In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

Organs for transplantation are a scarce resource. Markedly, the transplant community’s primary challenge is the stark disparity between the number of patients awaiting deceased donor organ transplants and the rate at which organs become available. However, the allocation of a limited number of organs poses another constant challenge: maintaining an equilibrium between renal transplant utility and equity, that is, striking a balance between the utilitarian argument of medical efficiency and the principle of equity. In this comprehensive overview, the authors delve (...) into the challenge of maintaining an acceptable balance between equity and efficiency and elaborate on some of the factors that might inform a decisionmaker’s evaluation of the extent to which a given allocation scheme is efficient or equitable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume and Davidson on Pride Pall S. krdal In reading the Treatise one has to be alive to the fact that Hume gives certain crucial words new meanings. He does not always draw the reader's attention to this and sometimes explicitly claims to be using terms with their ordinarymeaningswhen heis clearlygiving the words special technical uses by expanding or contracting their usual meanings. "Passion," "love," "hatred," "pride," and "humility" (...) have special meanings in Hume's psychological scheme. He also sometimes uses important terms with more than one meaning, as in the case of "justice." The word "justice" both is the title of Book 3, Part 2 of the Treatise, which deals with a number ofartificial virtues, and alsorefers to the much narrower concept ofthe respectfor property. The following two questions should be borne in mind: "What did Hume have to say about what he calls justice, passion, love, etc.?" and "What did Hume have to say about what we refer to by the same terms?" Thus, Books 2 and 3 of the Treatise may be described as attempting to show how unbiased evaluations are possible, in spite ofthe fact that our passions are naturally biased and judgments of virtue and vice are passions. This clearly involves a discussion of how we can do justice to our own and other peoples' qualities ofmind or character. Hume, therefore, had something very important to say about what we call justice, but his contribution to issues involving what we call justice today is obscured by his terminology. In what follows I shall be claiming that Hume's account of pride throws light upon what we call pride although he clearly was not aiming to do so. Donald Davidson, in a footnote to "Hume's Cognitive Theory of Pride," points out that there is a close similarity between his reading ofHume on pride andmy own treatment ofprideinPassionand Value. Since the two accounts were arrived at completely independently, each ofus can see the other'sinterpretation as some confirmation ofhis own. The accounts are not identical, however, and without aiming at a comprehensive comparison, I shall attempt to decide whether one or two ofthe apparent differences between our accounts indicate substantial disagreement. I shall try not to confuse the following four questions: "What did Hume thinkconstitutes pride?""Whatlesson do I, and whatlesson does Davidson think one could learn from Hume's account?" and finally, "What constitutes pride?" Davidson and I agree that a lesson is to be Volume XV Number 2 387 PALL ARDAL learned from Hume's account, but disagree about the nature of this lesson because of a disagreement about the nature of pride. Both Davidson and I think that Hume's claim that pride is a simple impression is an embarrassment to him. It does not have any immediate appeal and also leads him to clearly misrepresent the relation between pride and its object. Pride has self as its object in the sense that it is a form of self-evaluation. Hume misrepresents this relation as a causal relation between pride and the thoughtofoneself. Whenever one is proud, one's thought is drawn to oneself, but according to Hume it couldhave been otherwise. Itjust so happens that the feeling ofpride makes one think of oneself, that when you are already proud your thought turns to yourself. Hume, it is true, does not always represent the relation between the object of pride and the passion itself as contingent. Much ofwhathe says suggests the more plausible view that pride is essentially a form of self-evaluation, but the lesson we can learn from Hume we can learn in spite ofhis atomism, so let us follow Davidson and disregard the atomism, while admitting its presence in Hume's account. Davidson points out that the pride Hume is talking about is pride for a reason. I follow Hume in talking about the cause ofpride, or what makes one proud, but the two accounts seem to me to be very similar on this issue. The two accounts are alsoin agreement thatHume should have said that the proud person must see the cause of pride as having value rather than as an independent source of pleasure, but we are... (shrink)

