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)

Es wird gezeigt, dass Wittgenstein in seiner Frühphilosophie ein nicht-axiomatisches Beweisverständnis entwickelt, für das sich das Problem der Begründung der Axiome nicht stellt. Nach Wittgensteins Beweisverständnis besteht der Beweis einer formalen Eigenschaft einer Formel – z.B. der logischen Wahrheit einer prädikatenlogischen Formel oder der Gleichheit zweier arithmetischer Ausdrücke – in der Transformation der Formel in eine andere Notation, an deren Eigenschaften sich entscheiden lässt, ob die zu beweisende formale Eigenschaft besteht oder nicht besteht. Dieses Verständnis grenzt Wittgenstein gegenüber einem axiomatischen (...) Beweisverständnis ab. Sein Beweisverständnis bedingt ein Programm der Grundlegung der Mathematik, das eine Alternative zu den Ansätzen des Logizismus, Formalismus und Konstruktivismus darstellt. Wittgensteins Ansatz steht im Widerspruch zu den Ergebnissen der Metamathematik, da er die Möglichkeit der Formulierung von Entscheidungsverfahren in der Prädikatenlogik und Arithmetik voraussetzt. Um seinem Ansatz gegenüber der traditionellen Metamathematik Recht zu geben, müsste gezeigt werden, dass sein Beweisverständnis im Bereich der Logik und Arithmetik – der traditionellen Metamathematik zum Trotz – realisierbar ist. (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)

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)

This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context.

The young Wittgenstein called his conception of logic “New Logic” and opposed it to the “Old Logic”, i.e. Frege’s and Russell’s systems of logic. In this paper the basic objects of Wittgenstein’s conception of a New Logic are outlined in contrast to classical logic. The detailed elaboration of Wittgenstein’s conception depends on the realization of his ab-notation for first order logic.

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)

We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal.

It is a widely held principle that no one is able to do something that would require the past to have been different from how it actually is. This principle of the fixity of the past has been presented in numerous ways, playing a crucial role in arguments for logical and theological fatalism, and for the incompatibility of causal determinism and the ability to do otherwise. I will argue that, assuming bivalence, this principle is in conflict with standard views about (...) knowledge and the semantics for ‘actually’. I also consider many possible responses to the argument. (shrink)

Rigidity does interesting philosophical work, with important consequences felt throughout metaphysics, epistemology, philosophy of mind, and so on. The authors’ aim in this article is to show that rigidity has yet another role to play, with surprising consequences for the problem of free will and determinism, for the phenomenon of rigidity has the upshot that some metaphysically necessary truths are up to us. The significance of this claim is shown in the context of influential arguments against free will. The authors (...) show that some virtually indisputable inference rules employed in formulations of the Consequence Argument, as well as in fatalistic arguments, fail with a variety of counter-examples. Along the way, the authors compare the present arguments to other, similar arguments made in recent years. (shrink)

What the rational thing to do in the face of disagreement by an epistemic peer is has been much discussed recently. Those who think that a peer’s disagreement is itself evidence against one’s belief, as many do, are committed to a special form of epistemic dependence. If such disagreement is really evidence, it seems reasonable to take it into account and to adjust one’s belief accordingly. But then it seems that the belief one ends up with depends, in part, on (...) what someone else believes, even if one does not know why that someone believes what he does. While the practical impossibility of finding actual cases of peer disagreement has been often noted, its conceptual possibility has gone unquestioned. Here we challenge this consensus and argue, first, that, strictly speaking, peer disagreement is impossible and, second, that cases of – all-too-common – near-peer disagreement present no special puzzle and require nothing more than adhering to standard principles of sensible epistemic conduct. In particular, we argue that in such cases there is no good reason to adopt the widely accepted principle that evidence of evidence is evidence. If so, even if one takes a near-peer’s disagreement as a reason for reexamining one’s belief, one is not epistemically dependent in the sense one would be if that disagreement were evidence concerning the matter in question. (shrink)

“Christianity is a μίμησις of the divine nature.” This definition of what it means to be a Christian, given by St Gregory of Nyssa in his letter De pro- fessione Christiana, employs a term commonly translated as “imitation” or “representation.” Even a brief study of some of the seminal sources of classical Greek thought, however, will show that the concept of mimesis surpasses any of these translations and effortlessly crosses the boundaries of the sphere of aesthetics, towards the fundamental questions (...) of epistemology, metaphysics, ontology, and ethics. An analysis of Gregory’s letter, together with the related treatise De perfectione, will then show his nuanced familiarity with the subtleties of mimesis, which he consciously employs to arrive at his definition of Christianity. With this in hand, I will argue that even some of Gregory’s most perplexing scriptural exegesis in his homilies on the Song of Songs grows out of his coherent concept of mimesis, which ultimately is of fundamental importance to Gregory’s anthropology, cosmology, and Christology. (shrink)

