The model function for induction of Goodmans's composite predicate "Grue" was examined by analysis. Two subpredicates were found, each containing two further predicates which are mutually exclusive (green and blue, observed before and after t). The rules for the inductive processing of composite predicates were studied with the more familiar predicate "blellow" (blue and yellow) for violets and primroses. The following rules for induction were violated by processing "grue": From two subpredicates only one (blue after t) appears in the conclusion. (...) As a statement about a future and unobserved condition this subpredicate, however, is not projectible for induction whereas the only suitable predicate (green before t) does not show up in the conclusion. In a disjunction "a v b" where "a" is true and "b" false the disjunction is true. When, however, the only true component is dropped, what remains is necessarily false. An analogous mistake may be observed in the processing of "grue", where the only true component (green) was dropped in the conclusion. - As a potent criterion for correct inductions a check of the necessity of the conclusions is recommended. (shrink)
The probabilistic corroboration of two or more hypotheses or series of observations may be performed additively or multiplicatively . For additive corroboration (e.g. by Laplace's rule of succession), stochastic independence is needed. Inferences, based on overwhelming numbers of observations without unexplained counterinstances permit hyperinduction , whereby extremely high probabilities, bordering on certainty for all practical purposes may be achieved. For multiplicative corroboration, the error probabilities (1 - Pr) of two (or more) hypotheses are multiplied. The probabilities, obtained by reconverting the (...) product, are valid for both of the hypotheses and indicate the gain by corroboration.. This method is mathematically correct, no probabilities > 1 can result (as in some conventional methods) and high probabilities with fewer observations may be obtained, however, semantical independence is a prerequisite. The combined method consists of (1) the additive computation of the error probabilities (1 - Pr) of two or more single hypotheses, whereby arbitrariness is avoided or at least reduced and (2) the multiplicative procedure . The high reliability of Empirical Counterfactual Statements is explained by the possibility of multiplicative corroboration of "all-no" statements due to their strict semantical independence. (shrink)
When perceiving a face, we can easily decide whether it belongs to a human or non-human primate. It is thought that face information is represented by neurons in the macaque temporal cortex. However, the precise encoding mechanisms used by these neurons remain unclear. Here we use face stimuli of humans, monkeys and monkey-human hybrids (morphs) to gain a better understanding of these mechanisms, in particular of the categorization of faces into different species, and how learning affects representation of these stimuli.
It was conjectured by Halpern and Kapron (Annals of Pure and Applied Logic, vol. 69, 1994) that frame satisfiability of propositional modal formulas is incomparable in expressive power to both Σ 1 1 (Ackermann) and Σ 1 1 (Bernays-Schonfinkel). We prove this conjecture. Our results imply that Σ 1 1 (Ackermann) and Σ 1 1 (Bernays-Schonfinkel) are incomparable in expressive power, already on finite graphs. Moreover, we show that on ordered finite graphs, i.e., finite graphs with a successor, Σ 1 (...) 1 (Bernays-Schonfinkel) is strictly more expressive than Σ 1 1 (Ackermann). (shrink)
We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized to an oracle in C. We show that this is not always (...) true. However, after studying the problem from a general point of view, we derive sufficient conditions on C such that FO(Q) captures L C . These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures L NP . This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform. vol. 18, 1993]. (shrink)
Gottlob Frege (1848-1925) was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Frege's logical works were revolutionary, and are often taken to represent the fundamental break between contemporary approaches and the older, Aristotelian tradition. He invented modern quantificational logic, and created the first fully axiomatic system for logic, which was complete in its treatment of propositional and first-order logic, and also represented the first treatment of higher-order logic. (...) In the philosophy of mathematics, he was one of the most ardent proponents of logicism, the thesis that mathematical truths are logical truths, and presented influential criticisms of rival views such as psychologism and formalism. His theory of meaning, especially his distinction between the sense and reference of linguistic expressions, was groundbreaking in semantics and the philosophy of language. He had a profound and direct influence on such thinkers as Russell, Carnap and Wittgenstein. Frege is often called the founder of modern logic, and he is sometimes even heralded as the founder of analytic philosophy. (shrink)
: Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.