This little volume is a critical introduction to the phenomenological scene through discussion of the ideas of some of its more prominent exponents and an extensive analysis of the thought of its founder. About two thirds of the book is devoted to Husserl. It traces the evolution of Husserl's philosophy from an early interest in the psychological presuppositions of number, to the phenomenological analysis of acts of meaning, and finally to his unsuccessful attempt to construct a comprehensive system embracing the (...) presuppositions of knowledge in general. Previous acquaintance with Husserl's work is not necessary as Pivcevic gives critical expositions of the main theses of Husserl's Philosophy of Arithmetic, Logical Investigations, Ideas for a Pure Phenomenology and Phenomenological Philosophy, and The Crisis of European Sciences and Phenomenological Philosophy. He is careful to establish particular philosophic problems through reference to the views of other philosophers, and on different issues the position of Husserl is compared and contrasted to those of Frege, Brentano, Russell, Descartes, the positivists, the classical Empiricists and Kant. The presentation of Sartre's views includes brief discussions of European romanticism, Hegel, and Marx. Besides Sartre, Scheler and Heidegger are discussed to illustrate differing applications of the phenomenological method. The final chapter is a brief critique of the limitations of this method: the problem of "other egos," the marked tendency to anti-rationalism, and the failure to achieve its aim of a presupposition-less philosophy. Pivcevic is not totally sympathetic with the phenomenologists, which is perhaps what makes possible this critical exposition of their views. But at times one wishes his clear prose would follow a philosopher's thought further into its more obscure dimensions. His criticisms are often the standard, uninspired objections to what are generally accepted as unjustified excesses of existentialist thought. Nevertheless, this is an excellent introduction to phenomenology and especially valuable for its exposition of the thought of Edmund Husserl.--M. S. H. (shrink)