In a recent paper, Pruss proves the validity of the rule beta-2 relative to Lewis’s semantics for counterfactuals, which is a significant step forward in the debate about the consequence argument. Yet, we believe there remain intuitive counter-examples to beta-2 formulated with the actuality operator and rigidified descriptions. We offer a novel and two-dimensional formulation of the Lewisian semantics for counterfactuals and prove the validity of a new transfer rule according to which a new version of the consequence argument can (...) be formulated. This new transfer rule is immune to the counter-examples involving the actuality operator and rigidified descriptions. However, we show that counter-examples to this new rule can also be generated, demanding that the Lewisian semantics be generalized for higher dimensions where counter-examples can always be generated. (shrink)

In a recent article, P. Roger Turner and Justin Capes argue that no one is, or ever was, even partly morally responsible for certain world-indexed truths. Here we present our reasons for thinking that their argument is unsound: It depends on the premise that possible worlds are maximally consistent states of affairs, which is, under plausible assumptions concerning states of affairs, demonstrably false. Our argument to show this is based on Bertrand Russell’s original ‘paradox of propositions’. We should then opt (...) for a different approach to explain world-indexed truths whose upshot is that we may be morally responsible for some of them. The result to the effect that there are no maximally consistent states of affairs is independently interesting though, since this notion motivates an account of the nature of possible worlds in the metaphysics of modality. We also register in this article, independently of our response to Turner and Capes, and in the spirit of Russell’s aforementioned paradox and many other versions thereof, a proof of the claim that there is no set of all true propositions one can render false. (shrink)

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)

Several theists, including Linda Zagzebski, have claimed that theism is somehow committed to nonvacuism about counterpossibles. Even though Zagzebski herself has rejected vacuism, she has offered an argument in favour of it, which Edward Wierenga has defended as providing strong support for vacuism that is independent of the orthodox semantics for counterfactuals, mainly developed by David Lewis and Robert Stalnaker. In this paper I show that argument to be sound only relative to the orthodox semantics, which entails vacuism, and give (...) an example of a semantics for counterfactuals countenancing impossible worlds for which it fails. (shrink)

We present a new puzzle about logical truth, necessity, and moral responsibility. We defend one solution to the puzzle. A corollary of our preferred solution is that prominent arguments for the incompatibility of determinism and moral responsibility are invalid.

I argue that we can achieve certainty about some empirical propositions. When someone is having a migraine and attending to it, she can be certain that she is in pain. I show that examples intended to undermine claims of certainty or to raise doubts about the reliability of introspection do not touch such cases. Traditional foundationalists have held that epistemically certain beliefs can serve as the basis for all one’s other justified beliefs. This is not so, because those beliefs that (...) are certain are spread too thinly to serve as broad justificatory foundations. Certainty has a different role. The best explanation for the existence of epistemically certain empirical beliefs is experientialism, the view that nondoxastic sensory experiences can justify beliefs. Experientialism then offers a framework for showing how the stream of sensory experiences can provide an adequate basis for the justification of our ordinary beliefs about the external world. (shrink)

We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point (...) on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case. (shrink)

Patrick Grim advances arguments meant to show that the doctrine of divine omniscience—the classical doctrine according to which God knows all truths—is false. In particular, we here have in mind to focus on two such arguments: the set theoretic argument and the semantic argument. These arguments due to Grim run parallel to, respectively, familiar paradoxes in set theory and naive truth theory. It is beyond the purview of this article to adjudicate whether or not these are successful arguments against the (...) classical doctrine of omniscience. What we are here interested in is a way in which these arguments can be generalized. In particular, we show how generalizations of those arguments can target, explicitly, alternatives to the classical doctrine of omniscience, including what we here call 'restricted omniscience' and 'open future open theism'. As a corollary, considerations of Grim-style arguments do not support these alternatives to the classical doctrine of omniscience over the classical doctrine. We conclude that what is paradoxical is not the classical doctrine of omniscience just as such; rather, what is paradoxical is a core commitment shared by the classical doctrine and its more modest alternatives, namely, the thesis that God is a perfectly logical reasoner. (shrink)

This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.

