This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

Reviewed Works:Gaisi Takeuti, Proof Theory.Georg Kreisel, Proof Theory: Some Personal Recollections.Wolfram Pohlers, Contributions of the Schutte School in Munich to Proof Theory.Stephen G. Simpson, Subsystems of $\mathbf{Z}_2$ and Reverse Mathematics.Solomon Feferman, Proof Theory: A Personal Report.

In der jüngeren Philosophie entbrannte eine Debatte über den Geltungsanspruch ethischer Aussagen. Wie sich zeigen lässt, treten im Spektrum der metaethischen Positionen dialektische Begründungsmuster im Stile der vorkantischen Metaphysik auf. Die vorliegende Studie untersucht, ob sich die Metaethik auf ähnliche Weise „therapieren“ lässt wie die Metaphysik zu Kants Zeiten. Sie entwickelt dazu eine transzendentale Architektonik der Moralphilosophie und übergibt die Aufgabe der Moralforschung an empirische Wissenschaften, wobei eine pragmatistische Methodologie gewährleistet, dass diese unterschiedlichen Perspektiven nahtlos ineinander übergehenden.

The traditional picture of logic takes it for granted that "valid arguments have a fundamental epistemic significance", but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments and acts (...) of inferences and that we have to reconsider the latter notion to arrive at the desired explanation. More precisely, the notions should be developed so that the following relationship holds: one gets in possession of a ground for a conclusion by inferring it from premisses for which one already has grounds, provided that the inference in question is valid. The paper proposes explications of the concepts of ground and deductively valid inference so that this relationship holds as a conceptual truth. Logical validity of inference is seen as a special case of deductive validity, but does not add anything as far as epistemic significance is concerned—it resides already in the deductively valid inferences. (shrink)

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus (...) the base theory are called the ‘Big Five’ and the associated equivalences are robust following Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’s higher-order RM, we obtain two new and long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems. Thus, higher-order RM is much richer than its second-order cousin, boasting at least two extra ‘Big’ systems. (shrink)

There are two main routes to a concept of (generalized) quantifier. The first starts from first‐order logic, FO, and generalizes from the familiar ∀ and ∃ occurring there. The second route begins with real languages, and notes that many so‐called noun phrases, a kind of phrase which occurs abundantly in most languages, can be interpreted in a natural and uniform way using quantifiers.

We characterize the generalized quantifiers Q which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$ , the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

This article provides an introduction to Kaplan-style semantics. The formal semantics usually employs the notion of a model, which, besides supplying the sets utterance contexts, circumstances and a domain M 0 of individuals, and also interprets the nonlogical atomic expressions of the language. Standard compositionality applies only to character: Funct makes immediate sense, since character assigns a semantic value directly to expressions. For semantic functions taking contextual arguments, the notion of compositionality must be revised. Notions of contextual compositionality apply directly (...) to the functional version of the setup that starts with the function char, and defines the other semantics from it. The point of evaluation semantics is indeed basic: if it is compositional, so is the content semantics, and the character semantics. (shrink)

Dag Prawitz The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 119-122, March 2013.

A semantics may be compositional and yet partial, in the sense that not all well-formed expressions are assigned meanings by it. Examples come from both natural and formal languages. When can such a semantics be extended to a total one, preserving compositionality? This sort of extension problem was formulated by Hodges, and solved there in a particular case, in which the total extension respects a precise version of the fregean dictum that the meaning of an expression is the contribution it (...) makes to the meanings of complex phrases of which it is a part. Hodges' result presupposes the so-called Husserl property, which says roughly that synonymous expressions must have the same category. Here I solve a different version of the compositional extension problem, corresponding to another type of linguistic situation in which we only have a partial semantics, and without assuming the Husserl property. I also briefly compare Hodges' framework for grammars in terms of partial algebras with more familiar ones, going back to Montague, which use many-sorted algebras instead. (shrink)

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed (...) in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier. (shrink)

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to (...) prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma. (shrink)

