Showing papers in "History and Philosophy of Logic in 1990"
TL;DR: In this paper, a vorliegende Studie will diese Lucke unter Konzentration auf zwei Gegenstandsbereiche teileweise ausfullen: (1) den historischen Entstehungskontext von Zermelos ersten Arbeiten uber die Grundlagen der Mengenlehre; (2) die Vorgeschichte und naheren Umstande des 1907 an Zermelo verliehenen Lehrauftrages fur mathematische
Abstract: Zermelos Zeit in Gottingen (1897–1910) kann als wissenschaftlich fruchtbarste Periode in seiner Karriere angesehen werden. Gleichwohl stehen bisher Untersuchungen aus. die eine Einbettung von Zermelos Werk in den biographischen und sozialen Kontext ermoglichen Die vorliegende Studie will diese Lucke unter Konzentration auf zwei Gegenstandsbereiche teileweise ausfullen: (1) den historischen Entstehungskontext von Zermelos ersten Arbeiten uber die Grundlagen der Mengenlehre; (2) die Vorgeschichte und naheren Umstande des 1907 an Zermelo verliehenen Lehrauftrages fur mathematische Logik und verwandte Gegenstande. mit dem ein erster Schritt zur Institutionalisierung dieses Faches als mathematischer Teildisziplin gemacht wurde. Beides wird in enger Verbindung zur ersten Phase des Hilbertschen Programms zur Grundlegung der Mathematik gesehen. Es wird aber auch gezeigt, das fur die Erteilung des Lehrauftrages neben diesen systematischen und forschungspolitischen Motiven auch personliche und institutspolitische R...
23 citations
TL;DR: The authors argued that negative categorical statements are not to be accorded existential import insofar as they figure in the square of opposition, and the logic proper provides much thinner evidence than has been supposed for what appears to be the received view.
Abstract: Two main claims are defended. The first is that negative categorical statements are not to be accorded existential import insofar as they figure in the square of opposition. Against Kneale and others, it is argued that Aristotle formulates his o statements, for example, precisely to avoid existential commitment. This frees Aristotle's square from a recent charge of inconsistency. The second claim is that the logic proper provides much thinner evidence than has been supposed for what appears to be the received view, that is, for the view that insofar as they occur in syllogistic negative categoricals have existential import. At most there is a single piece of evidence in favor of the view–a special case of echthesis or the setting out of a case in proof.
13 citations
TL;DR: In this article, it is argued that this way of framing the contrast is not Aristotelian, and that an interpretation involving modal copulae allows us to see how these principles, and the modal system as a whole, are to be understood in light of close and precise connections to Aristotle's essentialist metaphysics.
Abstract: Aristotle founds his modal syllogistic, like his plain syllogistic, on a small set of ‘perfect’ or obviously valid sylligisms. The rest he reduces to those, usually by means of modal conversion principles. These principles are open to more than one reading, however, and they are in fact invalid on one traditional reading (de re), valid on the other (de dicto). It is argued here that this way of framing the contrast is not Aristotelian, and that an interpretation involving modal copulae allows us to see how these principles, and the modal system as a whole, are to be understood in light of close and precise connections to Aristotle's essentialist metaphysics.
7 citations
TL;DR: In this paper, the authors claim that the peculiarities of Hilbert's 1925 classic paper "On the infinite" are due to a tension between two incompatible semantical approaches to numerical statements of elementary arithmetic, and accordingly two incompatible metaphysical conceptions of the natural numbers.
Abstract: Serious difficulties attend the reading of David Hilbert's 1925 classic paper ‘On the infinite’. I claim that the peculiarities of presentation plaguing certain parts of that paper, as well as of the earlier ‘On the Foundations of Logic and Arithmetic’ (1904), are due to a tension between two incompatible semantical approaches to numerical statements of elementary arithmetic, and accordingly two incompatible metaphysical conceptions of the natural numbers. One of these approaches is the referential, or model-theoretical one; the other is the iterativist's approach. I draw out the two tendencies in these works, with more attention paid to Hilbert's iterativistic tendency because of the unfamiliarity of iterativism generally. I begin with an exposition of this view.
6 citations
4 citations
TL;DR: The connexions that exist between the logical doctrines of G. Frege and R. Lotze are, as shows their common treatment of natural language, deeper than is generally admitted as discussed by the authors.