Cognitive scaffolding refers to external structures that change the cognitive demands of tasks. This dissertation begins by reviewing the literature around scaffolding—discussing key thinkers such as Hutchins (1995a), Clark (1997) and Sterelny (2003, 2010). I then develop the above scaffolding characterisation by drawing a distinction between shallow and deep scaffolding. Shallow scaffolding primarily involves cues, whereas deep scaffolding involves the significant offloading of cognitive work. By appealing to the complementarity thesis, I show that deep scaffolds can be explained through a (...) model of cognitive extension (Menary, 2006; Sutton, 2010). Most crucially, despite the abundance of benign/ neutral examples in the literature, I argue that scaffolding can also be ‘hostile’ to agents’ interests—benefitting one agent while undermining another. I draw on Sterelny (2003) to clarify my use of the word ‘hostile’ and also specify what I mean by ‘interests’. I then review authors with similar concerns to my own—i.e., the lack of discussion around external structures that negatively impact agents (Aagaard, 2020; Liao & Huebner, 2020; Slaby, 2016). Despite these authors’ concerns, the idea of ‘hostile scaffolding’ is not quite reached. I then present examples of shallow hostile scaffolding by reviewing how sunglasses (Viola, 2022) and casino interior design and ambience (Friedman, 2000; Schüll, 2012) can be hostile. Most importantly, I show that deep hostile scaffolding is a genuine concern (and not only a theoretical one) by reviewing gambling machines that use player tracking systems and virtual reel mapping (Schüll, 2012), as well as Twitter’s use of gamification (Nguyen, 2021). I close the dissertation by discussing avenues for future hostile scaffolding research. This includes the ethical implications deep hostile scaffolding, scaffolding’s role in forming addictive behavior, other instance of gamification, ‘racist scaffolding’, and hostile scaffolding in developing technologies (such as VR, dark patterns and AI). (shrink)

In Simultaneity and Delay: A Dialectical Theory of Staggered Time, the Canadian philosopher Jay Lampert challenges theories that define time in terms of absolute simultaneity and continuous succession. To counter these theories he introduces an alternative: the dialectic of simultaneity and delay. According to Lampert, this dialectic constitutes a temporal succession that is no longer structured as a continuous line, but that is built out of staggered time-flows and delayed reactions. The bulk of the book consists of an (...) attempt to give a conceptual order to the ‘unsystematic analyses of simultaneity and delay sprinkled through the history of philosophy’ (2). This conceptual analysis leads us through ancient (Plato and Plotinus), medieval (Origen) and late modern issues (Kant, Hegel and Lessing), as well as scientific discussions (Einstein, McTaggart), and culminates in the central chapter of the book, which attempts to show ‘how the problems of the great simultaneity philosophers - Husserl and Bergson - might be solved by the great delay philosophers - Derrida and Deleuze’ (147). In this review, I will focus on three points. 1. The problem of synchronization. 2. The problem of synthesis. 3. Theproblem of localization. (shrink)

Wilfrid Sellars's "Empiricism and the Philosophy of Mind" begins with an argument against sense-datum epistemology. There is some question about the validity of this attack, stemming in part from the assumption that Sellars is concerned with epistemic foundationalism. This paper recontextualizes Sellars's argument in two ways: by showing how the argument of EPM relates to Sellars's 1940s work, which does not concern foundationalism at all; and by considering the view of H.H. Price, Sellars's teacher at Oxford and the only classical (...) datum theorist to receive substantive comment in EPM. Timm Triplett has claimed that Sellars's discussion simply begs the question against Price, but this depends on the mistaken assumption that Sellars's concern is with foundationalism. On the contrary, Sellars's argument concerns the assumption that the innate capacity for sensory experience counts as "thinking in presence" in the way needed for empiricist accounts of content acquisition. Price's distinction between noticing universals and being aware of them encapsulates the tensions empiricists face here. (shrink)

Das Problem, wie der Völkermord an den Juden in der Nazizeit in der SBZ/ DDR behandelt wurde, war Gegenstand zahlreicher wissenschaftlicher Abhandlungen von Philologen und Historiken. In den Jahren 1990-2000 wurden die Arbeiten von u.a. Mario Kessler, Michael Wolffsohn, Angelika Timm, Thomas Jung und Ulrike Offenberg veröffentlicht. Die dort formulierten Thesen bilden die Grundlage für diesen Artikel. Er schildert die Lage der jüdischen DDR-Bewohner, die sich nach Kriegsende für das Leben in Deutschland entschieden haben. Es wurde versucht, die Kluft (...) zu veranschaulichen, die zwischen der von der Regierung betriebenen Propaganda und dem ostdeutschen Alltag bestand. Die im Artikel dargestellten politischen und ideologischen Grundlagen des ostdeutschen Staates, wie auch die Realität, in welcher die jüdischen DDR-Bürger gelebt haben, bilden wichtige Kontexte für das Schaffen von Jurek Becker. Die angedeuteten Probleme, die die jüdische Bevölkerung zu bewältigen hatte, wie etwa die Unmöglichkeit der Identitätsfindung in der scheinbar homogenen, in Wirklichkeit jedoch deutlich geteilten ostdeutschen Gesellschaft, kommen hier zur Sprache. Da Becker selbst von diesem Problem betroffen war, inspirierte ihn dies bei der Arbeit an seinen Romanen, Erzählungen und anderen literarischen Texten, die oft autobiographische Züge aufweisen. Dies wurde am Beispiel von Bronsteins Kinder dargestellt. Die existenziellen Probleme der jüdischen Bevölkerung wurden hier in Verbindung mit der Kritik der ostdeutschen Gesellschaft geschildert. (shrink)