This long, diffuse, and intermittently quite interesting work is an attempt to explain and justify the process by which the concept of truth has been transformed in modern and contemporary philosophy from a doctrine of adequation via one of the true vision of things to that of the activity of the free subject. The main figures in this process are Descartes, Leibniz, Kant, and Hegel, although considerable space is devoted to Fichte, Schelling, Nietzsche, Husserl, Heidegger, Wittgenstein, and even K. O. (...) Apel and Hans Wagner. Despite the discussion of Wittgenstein and an odd comment or so about Frege, modern philosophy means primarily modern German philosophy; there is, however, some peripheral treatment of contemporary analytical thought. One may perhaps point to three main aspects of the author’s program. The first is his acceptance of the currently fashionable view that semantics is the orientating branch of philosophy. This thesis follows from the author’s acceptance of the historical rejection of Cartesian and Leibnizian intuition in favor of Kantian and Hegelian emphasis upon rationality as categoricity or conceptualism. The second aspect is Simon’s regular attempt to assimilate semantics into the doctrine of reflection. Hence the emphasis upon concepts shifts from that of objects to a second-order reflection upon the nature of conceptual thinking. The third aspect is in many ways the most comprehensive; at least Simon’s book culminates in its articulation. This is the attempt to assimilate the doctrine of reflection into a conception of freedom as a doctrine of both Recht and communication. I found especially valuable the emphasis upon the contribution of the subject to "objective" conceptions of meaning and truth. There are very good pages on the modern doctrine of intuition and the changing role assigned to imagination. The regular emphasis placed by Simon on negation as determination, and hence upon negation of negation as a crucial element even within non-dialectical logic, is also valuable. I was less persuaded by the treatment of intersubjectivity as an expression of a hermeneutics of communication within the context of a doctrine of individual rights. This Fichtean revision of Habermas and Apel may be the latest chapter in German academic philosophy of the traditional stream, but it does not follow even on Simon’s quasi-Hegelian principles that the text is sound. The deepest problem facing the author is to prevent the concept of truth from dissolving altogether into subjective reflection upon the limitations of objective definitions of truth. Stated somewhat differently, the author fails to persuade this reader that truth is actually the consensus of the community. Whatever the "positivist" deficiencies of Descartes and Leibniz, they do provide us with a concept of truth that is not easily mistaken for ideology.—S.R. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:CRUCIFIXION: ACCIDENT OR DESIGN? Sebastian Moore, O.S.B. Downside Abbey Lastyear I was visited by an old friend from my Liverpool days. Mike and I had worked together with the young of the parish, and one summer the two of us took a couple of boys camping in France, a trial of patience which made us known to each other at some depth. He was in fact a passionately convinced (...) Catholic, and very pastoral with it. We had many and long theological discussions. On his recent visit to Downside, I was showing him and some friends round our church, when, à propos ofquite what I don't remember, he made a remark about the Catholic cult of suffering, and said, "and it's all because ofthe crucifixion, isn't it? Why don't we recognize that the crucifixion was an accident?" I didn't react, because I was dumbfounded at what seemed to be the denial ofthe mystery ofthe cross by a friend who I had no reason to believe had stopped being a Catholic. And I'm sure he hasn't. He would certainly have told me ifhe had. It has taken me over a year to figure out what is going on here, and this would not have been possible without intensive study of the work of René Girard. The only alternative to saying that the crucifixion was an accident is to say that it was "ordained by God." It is only as ordained by God that this horrible event could be presented to people as a divine affirmation that suffering is to be sought and espoused. Also, it is only as ordained by God that the crucifixion can be thought of as "appeasing God" in some way. Both the moral and the dogmatic implication ofthe crucifixion—that suffering is a good thing, and that God required a victim—are grounded in the notion that God required the crucifixion ofhis Son. And it is because this idea is quite unacceptable, unworthy of God and ofthe human mind made in God's image, that my friend concludes that the crucifixion was not required by God, not written into the eternal scheme ofthings, but an accident. 156Sebastian Moore Once the problem is clarified in these terms, more and more instances come to mind, of rejection, by Christians and others, of the whole divine ordination ofthe cross. Gore Vidal refers to Christianity as "that lugubrious religion." Mary Hunt, a prominent American Catholic feminist, introduced a workshop with the statement, "we are not about naked guys nailed to crosses, blood all over the place!" Where divine ordering is accepted the special painfulness and darkness ofthe idea is noted. Anselm refers to the doctrine as the Narnia Chronicles, to commend the notion, by referring to a dark law written into the nature ofthings beyond our comprehension. The children's suggestion that Asian should simply not comply with the Snow Queen's requirement is rejected by Asian with a mysterious impatience—the children justhaven't grown up, like the romantics who, according to Matthew Arnold, "did not know enough." Again, during a day on "the roots of violence" that I led here yesterday there was much questioning about the role ofGod die Father in the crucifixion. The problem is crystallized in the famous cry from the cross, "Why have you forsaken me?" To my confessedly lame suggestion that Jesus was "only quoting a psalm," someone remarked, "to quote that psalm at thatjuncture and not mean it, is somewhatmisleading!" Again it's the same problem. How can God be understood as ordaining the crucifixion, compatibly with the notion of God as all-loving which Christianity of course affirms? Hans Urs Von Balthasar, who is 90% pro-Girard, in the other 10% wants to hold on to something he calls "the wrath of God." Thus the seeming necessity of having God require and ordain the crucifixion is the main crisis of Christian belief. The six-year-old Willie, in Thomas Klise's amazing novel, The Last Western, puts the problem with childlike directness. He keeps interrupting catechism class with, "Why did God kill Jesus?" and is shushed by Sister. I recall that Illtyd Trethowan said to... (shrink)

The article discusses the reasons why modern formal semantics of natural language is an integral part of a larger philosophical research program for the study of the nature of intentionality. The purpose of this article is to show how research in the field of formal semantics of natural language became the implementation of a large philosophical research program that is focused on the nature of intentional objects, which since the time of F. Brentano have been considered an integral part of (...) mental phenomena. Mental phenomena are opposed to physical phenomena due to their inherent intentionality. In modern philosophy, it is commonly considered that the comprehension of the nature of intentionality is the key to understanding the nature of consciousness, and, consequently, the nature of man. The article shows that formal semantics represents the structure of the meaning of linguistic expressions, with the help of which human thought is formed, which is one of the main objects of intentional consciousness. In formal semantics, this representation is carried out in a rigorous, systematic, and empirically verifiable way on the basis of natural languages, that is, on those symbolic systems that real people use in real practice to shape their thoughts. The research is based on the works of G. Frege, who laid the foundation for the formal-semantic methodology. Thus, the answer is given to the question of why modern formal linguistics, which originates in the analytical philosophy of the 20th century, is a philosophically significant discipline. This response is a reaction to the objections of skeptics who believe that modern analytic philosophy cannot be considered a direction that has any serious philosophical significance. (shrink)

The purpose of this paper is mainly to give a model of paraconsistent logic satisfying the "Frege comprehension scheme" in which we can develop standard set theory (and even much more as we shall see). This is the continuation of the work of Hinnion and Libert.

