at recent findings in the field of cognitive science. First I will examine the role that appeals to common intuition play in philosophical arguments about the moral worth of animals, focusing on the work of Carl Cohen and Richard Posner. After describing the theory of (...) Moral Disengagement—which has been used to explain how people live with themselves when they commit acts that they themselves believe are wrong—I will review the recent empirical research that details the nature of the moral disengagement that accompanies animal treatment. This includes studies that reveal that those who eat animals engage in the following behaviors: They minimize the nature of the harm of killing animals by casting the practice in a positive light; They obscure personal responsibility by blaming others for harms; They minimize the effect of the conduct on the animals by avoiding reference to the animal origins of meat; They derogate vegetarians as a way of avoiding feelings of guilt for their own practices. Perhaps most importantly, a number of empirical studies have shown that people’s intuitions about the moral worth of animals are shaped by their practice of eating animals—and not the other way around. In addition to the studies on moral disengagement, there is another body of research that has attempted to discover what accounts for the individual differences in attitudes people have toward the moral worth of animals. These studies have linked masculinity and traits like Authoritarianism, Social Dominance Orientation and Right Wing Authoritarianism to the tendency to disregard the moral worth of animals. I will briefly summarize this body of research and will then conclude by arguing that this data should cause us to be highly skeptical of the value that the common intuition that people who eat animals have about the moral status of animals may have in helping us understand the actual moral worth of animals. (shrink)

Phenomenology’s return to lived experience and “to the things themselves” is often contrasted with the synthesized perspective of science and its “view from nowhere.” The extensive use of neuropsychological case reports in Merleau-Ponty’s Phenomenology of Perception, however, suggests that the relationship between phenomenology and science is more complex than a sheer opposition, and a fruitful one for the praxis of medicine. Here, I propose a new reading of how Merleau-Ponty justifies his use of Adhémar Gelb and Kurt Goldstein’s reports on (...) Johann Schneider for his phenomenology of embodied perception. I argue that for Merleau-Ponty these neuropsychological case reports represent a coherent deformation of the intercorporeally expressed existence of Schneider that through speech fall again onto the common ground of perception, thereby allowing Merleau-Ponty to understand, in the equivalent sense delivered by language, Schneider’s total being and fundamental illness. I then discuss what Merleau-Ponty’s method implies for a phenomenological praxis of medicine, and for the role of science in this praxis. (shrink)

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...) in fact, reasonable and related to Gödel’s proof. (shrink)

Es wird anhand von Fallbeispielen aus der Geschichte der Farbenlehre inwissenschaftstheoretische Probleme eingeführt. Das Buch dient als Grundlagefür eine anwendungsbezogene Lehre und als Einführung in folgende ThemenbereicheNewton vs. Goethe; Theorie und Experiment, Colormetrie; Empfindungsmessung;Helmholtz vs. Hering; Theorienevaluation, Psychologische Farbenlehre; Phänomenologie,Farbausschluss; Beweistheorie, Farbdefinitionen; Theorien- und Begriffsbildung.Neben Aufgaben, Texten und Lösungsvorschlägen finden sich eine Bibliographiesowie Einleitungen zu den behandelten Fragen und Lösungsvorschlägen derjeweiligen Themen.

This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his (...) method to a subset of first-order formulas, namely, formulas that do not contain dyadic sentential connectives within the scope of any quantifier. (shrink)

In contrast to Hintikka’s enormously complex distributive normal forms of first- order logic, this paper shows how to generate minimized disjunctive normal forms of first-order logic. An effective algorithm for this purpose is outlined, and the benefits of using minimized disjunctive normal forms to explain the truth conditions of propo- sitions expressible within pure first-order logic are presented.

In a previous issue of this journal, Michael Hicks challenges my critique of Wilfrid Sellars’s arguments against the given and against the foundationalist epistemology that relies on the idea of a sensory given. I had argued that Sellars’s well-known claim that the given is a myth does not succeed because at a critical juncture he misconstrued sense-datum theorists such as Bertrand Russell and H. H. Price. In his response to my argument, Hicks makes the striking claim that Sellars was not (...) targeting foundationalism at all in his discussion of the myth of the given. Hicks reconstructs a key argument in “Empiricism and the Philosophy of Mind” (EPM) in a way intended both to avoid any reference to foundationalism and to do a more effective job than does Sellars’s original argument in uncovering a dilemma for traditional empiricism. The present paper challenges Hicks on two fronts. First, it argues that Hicks’s reconstruction is not more successful than Sellars’s original argument. Second, a review of relevant passages in makes clear that the critique of foundationalism is a prominent aspect of Sellars’s multi-faceted attack on the given. The conclusion reasserts the significance of Sellars’s place in the history of twentieth-century analytic philosophy. (shrink)