Abstract: Die Zusammenhange die zwischen G. Freges und R. H. Lotzes logischen Lehren bestehen, sind, wie die gemeinsame Beurteilung der Gebrauchssprache zeigt, noch tiefer als allgemein angenommen. Insbesondere die von Frege konzipierte logische Sprachkritik ist in drei Punkten von Lotze beeinflust. Lotze fordert namlich die strenge Trennung von Logik und Gebrauchssprache. Daneben spielt der Begriff des Logischeinfachen eine zentrale Rolle in seiner Logik. Schlieslich unterscheidet er den objektiven Gedanken von seiner Farbung. The connexions that exist between the logical doctrines of G. Frege and R. H. Lotze are, as shows their common treatment of natural language, deeper than is generally admitted. In particular, the logical criticism of language conceived by Frege is influenced in three points by Lotze. Firstly, Lotze postulates the strict separation of logic and natural language. Furthermore, the idea of logical simplicity plays an important role in his logic. Finally, he distinguishes objective thought from i...
3 citations
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TL;DR: A bibliographical search through the major libraries of Italy has revealed a large and various collection of writings of logic published during the 19th century before the rise of Peano and his school as discussed by the authors.
Abstract: A bibliographical search through the major libraries of Italy has revealed a large and various collection of writings of logic published during the 19th century before the rise of Peano and his school (from the 1880son). A survey of current findings is provided.
3 citations
TL;DR: A short seventeenth-century text, sometimes cited as one of the first essays in mathematical logic, is introduced, translated and evaluated in this article, which may be viewed as a preliminary step toward the formalization of logic.
Abstract: A short seventeenth-century text, sometimes cited as one of the first essays in mathematical logic, is introduced, translated and evaluated Although by no means sharing the depth and magnitude of the investigations by Leibniz being undertaken at the same time, and although in particular not yet applying algebraic symbolism to logical structures, the treatise is of historical interest as an early published attempt to trace out analogies between logical and mathematical form, and may be viewed as a preliminary step toward the formalization of logic
2 citations
TL;DR: In this paper, it was shown that research into the history of logic in the nineteenth century involves journals and periodicals which are normally not considered as standard sources for logic or its history.
Abstract: By using examples drawn from the periodical Nature, I show that research into the history of logic in the nineteenth century involves journals and periodicals which are normally not considered as standard sources for logic or its history
2 citations
Journal Article•
2 citations
TL;DR: Post's Nachlass has recently been made available to the public in an archive in the U.S.A. as mentioned in this paper, with a short summary of his life and career.
Abstract: Post's Nachlass has recently been made available to the public in an archive in the U.S.A. After a short summary of his life and career, this article indicates the character and content of the manuscripts, and their significance is assessed. Two short passages are transcribed; and. as a separate item, a paper of the 1930s on the paradoxes is reproduced.
TL;DR: The algebra of sets has, basically, two different types of symbols as mentioned in this paper : one type of symbol (∩, ∪, +, −) defines another set from two other sets, and a second type (∆, ⊂, =, ≠) makes a proposition about two sets.
Abstract: The algebra of sets has, basically, two different types of symbols. One type of symbol (∩, ∪, +, −) defines another set from two other sets. A second type of symbol (⊆, ⊂, =, ≠) makes a proposition about two sets. When the construction of these two types of symbols is based on the same four-dot matrix as the logic symbols described in a previous paper, the three symbol types then dovetail together into a harmonious whole that greatly simplifies derivation in the algebra of sets.
TL;DR: In this article, a uniform syntactic description is given of acceptable instances of the comprehension schema, which include all of the axioms mentioned, and which in their turn are theorems of the usual versions of Zermelo-Fraenkel-Skolem (ZFS) set theory.
Abstract: Unrestricted use of the axiom schema of comprehension, ‘to every mathematically (or set-theoretically) describable property there corresponds the set of all mathematical (or set-theoretical) objects having that property’, leads to contradiction. In set theories of the Zermelo–Fraenkel–Skolem (ZFS) style suitable instances of the comprehension schema are chosen ad hoc as axioms, e.g.axioms which guarantee the existence of unions, intersections, pairs, subsets, empty set, power sets and replacement sets. It is demonstrated that a uniform syntactic description may be given of acceptable instances of the comprehension schema, which include all of the axioms mentioned, and which in their turn are theorems of the usual versions of ZFS set theory. Well then, shall we proceed as usual and begin by assuming the existence of a single essential nature or Form for every set of things which we call by the same name? Do you understand? (Plato, Republic X.596a6; cf. Cornford 1966, 317)