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which (...) defines a model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (shrink)

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These (...) ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom. (shrink)

One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted , that is, which are being used to express judgements . There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of (...) the judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement , rather than – as in the typical contemporary view – of consequence relations amongst certain objects (propositions or well-formed formulae). The paper also explains why Frege's use of the judgement stroke is not in conflict with his avowed anti-psychologism, and why Wittgenstein's criticism of the judgement stroke as 'logically quite meaningless' is unfounded. The key point here is that while the judgement stroke has no content , its use in logic and mathematics is subject to a very stringent norm of assertion. (shrink)

The INBIOSA project brings together a group of experts across many disciplines who believe that science requires a revolutionary transformative step in order to address many of the vexing challenges presented by the world. It is INBIOSA’s purpose to enable the focused collaboration of an interdisciplinary community of original thinkers. This paper sets out the case for support for this effort. The focus of the transformative research program proposal is biology-centric. We admit that biology to date has been more fact-oriented (...) and less theoretical than physics. However, the key leverageable idea is that careful extension of the science of living systems can be more effectively applied to some of our most vexing modern problems than the prevailing scheme, derived from abstractions in physics. While these have some universal application and demonstrate computational advantages, they are not theoretically mandated for the living. A new set of mathematical abstractions derived from biology can now be similarly extended. This is made possible by leveraging new formal tools to understand abstraction and enable computability. [The latter has a much expanded meaning in our context from the one known and used in computer science and biology today, that is "by rote algorithmic means", since it is not known if a living system is computable in this sense (Mossio et al., 2009).] Two major challenges constitute the effort. The first challenge is to design an original general system of abstractions within the biological domain. The initial issue is descriptive leading to the explanatory. There has not yet been a serious formal examination of the abstractions of the biological domain. What is used today is an amalgam; much is inherited from physics (via the bridging abstractions of chemistry) and there are many new abstractions from advances in mathematics (incentivized by the need for more capable computational analyses). Interspersed are abstractions, concepts and underlying assumptions “native” to biology and distinct from the mechanical language of physics and computation as we know them. A pressing agenda should be to single out the most concrete and at the same time the most fundamental process-units in biology and to recruit them into the descriptive domain. Therefore, the first challenge is to build a coherent formal system of abstractions and operations that is truly native to living systems. Nothing will be thrown away, but many common methods will be philosophically recast, just as in physics relativity subsumed and reinterpreted Newtonian mechanics. -/- This step is required because we need a comprehensible, formal system to apply in many domains. Emphasis should be placed on the distinction between multi-perspective analysis and synthesis and on what could be the basic terms or tools needed. The second challenge is relatively simple: the actual application of this set of biology-centric ways and means to cross-disciplinary problems. In its early stages, this will seem to be a “new science”. This White Paper sets out the case of continuing support of Information and Communication Technology (ICT) for transformative research in biology and information processing centered on paradigm changes in the epistemological, ontological, mathematical and computational bases of the science of living systems. Today, curiously, living systems cannot be said to be anything more than dissipative structures organized internally by genetic information. There is not anything substantially different from abiotic systems other than the empirical nature of their robustness. We believe that there are other new and unique properties and patterns comprehensible at this bio-logical level. The report lays out a fundamental set of approaches to articulate these properties and patterns, and is composed as follows. -/- Sections 1 through 4 (preamble, introduction, motivation and major biomathematical problems) are incipient. Section 5 describes the issues affecting Integral Biomathics and Section 6 -- the aspects of the Grand Challenge we face with this project. Section 7 contemplates the effort to formalize a General Theory of Living Systems (GTLS) from what we have today. The goal is to have a formal system, equivalent to that which exists in the physics community. Here we define how to perceive the role of time in biology. Section 8 describes the initial efforts to apply this general theory of living systems in many domains, with special emphasis on crossdisciplinary problems and multiple domains spanning both “hard” and “soft” sciences. The expected result is a coherent collection of integrated mathematical techniques. Section 9 discusses the first two test cases, project proposals, of our approach. They are designed to demonstrate the ability of our approach to address “wicked problems” which span across physics, chemistry, biology, societies and societal dynamics. The solutions require integrated measurable results at multiple levels known as “grand challenges” to existing methods. Finally, Section 10 adheres to an appeal for action, advocating the necessity for further long-term support of the INBIOSA program. -/- The report is concluded with preliminary non-exclusive list of challenging research themes to address, as well as required administrative actions. The efforts described in the ten sections of this White Paper will proceed concurrently. Collectively, they describe a program that can be managed and measured as it progresses. (shrink)

