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6 results on '"Grundgesetze"'

1. Frege’s Theory of Types

Author

Bruno Bentzen

Subjects
Type theory, Frege, Grundgesetze, Martin-Löf type theory, Frege’s theory of function levels, Logic, BC1-199, Philosophy (General), B1-5802
Abstract

Abstract It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies Church’s simple theory of types. This claim roughly states that Frege presupposes a type of functions in the sense of simple type theory in the expository language of Grundgesetze. However, this view makes it hard to accommodate function names of two arguments and view functions as incomplete entities. I propose and defend an alternative interpretation of first-level function names in Grundgesetze into simple type-theoretic open terms rather than into closed terms of a function type. This interpretation offers a still unhistorical but more faithful type-theoretic approximation of Frege’s theory of levels and can be naturally extended to accommodate second-level functions. It is made possible by two key observations that Frege’s Roman markers behave essentially like open terms and that Frege lacks a clear criterion for distinguishing between Roman markers and function names.

Published
2023
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2. Frege's case for the logicality of his basic laws

Author

Yates, Alexander, Milne, Peter, and Sullivan, Peter M.

Subjects
160, Frege, Logic, Epistemic analyticity, Philosophy of logic, History of analytic philosophy, Basic laws, Grundgesetze
Abstract

Frege wanted to show that arithmetical truths are logical by proving them from purely logical basic laws. But how do we know that these basic laws are logical? Frege uses generality and undeniability to make a prima facie case for logicality—if a truth is general and undeniable, then it's likely logical. I argue that Frege could, did, and had to make a deeper case for why we're right in recognizing his basic laws as logical. Implicit in his work is a view of logical laws as epistemically analytic—his arguments for his basic laws serve to elicit a reflective awareness of the fact that understanding them is sufficient for recognizing them to be true. This view both fits with Frege's comments concerning the connection between logic, truth, and normativity, and serves to explain why and in what sense he took logic to be general and conceptually undeniable. In my view, semantics must play a distinctive role in any rational reconstruction of Frege's case for logicality—the aforementioned “reflective awareness” must be an explicit appreciation of how the truth of formulas expressing Frege's laws follows quickly from his stipulations governing terms which figure in those formulas. Opposing this view is the elucidatory interpretation of Thomas Ricketts, Warren Goldfarb, and Joan Weiner, which holds that Frege's arguments for his basic laws can't be taken at face value, and must serve the merely elucidatory purpose of easing us into the language. Another reading is the correctness interpretation of Richard Heck and Jason Stanley, which holds that Frege's primary purpose in his arguments is justifying the claim that Frege's axioms, qua formulas, are true. I argue against both of these interpretations, and in doing so clarify the role and limits of semantics in Frege's enterprise.

Published
2017

3. FRAGMENTS OF FREGE’S GRUNDGESETZE AND GÖDEL’S CONSTRUCTIBLE UNIVERSE

Author

WALSH, SEAN

Subjects
Pure Mathematics, Mathematical Sciences, Frege, Grundgesetze, constructibility, abstraction principles, math.LO, 03F35, 03F25, 03E45, 03-03, Computation Theory and Mathematics, Philosophy, General Mathematics, Theory of computation, Applied mathematics, Pure mathematics
Abstract

Abstract: Frege’sGrundgesetzewas 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 theGrundgesetzeformed 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 theGrundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of theGrundgesetzewhich 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 Gödel’s constructible universe of sets and to Kripke and Platek’s idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen’s fine structure theory). The axioms of theGrundgesetzeare examples ofabstraction 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 (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.

Published
2016

4. FRAGMENTS OF FREGE'S GRUNDGESETZE AND GODEL'S CONSTRUCTIBLE UNIVERSE

Author

Walsh, S

Subjects
constructibility, 03E45, Philosophy, math.LO, Frege, Grundgesetze, General Mathematics, 03F35, 03F25, abstraction principles, Computation Theory and Mathematics, Pure Mathematics, 03-03
Abstract

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.

Published
2016

5. Alfonso the Xth in Cadix. Constitutional vision of a monarch from the Middle Ages (I)

Author

Martínez Martínez, Faustino

Subjects
Verfassung, Mediävistik, Historicisme, Historicism, Grundgesetze, 1812. Medievalisme, Cortes von Cádiz, Lois Fondamentales, Medievalism, Alfons X, Cortes de Cádiz, Constitution, Medievalismo, Historizismus, Fundamental Laws, Cortes de Cadix, Alfonso X, Alfonso the Xth, Cortes of Cadix, Historicismo, Leyes Fundamentales, Constitución
Abstract

Se estudia en este trabajo el papel que corresponde como figura política, legendaria y mítica, pero también como legislador, a Alfonso X el Sabio, en el peculiar contexto del siglo XVIII y comienzos del XIX, especialmente en las Cortes de Cádiz (1810-1814). This essay studies the perception of Alfonso the Xth’s roll not only as politician, legendary figure and myth, but also as legislator. This presence of his political legacy and legislative works as intellectual topic is remarkable during the XVIIIth century and the beginning of XIXth century, specially in the Cortes of Cadix (1810-1814). aussi comme législateur, à Alfonso X Le Sage, dans le contexte particulier du XVIIIème siècle et les débuts du XIXème, spécialement dans les Cortes de Cadix (1810-1814). Dieser Beitrag untersucht die Alfons X. (dem Weisen) zukommende Rolle als politische, legendenumwobene und mystische Figur sowie als Gesetzgeber im besonderen Kontext des 18. und beginnenden 19. Jahrhunderts der Cortes de Cádiz (1810-1814).

Published
2012

6. Alfons X. in Cádiz. Ein verfassungsgeschichtlicher Blick auf einen mittelalterlichen Monarchen

Author

Martínez Martínez, Faustino and Martínez Martínez, Faustino

Abstract

Dieser Beitrag untersucht die Alfons X. (dem Weisen) zukommende Rolle als politische, legendenumwobene und mystische Figur sowie als Gesetzgeber im besonderen Kontext des 18. und beginnenden 19. Jahrhunderts der Cortes de Cádiz (1810-1814)., This essay studies the perception of Alfonso the Xth’s roll not only as politician, legendary figure and myth, but also as legislator. This presence of his political legacy and legislative works as intellectual topic is remarkable during the XVIIIth century and the beginning of XIXth century, specially in the Cortes of Cadix (1810-1814)., Se estudia en este trabajo el papel que corresponde como figura política, legendaria y mítica, pero también como legislador, a Alfonso X el Sabio, en el peculiar contexto del siglo XVIII y comienzos del XIX, especialmente en las Cortes de Cádiz (1810-1814)., aussi comme législateur, à Alfonso X Le Sage, dans le contexte particulier du XVIIIème siècle et les débuts du XIXème, spécialement dans les Cortes de Cadix (1810-1814).

Published
2012

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