Depuis la premiere edition de cet ouvrage en 1958, l'interet porte a la perception de la musique n'a cesse de croitre. Le domaine de recherche tel qu'il se presente aujourd'hui est, heureusement, degage des clivages institutionnels. Il associe psychologues, acousticiens, theoriciens de la musique, informaticiens, neuro-scientifiques et tous ceux qui contribuent au progres de cette discipline [...] L'ouvrage de R. Frances, par l'influence qu'il a exerce, est devenu un classique. Il analyse de maniere detaillee, entre autre, les effets de l'harmonie (...) tonale sur les perceptions des formes melodiques, le processus engage dans l'attribution d'une tonalite, et la representation abstraite des notes et de leur relation avec une tonalite. On y trouvera egalement plusieurs descriptions des differents effets melodiques sur la memoire et de la perception categoriale des notes et des intervalles, ainsi que de stimulantes discussion a propos des schemes melodiques et du role de l'attention dans la perception auditive. L'ouvrage respectte un equilibre judicieux entre les aspects culturels et les aspects biologiques de la question. Tandis qu'il insiste sur l'influence de la duree de l'ecoute et de la pratique du melomane sur la perception de la musique, R. Frances s'appuie sur la litterature concernant les amusies pour illustrer l'analyses des types d'abstractions cognitives executees pour l'auditeur. Prof. Diana Deutsch (Department of Psychology, University of California, San Diego). (shrink)

This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a given type (...) , Σ is a higher-order operator denoting a function from items of the given type to objects in the range of the first-order variables, and E is an equivalence relation over items of the given type. In what follows, I will sometimes omit the initial quantifiers.Frege [1884, 1893] himself employed three abstraction principles. One of them, used for illustration, comes from geometry: "The direction of l1 is identical to the direction of l2 if and only if l1 is parallel to l2."Call this the direction principle. The second was dubbed N= in [Wright, 1983] and is now called Hume’s principle: Formula where F≈G is an abbreviation of the second-order statement that there is a one-to-one relation mapping the F’s onto the G’s. In words, states that the number of F is identical to the number of G if and only if F is equinumerous with G. Unlike the direction-principle, the relevant variables, F, G here are second-order. Georg Cantor …. (shrink)

The traditional understanding of abstraction operates on the basis of the assumption that only entities are subject to thought processes in which particulars are disregarded and commonalities are lifted out (the so-called method of genus proximum and differentia specifica). On this basis Frege criticized the notion of abstraction and convincingly argued that (this kind of) “entitary- directed” abstraction can never provide us with any numbers. However, Frege did not consider the alternative of “property- abstraction.” In this (...) article an argument for this alternative kind of abstraction is formulated by introducing a notion of the “modal universality” of the arithmetical and by developing it in terms of the distinction between type-laws (laws for entities – applicable to a limited class of entities) and modal laws (obtaining for every possible entity without any restriction). In order to substantiate this argument a case is made for the acceptance of an ontic foundation for the arithmetical (and other modes or functions of reality – with special reference to Cassirer, Bernays, Gödel and Wang), which, in the final section, serves to give an ontological account of (i) the connections between the arithmetical and other aspects of reality and (ii) the applicabality of arithmetic. In the course of the argument the impasse of logicism is briefly highlighted, while a few remarks are made with regard to the logical subject- object relation in connection with Frege's view that number attaches to a concept. S. Afr. J. Philos. Vol.22(1) 2003: 63-80. (shrink)