The relevance of the problem under study is due to the need to improve the practical skills and competencies of students in the course of training in order to prepare them for competition with other job seekers in employment. In this regard, this article is aimed at identifying the expediency of students’ participation in competitive selections and grants as a factor that creates conditions for effective practice-oriented learning. The leading method for the study of this problem is a pedagogical experiment, (...) which allows us to comprehensively consider the relationship between students’ participation in competitions and their further self-realization and development as young professionals. The article presents the results of monitoring the behavior of students who received an offer to participate in the university-wide competitive selection of student projects, implemented as part of the Program for Supporting Applied Research and Development of Students and Postgraduates of Petrozavodsk State University. The materials of the article can be useful for teaching staff in building a scheme for teaching students and services involved in the recruitment of personnel for large and medium-sized companies that require young professionals to have practical work skills, teamwork skills and creativity. (shrink)

According to Peter Aczel, the inconsistency of Frege’s system in Grundgesetze is due, not to the introduction of sets, as is usually thought, but to the introduction of the Horizontal. His argument is that the principles governing sets are intuitively correct and therefore consistent, while the scheme introducing the Horizontal amounts to an internal definition of truth conflicting with Tarski’s classic result on the undefinability of truth in the object language. The aim of this paper is to show that (...) the Horizontal is innocent: Aczel’s diagnosis is based on a mistaken view of the structure underlying Frege’s ideal language. (shrink)

Summary. This article develops two principal points. First, the so-called rivals of logical foundations, associated with Zermelo, Hilbert, and Brouwer, are here regarded as variants; specifically: to simplify, refine, resp. extend Frege’s scheme. Each of the variations is seen as a special case of a familiar strategy in the pursuit of knowledge. In particular, the extension provided by Brouwer’s intuitionistic logic concerns the class of propositions considered: about incompletely defined objects such as choice sequences. In contrast, Frege or, for (...) that matter, Aristotle thought that only propositions about precisely defined terms lent themselves to anything like logical theory.. This view of intuitionistic logic is fairly consistent with the mature Brouwer’s own research practice, but not at all with his rhetoric. Secondly, the article explains the silent majority’s scepticism about all variants of logical foundations, and, in note 7, sketches a genuine alternative which is almost explicit in the manifesto [#3] of Bourbaki with the aim of exploring the architecture of mathematics and our intuitive resonances to it. The alternative sees the weakness of logical foundations not in any logical defects, such as lack of precision, but in its sterility: stress on the logical elements of mathematics and mathematical reasoning draws attention away from discoveries of more significant features; more significant even for validity and reliability, though these matters are less prominent here than in logical foundations. The weakness in question is relevant beyond the present topic of foundations because this kind of weakness affects the whole analytic tradition in philosophy. Finally, for reasons explained there, an Appendix goes into some relations between the present paper and [17]. (shrink)

It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my (...) view that, say, Delta-3-1 comprehension axioms are not logical truths. What I am going to suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic". (shrink)

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further (...) conjectured that the more encompassing Δ₁¹-comprehension schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

This is the first single-volume edition and translation of Frege's philosophical writings to include his seminal papers as well as substantial selections from ...

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further (...) conjectured that the more encompassing Δ11-comprehension schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

"By looking at Frege's lectures on logic through the eyes of the young Carnap, this book casts new light on the history of logic and analytic philosophy. As two introductory essays by Gottfried Gabriel and by Erich H. Reck and Steve Awodey explain, Carnap's notes allow us to better understand Frege's deep influence on Carnap and analytic philosophy, as well as the broader philosophical matrix from which both continental and analytic styles of thought emerged in the 20th century."--BOOK (...) JACKET. (shrink)

Translation of Frege's 'Über Begriff und Gegenstand' (1892). Translation by Peter Geach, revised by Max Black.