: Frege's philosophical writings, including the "logistic project," acquire a new insight by being confronted with Kant's criticism and Wittgenstein's logical and grammatical investigations. Between these two points a non-formalist history of logic is just taking shape, a history emphasizing the Greek and Kantian inheritance and its aftermath. It allows us to understand the radical change in rationality introduced by Gottlob Frege's syntax. This syntax put an end to Greek categorization and opened the way to the multiplicity of expressions (...) producing their own intelligibility. This article is based on more technical analyses of Frege which Claude Imbert has previously offered in other writings (see references). (shrink)
The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such (...) a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. (shrink)
To fully respond to the demands of multiculturalism, a view of toleration would need to duly respect diversity both at the level of the application of principles of toleration and at the level of the justificatory foundations that a view of toleration may appeal to. The paper examines Rainer Forst’s post-Rawlsian, ‘reason-based’ attempt to provide a view of toleration that succeeds at these two levels and so allows us to tolerate tolerantly. His account turns on the view that a (...) constructivist requirement of generality and reciprocity provides a suitable criterion of toleration since a commitment to this requirement is part of what defines people as reasonable. But it is neither plausible nor coherent to build such a requirement into an idea of reasonableness from which an account of toleration starts. Thus, constructivism cannot provide a tolerant criterion of toleration, if such criterion, in order to overcome the ‘paradox’ of intolerant toleration, must escape reasonable disagreement. (shrink)
In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik . In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the (...) defectiveness of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege 'reckless' for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell-Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning. (shrink)
Between April and November 1912, Bertrand Russell and Ludwig Wittgenstein were engaged in a joint philosophical program. Wittgenstein‘s meeting with Gottlob Frege in December 1912 led, however, to its dissolution – the joint program was abandoned. Section 2 of this paper outlines the key points of that program, identifying what Russell and Wittgenstein each contributed to it. The third section determines precisely those features of their collaborative work that Frege criticized. Finally, building upon the evidence developed in the preceding (...) two sections, section 4 recasts along previously undeveloped lines Wittgenstein‘s logical–philosophical discoveries in the two years following his encounter with Frege in 1912. The paper concludes, in section 5, with an overview of the dramatic consequences the Frege-Wittgenstein critique had for Russell‘s philosophical development. (shrink)
This analysis of Frege's views on language metaphysics raised in On Sense Reference, (arguably one of the most important philosophical essays of the past hundred years) provides a thorough introduction to the function/argument analysis. It applies Frege's technique to the central notions of predication, identity, existence and truth, and Bertrand Russell's views throughout serve as a foil to Frege.
Last spring, as I was beginning a graduate seminar on Frege, I received a complimentary copy of this new translation of his masterwork, The Foundations of Arithmetic . I had ordered Austin's famous translation, well-loved for the beauty of its English and the clarity with which it presents Frege's overall argument, but known to be less than literal, and to sometimes supplement translation with interpretation. I was intrigued by Dale Jacquette's promise "to combine literal accuracy and readability for beginning students (...) and professional scholars alike," and to improve on Austin where the latter "does not always faithfully represent or seem to perfectly understand certain of Frege's German idioms." (v) Such a translation, complete with index, critical introduction, and commentary, and at a bargain price, seemed worthy of my students' attention. So, I mentioned to the class that this book might be worth looking into. (shrink)
Frege and Eucken were colleagues in the faculty of philosophy at Jena University for more than 40 years. At times they had close scientific contacts. Eucken promoted Frege's career at the university. A comparison of Eucken's writings between 1878 and 1880 with Frege's writings shows Eucken to have had an important philosophical influence on Frege's philosophical development between 1879 and 1885. In particular the classification of the Begriffsschrift in the tradition of Leibniz is influenced by Eucken. Eucken also influenced Frege's (...) choice of philosophical and logical terms. Finally, there are analogous positions concerning relations between concepts and their expressions in natural language, Frege was probably also influenced by Eucken's use of the term ?tone?. Eucken used Frege's arguments in his own fight against psychologism and empiricism. (shrink)