This study of Western philosophic systems, their types, history, relations, and projected future in the next half century, stems from Robert S. Brumbaugh’s forty-year fascination with the paradox of the many consistent overarching systems of ideas that are nevertheless mutually exclusive. Brumbaugh argues that when we isolate these systems’s patterns and look at them more abstractly, they consistently fall into four main types, and the interaction of these four types of explanation and order is a dominant theme in the history (...) of Western philosophy. In Brumbaugh’s view these four philosophic systems are not, as some critical historians and thinkers have claimed, so different that they are mutually unintelligible, forcing us to make a choice among them that is entirely arbitrary. But neither are they, as a majority of past thinkers and historians have hoped, simply parts of some single "right" or "orthodox" scheme. Their mutual understanding requires a method of transformation that interprets one to another without destroying their diversity. The history of Western philosophy from the fifth century A.D. to the present shows a pattern of alternating revolutions in systematic method and direction of explanation. Brumbaugh feels that the pattern is continuing in a change toward a revised Platonism, just beginning with the twenty-first century. He anticipates that it will be a Platonism of a new texture, one that has matured and learned a great deal in the course of the adventures of its ideas through space and time. (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)

In lieu of an abstract, here is a brief excerpt of the content:Going to our Happy Place: Idealism, Realism, and Nishida's Eutopia: A Response to Christian UhlChristopher S. Goto-JonesWords can be tricky things; their significance is often found in unexpected places. The word 'utopia,' for example, is usually considered to have originated in the early sixteenth century in Louvain when Thomas More fused two Greek words (ou, 'not,' and topos, 'place'). The result was a new, Greek-sounding compound, utopia, translatable as (...) 'Noplace.' Apparently, it was a small, newly discovered island where people did things differently from Europeans. By a happy coincidence, utopia was almost homophonous with another Greek compound, eutopia, which might be translated as 'Happyplace.' The word has come to be used to describe models of nonexistent societies that are imagined to be better than the ones in which we live.In his stimulating, thorough, and carefully crafted review of my recent book, Political Philosophy in Japan: Nishida, the Kyoto School, and Co-Prosperity, Christian [End Page 482] Uhl makes a great deal of my claim that Nishida's political project was utopian. He also takes issue with my characterization of Nishida as a political idealist. In the interests of clarification and in the spirit of engaged debate, I think that it is important to consider what these important (and often complicated) terms mean. In particular, the idea that 'idealist' and 'utopian' are coterminous should not be allowed to stand unchallenged. In a related way, it may be helpful to clarify what is meant by 'realist' in political thought, since Uhl claims that, far from being an idealist, Nishida was actually a realist "in a very fundamental sense of the word."Uhl is quite right to observe that I identify Nishida's political scheme as utopian in the sense that he provides a model of an ideal society as a means of demonstrating the problems in the society in which he lived.1 However, when it comes to the effectiveness of Nishida's utopian vision qua dissent in wartime Japan (or simply qua practice), the key problem slides from being utopianism itself to idealism. A utopia is an ideal society; the method required to realize that society is not necessarily part of the vision itself, although it may be implied in the structure. Consider a socialist utopia: for Marx, the road to this perfect society was paved by historical materialism; for a number of other socialists (including many Japanese socialists), the route to a broadly similar society was defined in humanist or even idealist terms. In a rather casual sense we might label the later Marx a realist and, say, Kawakami Hajime as an idealist. Nishida's utopia is no simpler,2 although it does, in fact, contain a sense of process: it is defined as a type of self-realization and it is premised upon an underlying faith in the power of the mind to transform both the self and the everyday world.I would contend that Nishida was an idealist (a thinker who works on the principle that the reality in which we live is somehow mind-correlative), although not a simple idealist, and also a political utopian. A number of his students, for example Nishitani and Kōyama, broadly shared their mentor's goals but differed in their construction of the paths to them. The quotation from Nishitani that Uhl selects reveals this clearly: he argues that if we do not "have both feet firmly grounded in the consciousness of the past as well as in the practice of the present, any idea will remain a mere concept, a utopian ideal."3 Nishitani is not taking issue with the utopia; he is challenging the process appropriate to its realization. He argues (in common with many of the other second-generation members of the Kyoto School) that an idealist philosophy that pays no attention to the significance of history can offer no blueprints for the achievement of its ideals in reality. The implied challenge to Nishida's thought here is to his understanding of the forces of historical change: idealist self-power (jiriki) is laid bare as ahistorical and inadequate as an engine for social change. Having said that, however, Nishitani does not make... (shrink)