This paper argues for a physicalistic interpretation of Wittgenstein's Tractatus Logico-Philosophicus. Wittgenstein's general conception of world and language analysis is interpreted and exemplified in relation to the historical background of the psychophysical analysis of sense data and, in particular, color analysis. Three of his main principles of analysis—the principle of independence, the context principle and the principle of atomism—are interpreted and justified on the background of physicalism. From his proof of color exclusion in the Tractatus, it is shown that Wittgenstein (...) had a detailed conception of how philosophy should fulfil the task of distinguishing between sense and nonsense using physicalistic presuppositions. (shrink)

In the following dialogue between TT - a foundationalist - and WdeV - a Sellarsian, we offer our differing assessments of the principle for observational knowledge proposed in Wilfrid Sellars's 'Empiricism and the Philosophy of Mind'. Sellars writes: 'For a Konstatierung "This is green" to "express observational knowledge", not only must it be a symptom or sign of the presence of a green object in standard conditions, but the perceiver must know that tokens of "This is green" are symptoms of (...) the presence of green objects in conditions which are standard for visual perception.' In the ensuing dialogue, TT argues that it sets the bar too high when knowledge about perceptual conditions is required for ordinary observational knowledge - that young children, for example, are implausibly excluded as knowers given Sellars's principle. WdeV defends Sellars's metaknowledge requirement against these charges. Results from developmental psychology are surveyed for what they show about the actual capabilities of young children. The implications of these results for the success of Sellars's principle are debated. (shrink)

It is controversial whether Wittgenstein's philosophy of mathematics is of critical importance for mathematical proofs, or is only concerned with the adequate philosophical interpretation of mathematics. Wittgenstein's remarks on the infinity of prime numbers provide a helpful example which will be used to clarify this question. His antiplatonistic view of mathematics contradicts the widespread understanding of proofs as logical derivations from a set of axioms or assumptions. Wittgenstein's critique of traditional proofs of the infinity of prime numbers, specifically those of (...) Euler and Euclid, not only offers philosophical insight but also suggests substantive improvements. A careful examination of his comments leads to a deeper understanding of what proves the infinity of primes. (shrink)

Antony Flew's ?A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365?78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But this analysis only (...) succeeds if the rejection of an explanatory theory entails that there are reasonable grounds for the rejection of the set of propositions which that theory was intended to explain. I argue against such an entailment, and thus against Bloor's attempt to relativize mathematical knowledge. (shrink)

Logical relativism is the view that a logical proposition is known just in case it is collectively endorsed in some culture. This striking and controversial view is defended by David Bloor and Richard C. Jennings. They cite in its support distinctive reasoning practices among the Azande as described by E. E. Evans-Pitchard. Jennings has challenged my critique of Bloor's logical relativism, claiming that my analysis is based on misunderstandings of Bloor and Evans-Pritchard. I argue that Jennings' clarifications of Bloor do (...) nothing to support the thesis of logical relativism, and that a direct examination of Evans-Pritchard's evidence suggests that the Azande reason just as we do. (shrink)

Anhand der genaueren Analyse von Newtons experimentum crucis und der Argumentation, die er auf dieses Experiment stützt, sowie Goethes Kritik hieran sollen im Folgenden zwei verbreitete Vorurteile revidiert werden: -/- 1. Newton ist kein Dogmatiker, der methodische Ansprüche vertritt, die er nicht einlösen kann, sondern gründet seinen Anspruch, experimentelle Beweise führen zu können, auf einer vorbildlichen Methodologie kausaler Erklärungen, was seine Kritiker allerdings übersehen. 2. Goethe ist kein Antiwissenschaftler, der einen einzigartigen Kontrapunkt zur vorherrschenden wissenschaftlichen Tradition bildet, sondern steht inmitten (...) traditioneller Auffassungen zur Farbenlehre, deren experimentelle und methodologische Grundlagen bezüglich eines Erklärungsanspruches denen Newtons unterlegen sind. (shrink)

In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...) Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (shrink)

In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; (...) its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers. (shrink)

This paper illustrates what a philosophical and a logical investigation of colors amounts to in contrast to other kinds of color analysis such as physical, physiological, chemical, psychological or cultural analysis of colors. Neither a philosophical nor a logical analysis of colors is concerned with specific aspects of colors. Rather, these kinds of color analysis are concerned with what one might call “logical foundations of color theory”. I will illustrate this first by considering philosophical and then logical analysis of colors.

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on (...) nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction. (shrink)