This volume contains all of Frege's extant unpublished writings on philosophy and logic other than his correspondence, written at various stages of his career.

We provide an overview of consistent fragments of the theory of Frege’s Grundgesetze der Arithmetik that arise by restricting the second-order comprehension schema. We discuss how such theories avoid inconsistency and show how the reasoning underlying Russell’s paradox can be put to use in an investigation of these fragments.

Dieser Band enthält die vier Arbeiten Freges: Begriffsschrift, eine der arithmetischen nachgebildeten Formelsprache, 1879; Anwendungen der Begriffsschrift, 1879; Über den Briefwechsel Leibnizens und Huggens mit Papin, 1881; Über den Zweck der Begriffsschrift, 1883; Über die wissenschaftliche Berechtigung einer Begriffsschrift, 1882. Frege's research work in the field of mathematical logic is of great importance for the present-day analytic philosophy. We actually owe to Frege a great amount of basical insight and exemplary research, which set up a new standard also in (...) other fields of knowledge. As the founder of mathematical logic he severely examindes the syllogisms on which arithmetic is built up. In doing so, Frege recognized that our colloquial language is inadequate to define logic structures. His notional language corresponded to the artaivicial logical language demandes by Leibniz. Frege's achievement in the field of logic were so important, that they radiated into the domain of philosophy and influenced the development of mathematical logic decisively. (shrink)

[Translation of Frege's 'Gedankengefüge' (1923)].

The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It (...) was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy. (shrink)

[Translation of Frege's 'Über das Trägheitsgesetz].

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

Translation of Frege's 'Über die Begriffsschrift des Herrn Peano und meine eigene' (1896). Translated by V.H. Dudman.

(Translation of Frege's Grundgesetze II, §§ 86-137).

Translation of Frege's 'Über den Zweck der Begriffsschrift' (1882).

(Translation of Frege's "Le nombre entier", 1895).

Dieser Band enthält die vier Arbeiten Freges: Begriffsschrift, eine der arithmetischen nachgebildeten Formelsprache, 1879; Anwendungen der Begriffsschrift, 1879; Über den Briefwechsel Leibnizens und Huggens mit Papin, 1881; Über den Zweck der Begriffsschrift, 1883; Über die wissenschaftliche Berechtigung einer Begriffsschrift, 1882. Frege's research work in the field of mathematical logic is of great importance for the present-day analytic philosophy. We actually owe to Frege a great amount of basical insight and exemplary research, which set up a new standard also in (...) other fields of knowledge. As the founder of mathematical logic he severely examindes the syllogisms on which arithmetic is built up. In doing so, Frege recognized that our colloquial language is inadequate to define logic structures. His notional language corresponded to the artaivicial logical language demandes by Leibniz. Frege's achievement in the field of logic were so important, that they radiated into the domain of philosophy and influenced the development of mathematical logic decisively. (shrink)

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 (...) has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles. (shrink)

Frege's essay "Der Gedanke.Eine logische Untersuchung" was first published in the Beitrage zur Philosophie des Deutschen Idealismus for 1918-1919 and is one of three related logical studies published as a complete work by Gunther Patzig entitled Logische Untersuchungen in Gottingen, 1966 .

Frege's philosophy of language includes detailed views on judgments. His formal logic - the Begriffsschrift - documents some of these views in the introduction and treatment of the judgment stroke. In current logic such an expression is either entirely ignored or, appearing as turnstile, plays an fundamentally different role. In this paper I put forward four claims: (i) Considering Frege's Begriffsschrift, it is methodologically palpable why the judgment stroke was omitted in nearly all logical systems developed after Frege. (...) (ii) The Frege-style inclusion of the judgment stroke in a formal language represents a partial implementation of results from speech act theory. (iii) A more comprehensive implementation, not limited to the judgment stroke, helps to satisfy justified expectations about formal languages and logical calculi. (iv) Such an implementation has not yet been achieved in the current logical mainstream. (shrink)