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a (...) widespread debate in Germany on ‘directions’ as a central notion in the theory of parallels. (shrink)

Moritz Pasch is usually seen today as a precursor of Hilbert. The Vorlesungen über neuere Geometrie is indeed one of the few works Hilbert referred to in the Grundlagen. Unfortunately, Hilbert's epoch-making book has eclipsed Pasch's achievement; so much so that Pasch's Vorlesungen has not been yet translated into English. But as Pollard emphasizes, it would be a mistake to reduce Pasch's research to his work on the foundations of geometry. Pasch published another book on the foundations of real analysis (...) and arithmetic in 1882, and two other works, respectively, in 1909 and in 1914 . Pasch thus aimed not only to analyze the foundations of geometry but also to elaborate a complete theory of mathematical knowledge. This ambition brought him close to the logicists . But, unlike them, Pasch was a resolute empiricist: he considered that ‘geometry and number theory are branches of empirical science’ 1 and that all mathematical concepts must be derived from a set of notions abstracted from our experience.Pasch made his whole career in Giessen — that is, outside of the major German research centers. Heavily burdened with administrative duties, Professor Pasch seems to have found time for his research only in 1917, when he retired at the age of 67. As Pollard notes, nearly half of Pasch's published output appeared after his sixty-eighth birthday! This volume is a translation of fourteen articles dating from the later period of Pasch's life, ordered by the time of their publication. The papers are preceded by a substantial introduction by Stephen Pollard, the editor and translator of all the papers. Aside from his correspondence with Frege, this volume is thus the first translation of Pasch's work …. (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)

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)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal (...) axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (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)

A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result.

I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which (...) it may successfully be applied. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

One of the remarkable results of Frege’s Logicism is Frege’s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume’s Principle (HP) without using Frege’s Basic Law V. This result was rediscovered by the Neo-Fregeans and their allies. However, when applied in developing a more advanced theory of mathematics, their fundamental principles—the abstraction principles—incur some problems, e.g., that of inflation. This paper finds alternative paths for such inquiry in extensionalism and object theory.

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in (...) this extended system. (shrink)

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...) what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (shrink)

In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of concept-correlates. Frege held that, where n > 2, there is a one—one correlation between each n-level function and an n—1 level function, and a one—one correlation (...) between each first-level function and an object. Applied to cardinals, the correlation offers new answers to some perplexing features of Frege's philosophy. It is shown that within Frege's concept-script, a generalized form of Hume's Principle is equivalent to Russell's Principle ofion — a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege's rejection of definition of cardinal number by Hume's Principle parallels Russell's objection to definition by abstraction. Frege's correlation thesis reveals that he has a way of meeting the structuralist challenge that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals. (shrink)

This paper presents a new interpretation of Frege's context principle on which it applies primarily to singular terms for abstract objects but not necessarily to singular terms for ordinary objects.

ABSTRACT In this paper, I argue that a number of influential Millian responses to Frege’s puzzle, which consist in denying that Frege’s data apply to natural languages, are not viable if logic is to play its role in legitimizing the logical appraisal of rational subjects. A notion of validity which does justice to the normativity of logic must make room for a distinction between valid inferences and enthymemes. I discuss the prospects of formal, relevant and manifest validity as candidates for (...) a notion which complies with this desideratum. Their success, or failure is argued to hang on the viability of a semantical account of de jure co-reference, which is in tension with standard Millian tenets. I conclude that these Millian theories face the following dilemma: either accept that there is no notion of logical validity which makes logic normative for reasoning, thus jeopardizing our well entrenched practices of rational appraisal; or accept that de jure co-reference is a real semantical relation. (shrink)