By a valid inference is here understood an inference that succeeds in its aim to justify its conclusion given that its premisses are already justified. For an inference to be valid it is thus not enough that the sentence asserted in the conclusion is a logical consequence of the sentences asserted in the premisses. A proof is understood as a succession of valid inferences that is closed (i.e. all its assumptions are discharged and all its free variables are bound by (...) some of the inferences). On the other hand, in mathematics the justification of an assertion has traditionally been taken to consist in a proof of it, which gives rise to an explanatory circle.In an attempt to avoid this circle, I have in previous works tried to explain the notion of ground for an assertion without referring to valid inferences. However, it turns out that the resulting notion of ground has unwanted consequences like the notion of proof in the so-called BHK-interpretation of intuitionistic logic. For instance, one is guaranteed a ground or a proof of an implication A → B as soon as there is in fact no ground or proof of A, regardless of whether one really knows this fact, and hence without having a real justification of the assertion of the implication.In the absence of a non-circular explanation of the notions of valid inference and proof, I suggest that we should try to formulate a number of principles of how these two epistemic notions depend on each other. A more careful explanation of the validity of an inference results in what I call a heuristic idea about the validity of inference, which is used as a guide to formulate precise principles. One such principle generalizes Gentzen’s idea that his introduction rules define the meaning of the logical constants. Another principle generalizes a previously noted property of Gentzen’s elimination rules that I have called that of satisfying the inversion principle. It results in a notion that I call non-creativity, which implies validity. (shrink)

This article points out problems in current dynamic treatments of anaphora and provides a new account that solves these by grafting Muskens' Compositional Discourse Representation Theory onto a partial theory of types. Partiality is exploited to keep track of which discourse referents have been introduced in the text (thus avoiding the overwrite problem) and to account for cases of anaphoric failure. Another key assumption is that the set of discourse referents is well-ordered, so that we can keep track of the (...) order in which they have been introduced, allowing a semantic characterization of anaphoric accessibility across stretches of discourse. Unlike other dynamic approaches, the system defines semantic values for unresolved anaphors. This leads to a clear separation of monotonic and non-monotonic content (in this case anaphoric resolution) and arguably provides a sound basis for a non-monotonic theory of anaphoric resolution. (shrink)

"Das Ringen ums Seyn" entwickelt eine Destruktion der Schriften Martin Heideggers von "Das Ende der Philosophie und die Aufgabe des Denkens" bis zu "Sein und Zeit". Dabei wird zunächst untersucht, ob die Annahme einer Kehre in diesem Denkprozess berechtigt ist. Im zweiten Teil wird vor diesem Hintergrund das "Denken des Selben" als zentrales Besinnungsmoment herausgestellt, um Rückschlüsse auf den Umgang Heideggers mit den Topoi Existenz, Eigentlichkeit und Apperzeption ziehen zu können.

All of metaethical positions today can be replaced by a universal architecture of moral philosophy, all but one: moral realism. Here, I use the term “metaethics” to refer to any theory of ethics concerning the groundwork of ethics, on the one hand, and the inquiry of the use of philosophical words, concepts or methods on the other. In this article, I will present my hypothesis that in moral philosophy, we do not need any specialized metaethics at all. Metaethics as a (...) discipline of philosophy is only required by the work of moral realists, who try to show us a realm of values and norms that exist (per se) naturally, non-naturally or supernaturally. How can they know? The effort of metaethical realists cannot be proven either in ontology or in the philosophy of language or in cognitive science or in any meta-science that works en plus to ethics, because even in every additional discipline, we have to accept the presupposition of a validity of judgments. So, let us try it the other way around; we have to find a way to found ethics by following its structures, and that means, based on David Velleman’s concepts: a) We have to search for a ubiquitous point of ethical theory in its foundation – here, no kind of value or norm can be found that is not based on a universal formal structure of normativity. b) We have to start an empirical inquiry to collect norms and values in actual use. MFT, moral psychology and moral sociology are in charge here. The combination of such an abstract groundwork with mere empirical study has to be legitimized again. Hence, I am going to try to sum up the main ideas of such a project to show the relevance of a new architecture of moral philosophy today. There is a line of reasoning that addresses the possibility of a transcendental critique in practical philosophy; therefore, it has to look into the different notions of “intuition” in moral methods like it was used by Sidgwick (Rashdall, Green, Ross, Brentano, McTaggart) and Moore on the one hand and Brentano and Bergson on the other. In my view, there is a way to combine these perspectives using the two-level-model of Hare, Singer, Greene, where “intuition” is used to categorize habits and customs of the common sense morality in general while a critical reflection uses act-utilitarian calculus to provide a universal decision – in the sense of “concrete reason” – for any possible actor in a singular situation (Hegel, Peirce, Bloch etc.). The change between these levels may be explained by means of a pragmatistic kind of continuum of research with an ideal summum bonum in the long run and a concept of common sense morality as can be found in every group or society. (shrink)