The dissertation seeks to gain a comprehensive understanding of Frege's watershed work of logic, the 1879 Begriffsschrift . The treatise is the first since Aristotle's instauration of the discipline to have succeeded in completely recasting logic. Very generally stated, the dissertation aims at making explicit the conceptual origins of modern logic. To do that will require that each of the achievements of Concept-script be examined from one or more vantage points: either within the perspective of Frege's larger and (...) guiding ambition, that of providing a logical justification of arithmetic; or with respect to the horizon of modernity; or in relation to later developments in logical theory and the philosophy of language, developments stimulated, albeit not exclusively, by Frege's very work. The vantage point adopted in each case will depend on the nature of the topic and the extent to which it requires to be made explicit. One perspective will have preponderance, that of Frege's relationship to his early modern ancestors, a group that includes philosophers, scientists, and mathematicians. Since modernity is constituted in a nearly constant critical, if not polemical, relationship to the Ancients, they will also at times be drawn into the discussion. The elucidation of Frege's breakthrough work will be guided by the following thesis: Frege's attempt to justify arithmetic logically and the fine work of clarification he produced in actualizing this project; his discovery of function-theoretic logic and his fashioning of a concept-script ; and his critique of natural language and the attendant semantics developed over the course of that critique constitute an unfolding and an extension of the primacy of universal method over ontology that is the signature of modern inquiry. (shrink)

This new book offers a comprehensive and accessible introduction to Frege's remarkable philosophical work, examining the main areas of his writings and demonstrating the connections between them. Frege's main contribution to philosophy spans philosophical logic, the theory of meaning, mathematical logic and the philosophy of mathematics. The book clearly explains and assesses Frege's work in these areas, systematically examining his major concepts, and revealing the links between them. The emphasis is on Frege's highly influential work in (...) philosophical logic and the theory of meaning, including the features of his logic, his conceptions of object, concept and function, and his seminal distinction between sense and reference. Frege will be invaluable for students of the philosophy of language, philosophical logic, and analytic philosophy. (shrink)

In this paper, the authors discuss Frege's theory of "logical objects" and the recent attempts to rehabilitate it. We show that the 'eta' relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the 'eta' relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses (...) unrestricted Comprehension for Logical Objects and banishes encoding formulas from Comprehension for Properties. The relative mathematical and philosophical strengths of the two theories are discussed. Along the way, new results in the theory of abstract objects are described, involving: the theory of extensions, the theory of directions and shapes, and the theory of truth values. (shrink)

It is Frege's third contribution that makes the point of departure for the present paper. Not merely did Frege show how to manipulate symbols more exactly; he also gave a searching account of what these symbols mean. Consider a philosophical problem that arises out of the simplest arithmetic. When we say that 5 = 2 + 3, what do we mean? Do we mean that 5 is identical with 2 + 3? But in some ways 5 and 2 + (...) 3 are obviously different. Or do we mean that 5 and 2 + 3 are equal but not identical, equality being a relation that falls short of complete identity? But in that case, some ordinary ways of speaking in mathematics must be false. Suppose a pupil is asked for the positive square root of 25. The phrasing of the question implies that there is one and only one. No doubt he may answer, '5'. But then it follows that the answer '2 + 3' is not allowable, since ex hypothesi 5 and 2 + 3, though equal, are not identical. Yet ordinarily the answer '2 + 3' would be regarded as strange but not as wrong. (shrink)

Urban infrastructures serve as the backbone of modern economies, mediating global exchanges and responding to urban demands. Yet, our comprehension of these complex structures, particularly within diverse socio-political terrain, remains fragmented. In bridging this knowledge gap, this study delves into “boundary objects”—entities enabling diverse stakeholders to collaborate without a comprehensive consensus. Central to our investigation is the hypothesis that oceanic infrastructural developments are instrumental in molding the interface of urban, industrial, and energy sectors within marine contexts. Our lens is (...) directed at the Gulf of Mexico, which is distinguished by its industrial depth and expansive marine grid. We highlight the Gulf Intracoastal Waterway’s (GIWW) paramount role in regional movement and the ecological facets of practices such as dredging, which is vital for transport and coastal conservation. A striking revelation of our study is the transformation of offshore structures in the Gulf into vibrant marine habitats. Emphasizing the intertwined nature of marine infrastructures, we denote oceans as pivotal platforms for impending urban expansion, especially as land resources wane. Our research aspires to validate the role played by oceans as a nexus in the urban–industrial–energy fusion. (shrink)