Abstract I argue against the two most influential readings of Frege's methodology in the philosophy of logic. Dummett's ?semanticist? reading sees Frege as taking notions associated with semantical content?and in particular, the semantical notion of truth?as primitive and as intelligible independently of their connection to the activity of judgment, inference, and assertion. Against this, the ?pragmaticist? reading proposed by Brandom and Ricketts sees Frege as beginning instead from the independent and intuitive grasp that we allegedly have on the latter (...) activity and only then moving on to explain semantical notions in terms of the nature of such acts. Against both readings, I argue, first, that Frege gives clear indication that he takes semantical and pragmatical notions to be equally primitive, such that he would reject the idea that either sort of notion could function as the base for a non-circular explanation of the other. I argue, secondly, that Frege's own method for conveying the significance of these primitive notions?an activity that Frege calls ?elucidation??is, in fact, explicitly circular in nature. Because of this, I conclude that Frege should be read instead as conceiving of our grasp of the semantical and pragmatical dimensions of logic as far more of a holistic enterprise than either reading suggests. (shrink)

In this note the claim is defended that Frege was a realist in the sense that he attributed causal efficacy to certain abstract objects. The arguments of Dummett and Sluga (cf. Inquiry, Vols. 18, 19, and 20 [1975–77]) to the contrary are criticized.

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)

Abstract Though it has been claimed that Frege's commitment to expressions in indirect contexts not having their customary senses commits him to an infinite number of semantic primitives, Terrence Parsons has argued that Frege's explicit commitments are compatible with a two-level theory of senses. In this paper, we argue Frege is committed to some principles Parsons has overlooked, and, from these and other principles to which Frege is committed, give a proof that he is indeed committed to an (...) infinite number of semantic primitives?an intolerable result. (shrink)

The volume is the first collection of essays that focuses on Gottlob Frege's Basic Laws of Arithmetic (1893/1903), highlighting both the technical and the philosophical richness of Frege's magnum opus. It brings together twenty-two renowned Frege scholars whose contributions discuss a wide range of topics arising from both volumes of Basic Laws of Arithmetic. The original chapters in this volume make vivid the importance and originality of Frege's masterpiece, not just for Frege scholars but for the study (...) of the history of logic, mathematics, and philosophy. (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)

We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, (...) in the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning. (shrink)

The main objective of this paper is to examine how theories of truth and reference that are in a broad sense Fregean in character are threatened by antinomies; in particular by the Epimenides paradox and versions of the so-called Russell-Myhill antinomy, an intensional analogue of Russell’s more well-known paradox for extensions. Frege’s ontology of propositions and senses has recently received renewed interest in connection with minimalist theories that take propositions (thoughts) and senses (concepts) as the primary bearers of truth and (...) reference. In this paper, I will present a rigorous version of Frege’s theory of sense and denotation and show that it leads to antinomies. I am also going to discuss ways of modifying Frege’s semantical and ontological framework in order to avoid the paradoxes. In this connection, I explore the possibility of giving up the Fregean assumption of a universal domain of absolutely all objects, containing in addition to extensional objects also abstract intensional ones like propositions and singular concepts. I outline a cumulative hierarchy of Fregean propositions and senses, in analogy with Gödel’s hierarchy of constructible sets. In this hierarchy, there is no domain of all objects. Instead, every domain of objects is extendible with new objects that, on pain of contradiction, cannot belong to the given domain. According to this approach, there is no domain containing absolutely all propositions or absolutely all senses. (shrink)

It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.