In the 12th century the "Book of the Soul" by the philosopher Avicenna was translated from Arabic into Latin. It had an immense success among scholastic writers and deeply influenced the structure and content of many psychological works of the Middle Ages. The reception of Avicenna's book is the story of cultural contact at an imipressively high intellectural level. The present volume investigates this successful reception using two approaches. The first is chronological, tracing the stages by which Avicenna's work was (...) accepted and adapted by Latin scholars. The second is doctrinal, analyzing the fortunes of key doctrines. The sense of the original Arabic text of Avicenna is kept in mind throughout and the degree to which his original Latin interpreters succeeded in conveying it is evaluated. (shrink)

Levinas did not present any new ethical theories; he did not even give any normative recommendations. But his phenomenological investigations help us to understand how the idea of ethics emerges and how we try to cope with it. The purpose of this paper is to suggest some implications from a reading of Levinas on how ethical challenges are handled within a management perspective. The paper claims that management, both in theory and in practice, is necessarily egocentric and thus ethically biased. (...) Therefore, ethics, defined as the idea that it is possible to do otherwise than pursue one's own interest, must have its source outside the scope of management. Levinas identifies this source as the call for responsibility from the Other, beyond the reach of managerial control. He also suggests how this idea of ethics is put into practice, especially in economic life: the existence of money makes it both possible and necessary to compare unique and incomparable others. Management tools may be used in the service of always more justice, as it also may be used for the opposite purpose. Instead of trying to draw a normative conclusion from Levinas, we may distinguish between two ways of experiencing reality: one through language and knowledge, covering both descriptive and normative knowledge, including business ethics, and the other through the particular events of encounters, from where this knowledge is continuously questioned through the call for responsibility, and thereby may provide possibilities for continuous improvements towards always more justice. (shrink)

The main question addressed in this chapter is how to analyze the modal ingredient in the concept of logical consequence or logical validity of an inference, here expressed by saying that the truth of the conclusion of a logically valid inference should follow by necessity of thought from the truth of the premisses. It is claimed that this modal ingredient is not taken care of by Tarski’s requirement, later developed in model theory, that for all interpretations of the non-logical terms (...) of the sentences, the conclusion is true if the premisses are. Instead we must bring in reasoning or the notion of proof in the analysis, and it is shown how one may develop a notion of proof or validity of an argument based on the constructive meaning of the logical constants. It is proposed that from a constructive point of view, a sentence is a consequence of a set of premisses if and only if there is a proof in the sense explained of the sentence from the set of premisses. Knowledge of such a proof compels us to hold the conclusion true given that we hold the premisses true. (shrink)

This is a reply to H. Ben-Yami, 'Generalized quantifiers, and beyond' (this journal, 2009), where he argues that standard GQ theory does not explain why natural language quantifiers have a restricted domain of quantification. I argue, on the other hand, that although GQ theory gives no deep explanation of this fact, it does give a sort of explanation, whereas Ben-Yami's suggested alternative is no improvement.

The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five (...) systems together are called the ‘Big Five’. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalizations and variations of the aforementioned third-order theorems fall far outside of the Big Five. (shrink)

We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...) as “almost countable”. This reduction of the uncountable to the finite/countable turns out to have a considerable logical and computational cost: we show that the aforementioned lemmas, and many related theorems, are extremely hard to prove, while the associated sub-covers are extremely hard to compute. Indeed, in terms of the standard scale (based on comprehension axioms), a proof of these lemmas requires at least the full extent of second-order arithmetic, a system originating from Hilbert–Bernays’ Grundlagen der Mathematik. This observation has far-reaching implications for the Grundlagen’s spiritual successor, the program of Reverse Mathematics, and the associated Gödel hierarchy. We also show that the Cousin lemma is essential for the development of the gauge integral, a generalization of the Lebesgue and improper Riemann integrals that also uniquely provides a direct formalization of Feynman’s path integral. (shrink)

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$. Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, (...) while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$. Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number. (shrink)

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may (...) consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting’s view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference. (shrink)