Abstract:This paper aims at clarifying the nature of Frege's system of logic, as presented in the first volume of the Grundgesetze. We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic.
“…We can distinguish between a slightly more semantic and a slightly more pragmatic interpretation of the assertion sign (with the majority of scholars falling on the pragmatic side; see e.g., Greimann, 2014). An example of the former is provided by Cadet and Panza (2015), who suggest that assertions in Frege's system can receive a slightly more semantic interpretation; in their words, the act of asserting something, or claiming truth or falsehood is rendered, within [Frege's Grundgesetze system], as the act of holding that a certain appropriate term or all the terms of a class of appropriate terms refer to the True (Cadet & Panza, 2015, p. 21).…”
Section: The Assertion Sign the Double Judgment Stroke And The Lingui...mentioning
confidence: 99%
“…the act of asserting something, or claiming truth or falsehood is rendered, within [Frege's Grundgesetze system], as the act of holding that a certain appropriate term or all the terms of a class of appropriate terms refer to the True (Cadet & Panza, 2015, p. 21).…”
Section: The Assertion Sign the Double Judgment Stroke And The Lingui...mentioning
In Die Grundgesetze der Arithmetik (I, §48), Frege introduces his rule of the fusion of horizontals, according to which if an occurrence of the horizontal stroke is followed by another occurrence of the same stroke, either in isolation or “contained” in a propositional connective, the two occurrences can be fused with each other. However, the role of this rule, and of the horizontal sign more generally, is controversial; Michael Dummett notoriously claimed, for instance, that the horizontal is “wholly superfluous” in Frege's logical system. In this paper, we challenge Dummett's view by providing a comprehensive analysis of the significance of the horizontal stroke. After some preliminary remarks, we argue that even if Frege's connectives in some sense “contain” the horizontal, yet they are total functions. Then, we take up the question of the sense expressed by the horizontal, and we claim that, unlike other sentential operators, the horizontal is not sense‐compositional. Finally, we consider the semantic and pragmatic aspects of Frege's horizontal in connection to his judgment stroke and the double judgment stroke. Contra Dummett, we argue that the horizontal is a special and indispensable element of Frege's logic.
“…We can distinguish between a slightly more semantic and a slightly more pragmatic interpretation of the assertion sign (with the majority of scholars falling on the pragmatic side; see e.g., Greimann, 2014). An example of the former is provided by Cadet and Panza (2015), who suggest that assertions in Frege's system can receive a slightly more semantic interpretation; in their words, the act of asserting something, or claiming truth or falsehood is rendered, within [Frege's Grundgesetze system], as the act of holding that a certain appropriate term or all the terms of a class of appropriate terms refer to the True (Cadet & Panza, 2015, p. 21).…”
Section: The Assertion Sign the Double Judgment Stroke And The Lingui...mentioning
confidence: 99%
“…the act of asserting something, or claiming truth or falsehood is rendered, within [Frege's Grundgesetze system], as the act of holding that a certain appropriate term or all the terms of a class of appropriate terms refer to the True (Cadet & Panza, 2015, p. 21).…”
Section: The Assertion Sign the Double Judgment Stroke And The Lingui...mentioning
In Die Grundgesetze der Arithmetik (I, §48), Frege introduces his rule of the fusion of horizontals, according to which if an occurrence of the horizontal stroke is followed by another occurrence of the same stroke, either in isolation or “contained” in a propositional connective, the two occurrences can be fused with each other. However, the role of this rule, and of the horizontal sign more generally, is controversial; Michael Dummett notoriously claimed, for instance, that the horizontal is “wholly superfluous” in Frege's logical system. In this paper, we challenge Dummett's view by providing a comprehensive analysis of the significance of the horizontal stroke. After some preliminary remarks, we argue that even if Frege's connectives in some sense “contain” the horizontal, yet they are total functions. Then, we take up the question of the sense expressed by the horizontal, and we claim that, unlike other sentential operators, the horizontal is not sense‐compositional. Finally, we consider the semantic and pragmatic aspects of Frege's horizontal in connection to his judgment stroke and the double judgment stroke. Contra Dummett, we argue that the horizontal is a special and indispensable element of Frege's logic.
“…This is precisely what it means to say that a Roman marker indicates. Now, the matter of the presence of Roman markers in the object language of the ideography is a delicate one, as Cadet and Panza [2] emphasize. Unlike small Greek letters, Roman letters can occur in a judgment in Grundgesetze, so while function names are only part of the expository language, Roman markers belong to the object language.…”
Section: Roman Letters and Indicationmentioning
confidence: 99%
“…Instead, function names appear only in the expository language of Grundgesetze, while the language of the formal system only contains names of values of functions, which are names for objects. 2 The objection is that when 'Φ(∆)' appears as a name of an object in the ideography, for example, we cannot say that the expression 'Φ' appears as a name of a function in Frege's object language. For him only an expression requiring completion such as 'Φ(ξ)' can be a function name.…”
It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik * I would like to thank Ansten Klev, Marco Panza, Marco Ruffino, Will Stafford, Göran Sundholm, Wim Vanrie, and two anonymous reviewers for their helpful comments on early drafts of this paper. Most of the material here has been presented in different online seminars and conferences over the past three years, in particular the OCIE Seminar in the History and Philosophy of Mathematics and Logic, Logicians in Quarantine, I Colóquio NULFA-NuLFiC de Lógica e Filosofia da Matemática, and Chá das 5. I also thank Alessandro Duarte, Erich Reck, and Edward Zalta for stimulating discussions. This research was partly supported by the AFOSR grant FA9550-18-1-0120 and the Lumina quaeruntur fellowship number LQ300092101 from the Czech Academy of Sciences.
“…Moreover, once epistemology is so conceived, proofs appear less as procedures for discovering actual truths than as arguments for obtaining conclusions starting from some assumptions or from other previously established conclusions. Hence, assessing the 6 Regarding my appraisal of Frege's conception of logic, I refer the reader to [6]. tools required to conduct a proof appears less as a way to estimate the objective role of a certain truth than as a way to evaluate the epistemic cost of reaching a certain conclusion within the relevant theory, namely the intellectual resources that are to be deployed within this theory to fix a certain thought.…”
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