In lieu of an abstract, here is a brief excerpt of the content:RUSSELLS ARGUMENT AGAINST FREGE'S SENSE-REFERENCE DISTINCTION PAWEL TURNAu Philosophy I Jagiellonian University Cracow, Poland I n "On Denoting"l Russell argued that Frege's theory of sense and reference was an "inextricable tangle", but, ironically, many readers found the argument even more knotry. In an effort to make sense of it, commentators were often driven to attribute to Russell quite obvious and simple fallacies. A different approach was taken (...) by Peter Geach, who suggested that Russell's argument could be given a consistent reading if it were construed as arguing, not against Frege's theory, but against Russell's own earlier theory, which he put forward in The Principles of Mathematics. It is not an implausible hypothesis, considering how prone Russell was to misrepresent' other philosophers' views and read his own powerful ideas into their writings. In order to justifY Geach's hypothesis properly, one would have to show the crucial difference between Russell's and Frege's theories that made only one of them susceptible to the attack in "On Denoting ". Russell explicitly considered the two theories to be "very nearly the same"; this self-interpretation would have to be questioned. Geach's hypothesis would be further strengthened by providing an actual exegesis of the "On Denoting" argument as directed against The Principles ofMathematics theory. In this paper 1 will try to accomplish both of these tasks. Russell is often accused of confusing use and mention. The accusation is not fair, however. It is true that he does not insist on this distinction; he does not employ it extensively. But he is certainly Russell's Argument against Frege's Distinction 53 aware of it; he simply does not think that it is an important logical distinction. Today this view may seem almost incomprehensible, but that only shows the strength of Frege's influence on our ideas. There are two striking symptoms of Russell's lack of concern for the use-mention distinction: firstly, he does not observe the convention which says that putting an expression in quotation marks produces a name of that expression; secondly, he does not feel any need to employ the pair of terms "sentence" and "proposition", but contents himself with the latter only. The claim about Russell's use of inverted commas needs to be proved; I will come to that later. Russell's assertions are often ambiguous between use and mention of expressions. For example, he asserts that "propositions do not contain words" (PaM, p. 47). However, a few pages later, he speaks of propositions that contain the phrase "any number" (p. 53). If this is not to be self-contradictory, then either the term "proposition" must sometimes signifY a linguistic expression, or the term "phrase" must sometimes be understood without implying that a phrase consists of words. This ambiguity is openly admitted in a later article, where he writes: [we shall be concerned with] the distinction between verbs and substantives, or, more correctly, between the objects denoted by verbs and the objects denoted by substantives. Since this more correct expression is long and cumbrous, I shall generally use the shorter expression to mean the same thing. Thus, when I speak of verbs, I mean the object denoted by verbs, and similarly for substantives.2. Russell's attitude towards the distinction between use and mention is not unjustified. It has its reason in his view oflanguage and the subject -matter of logic. In Russell's view, the subject-matter of logic has nothing to do with language. Words, both spoken and written, are entirely outside its realm. Logic, just as other sciences (except linguistics ), is concerned with the entities indicated by words, and not with words themselves. Even the relation of indicating, which holds between words and things in the world, is of no interest to it: "mean1 In Lf(, pp. 39-56. russ.lI: the Journal of the Bertrand Russell Archives McMaster Univecsity Library Press n.s. 11 (summer 1991): 51- 66 ISSN 0036-o16~ 2 B. Russell, "On the Relations of Universals and Particulars", Proceedings of the Aristotelian Society, n.s. 12 (r9II-12): 1-24; in Lf(, pp. 103-24 (at 107-8). 54 PAWEL TURNAU Russell's Argument... (shrink)

Abstract:This paper completes a series of three devoted to the notes that Russell made on reading Gottlob Frege’s works beginning in the summer of 1902. Notes in the two previous papers were used in the preparation of Appendix a of The Principles of Mathematics, “The Logical and Arithmetical Doctrines of Frege”. The bulk of the notes published here are on the formal proofs in Grundgesetze der Arithmetik, which begin at §53 and continue through the rest of Vol. 1. There is (...) no mention of these notes in published works of Russell. Additional notes were made in 1903 when Vol. 2 arrived. Brief notes found in Russell’s copy of Grundgesetze include some on Die Grundlagen der Arithmetik and preliminary notes on Grundgesetze. With material on the contradiction already published this completes the publication of Russell’s notes on Frege in the Russell Archives. (shrink)

Resumo: o propósito desse artigo é circunscrever e discutir a autocrítica, feita por Wittgenstein no período de 1933-39, a uma das teses mais fundamentais do Tractatus (TLP): ”Há uma e apenas uma análise completa da proposição” (3.25). Chamaremos esse procedimento peculiar de a “Grande análise”. Os argumentos de Wittgenstein contra a sustentabilidade da sua antiga tese podem ser encontrados em algumas passagens do livro Investigações Filosóficas (IF), bem como em passagens do Grande Datiloscrito (BT). Conforme será argumentado nesse artigo, essa (...) autocrítica é uma consequência direta da preocupação de Wittgenstein, no período intermediário, com duas importantes propostas feitas por Frege nos §§18-24 do Os Fundamentos da Aritmética (FA) em relação o seu entendimento da noção de “atribuição numérica”. Nosso objetivo principal será mostrar de que maneira o movimento de autocrítica feito por Wittgenstein em relação a sua antiga concepção de “Grande Análise” teria sido influenciado por suas preocupações, no período intermediário de sua obra, com essas duas propostas feita por Frege no FA a respeito da noção de “atribuição numérica”. A fim de completar tal tarefa, será apresentada inicialmente de modo breve uma versão de como seria a “Grande análise” no TLP. Em seguida, selecionaremos e discutiremos algumas passagens do IF nas quais Wittgenstein recusa a completude e unicidade envolvida em sua prévia concepção tratariana de “análise”. Procuraremos destacar com clareza uma noção que estaria envolvida de um modo essencial na autocrítica de Wittgenstein: a noção de “ver como/ver aspectos”. Como será argumentado, essa noção, assim como a autocrítica feita por Wittgenstein, foram ambas diretamente influenciadas pela noção fregiana de “atribuição numérica”. Para completar essa segunda tarefa, teremos de revisitar e discutir algumas noções pertencentes ao período inicial da filosofia de Wittgenstein -- como a de “propriedades internas e externas” e a distinção entre “dizer e mostrar” -- ambas noções discutidas no TLP. No decorrer de nossa discussão, apresentaremos uma crítica feita por Wittgenstein a Russell, mas também a Frege, sobre a concepção de generalidade, ou mais especificamente, sobre o uso do quantificador existencial, proposta por ambos. Abstract: The purpose of this article is to discuss Wittgenstein 1933-39 self-criticism of one of the most fundamental tractarian theses: There is one and only one complete analysis of the proposition (3.25). Wittgenstein's arguments against his previous thesis can be found in some passages of the Philosophical Investigation (PI) as well as in the Big Typescript (BT). As we shall argue, Wittgenstein's self-critique is directly related to two important ideas proposed by Frege in §§18-24 of The Foundations Arithmetic (FA) with respect to his construal of “number attributions”. Our main goal in this paper will therefore be to show how Wittgenstein's self-criticism of his old conception of the tractarian “analysis” would have been influenced by his concerns in the middle period of his work with these two proposals made by Frege in FA regarding the notion of “numerical attribution”. Our first task will then be to connect Wittgenstein's self-criticisms to his former conception of “analysis” in the Tractatus Logico-Philosophicus (TLP). To accomplish this first goal, we will have to briefly present a version of how such analysis would look like in TLP. Our next task will be to pinpoint some passages of the PI in which Wittgenstein rejects the completeness and uniqueness involved in his previous construal of the analytical process. We will try to circumscribe clearly a crucial notion involved in Wittgenstein's self-critique: the idea of “seeing as/seeing aspects”. Our claim is that this notion, as well as Wittgenstein's self-criticism, were both directly influenced by Frege's construal of the “number attribution” presented in FA. To accomplish this second task, we will have to revisit and discuss some further concepts from an early period of Wittgenstein's philosophy, like “internal and internal properties” and the “saying/showing distinction” in TLP. On our way towards accomplishing this goal, we will present a curious complaint by Wittgenstein regarding Russell's, as well as Frege's conception of generality, or more specifically, on the usage of the existential quantifier suggested by both these authors. (shrink)

Quantifiers in Frege's Grundgesetze like are not well-defined because the part Fx & Gx stands for a concept but the yoking conjunction is horizontalised and must stand for a truth-value. This standard interpretation is rejected in favor of a substitutional reading that, it is argued, both conforms better to the text and is well-defined. The theory of the horizontal is investigated in detail and the composite reading of Frege's connectives as made up of horizontals is rejected. The sense (...) in which the Grundgesetze has a many-valued but classical logic is explained by proving that its propositional fragment (under the standard interpretation) and Bochvar's 3-valued logic are instances of the same metatheoretic methods. Historically, the paper argues for a more naive but well-defined reading of the text. Theoretically, it provides a formally adequate statement of that semantics, as well as developing the abstract metatheory which embraces Bochvar's language and the Fregean fragment. (shrink)

This paper aims to determine what kind of problem Frege's famous “Julius Caesar problem” is. whether it is to be understood as the metaphysical problem of determining what kind of things abstract objects like numbers or value‐courses are, or as the epistemological problem of providing a means of recognizing these objects as the same again, or as the logical problem of providing abstract sortal concepts with a sharp delimitation in order to fulfill the law of excluded middle, or as (...) the semantic problem of fixing the referents of the corresponding abstract singular terms. It is argued that, for Frege, the Caesar problem is a bundle of related problems of which the semantic problem is the most basic one. (shrink)

ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term (...) class='Hi'>Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)

Commentators regularly attribute to Frege realist, idealist, and quietist responses to metaphysical questions concerning the abstract objects he calls ‘thoughts’. But despite decades of effort, the evidence offered on behalf of these attributions remains unconvincing. I argue that Frege deliberately avoids commitment to any of these positions, as part of a metaphysical separatist policy motivated by the fact that logic is epistemologically autonomous from metaphysics. Frege’s views and arguments prove relevant to current attempts to argue for epistemological autonomy, particularly that (...) of ethics. (shrink)

This article derives from a project attempting to show that Western formal logic, from Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In the present article, I will first discuss, in light of Frege’s honorary role as founder of the philosophy of mathematics, Reuben Hersh’s What is Mathematics, Really? Second, I will explore how the infamous section of Frege’s 1924 diary (specifically the entries from March 10 to April 9) supports (...) Hersh's claim regarding the link between political conservatism and the (historically and currently) dominant school of the philosophy of mathematics (to which Frege undeniably belongs). Third, I will examine Frege’s attempt at a more reader-friendly introduction to his philosophy of mathematics, The Foundations of Arithmetic. And finally, I will briefly analyze Frege’s Begriffsschrift to see how questions of race arise even at the heights of his logical abstraction. (shrink)

Abstract: The article presents and critically discusses Walton's (and Reed's and Macagno's) argument scheme approach to a theory of good argumentation. In particular, four characteristics of Walton's approach are presented: 1. It presents normative requirements for argumentation in the form of argument schemes, i.e. relatively concrete type descriptions. 2. These schemata are enthymematic, i.e. they omit some of the premises required by other approaches. 3. The actual argument schemes are usually supplemented by critical questions. 4. The method is inductive, (...) bottom-up, gaining the normative schemata by abstraction from empirically collected groups of similar arguments. These characteristics, among others, are then discussed on the basis of four adequacy conditions: AC1: effectiveness in achieving the epistemic goal of obtaining and communicating justified acceptable opinions; AC2: completeness in capturing the good argument types; AC3: efficiency in achieving the goals; AC4: justification of the argument schemes. The discussion then reveals a number of weaknesses in Walton's argument schemes; among other things, they are neither effective (in the defined sense) nor truly justified. Contributing factors to these problems include the schema approach, i.e. not looking at the form of arguments, and the lack of epistemological foundations in the development of good types of arguments. However, the critical analyses reveal a better alternative: an epistemological approach based on epistemological principles. The article concludes with a detailed analysis of the scheme Practical Inference, which confirms the general criticism in detail. (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 ...

Edmund Husserl was one of the very first to experience the direct impact of challenging problems in set theory and his phenomenology first began to take shape while he was struggling to solve such problems. Here I study three difficulties associated with Frege's use of sets that Husserl explicitly addressed: reference to non-existent, impossible, imaginary objects; the introduction of extensions; and 'Russell's paradox'.I do so within the context of Husserl's struggle to overcome the shortcomings of set theory and to (...) develop his own theory of manifolds. I define certain issues involved and discuss how Husserl's theory of manifolds might confront them. In so doing I hope to help bring Husserl's theories about sets and manifolds out of the realm of abstract theorizing and prompt further exploration of uncharted philosophical territory rich in philosophical implications. (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)

A Mathematical Review by John Corcoran, SUNY/Buffalo -/- Macbeth, Danielle Diagrammatic reasoning in Frege's Begriffsschrift. Synthese 186 (2012), no. 1, 289–314. ABSTRACT This review begins with two quotations from the paper: its abstract and the first paragraph of the conclusion. The point of the quotations is to make clear by the “give-them-enough-rope” strategy how murky, incompetent, and badly written the paper is. I know I am asking a lot, but I have to ask you to read the quoted passages—aloud (...) if possible. Don’t miss the silly attempt to recycle Kant’s quip “Concepts without intuitions are empty; intuitions without concepts are blind”. What the paper was aiming at includes the absurdity: “Proofs without definitions are empty; definitions without proofs are, if not blind, then dumb.” But the author even bollixed this. The editor didn’t even notice. The copy-editor missed it. And the author’s proof-reading did not catch it. In order not to torment you I will quote the sentence as it appears: “In a slogan: proofs without definitions are empty, merely the aimless manipulation of signs according to rules; and definitions without proofs are, if no blind, then dumb.”[sic] The rest of my review discusses the paper’s astounding misattribution to contemporary logicians of the information-theoretic approach. This approach was cruelly trashed by Quine in his 1970 Philosophy of Logic, and thereafter ignored by every text I know of. The paper under review attributes generally to modern philosophers and logicians views that were never espoused by any of the prominent logicians—such as Hilbert, Gödel, Tarski, Church, and Quine—apparently in an attempt to distance them from Frege: the focus of the article. On page 310 we find the following paragraph. “In our logics it is assumed that inference potential is given by truth-conditions. Hence, we think, deduction can be nothing more than a matter of making explicit information that is already contained in one’s premises. If the deduction is valid then the information contained in the conclusion must be contained already in the premises; if that information is not contained already in the premises […], then the argument cannot be valid.” Although the paper is meticulous in citing supporting literature for less questionable points, no references are given for this. In fact, the view that deduction is the making explicit of information that is only implicit in premises has not been espoused by any standard symbolic logic books. It has only recently been articulated by a small number of philosophical logicians from a younger generation, for example, in the prize-winning essay by J. Sagüillo, Methodological practice and complementary concepts of logical consequence: Tarski’s model-theoretic consequence and Corcoran’s information-theoretic consequence, History and Philosophy of Logic, 30 (2009), pp. 21–48. The paper omits definitions of key terms including ‘ampliative’, ‘explicatory’, ‘inference potential’, ‘truth-condition’, and ‘information’. The definition of prime number on page 292 is as follows: “To say that a number is prime is to say that it is not divisible without remainder by another number”. This would make one be the only prime number. The paper being reviewed had the benefit of two anonymous referees who contributed “very helpful comments on an earlier draft”. Could these anonymous referees have read the paper? -/- J. Corcoran, U of Buffalo, SUNY -/- PS By the way, if anyone has a paper that has been turned down by other journals, any journal that would publish something like this might be worth trying. (shrink)