MCMP Curated Collections

Carnap on Logic
Hand-Picked by Georg Schiemer

Rudolf Carnap is certainly one of the central figures in twentieth-century formal philosophy. His work covers a wide spectrum of topics and philosophical fields, ranging from the philosophy of logic and mathematics to the philosophy of language and from philosophy of science to probability theory. While Carnap’s contributions in these fields are diverse, there is a common method of doing philosophy that can be identified in his work. This is the use of logic in his approach to philosophical problems. Formal logic plays a central role in Carnap’s work throughout his career, from his early type-theoretic formulation of a constitution theory of knowledge in Der logische Aufbau der Welt to his mature work on ‘Wissenschaftslogik’, i.e. the logic of science, from the 1950s.The present collection focuses on Carnap’s work on logic as well as on different uses of logical methods in his philosophical work. It comprises seven talks by international scholars that investigate several of Carnap’s contributions from different phases in his intellectual career. The talks selected here were originally presented at the “Carnap on Logic” conference held at the MCMP in June 2013 and co-financed by the German Research Foundation. The central objective of this conference was to investigate from different perspectives Carnap’s work on logic, its philosophical understanding, as well as on various applications of logic in the philosophies of mathematics, language, and science (among other fields). Topics discussed in Munich included his early work on logical type theory and on the metatheory of axiomatic systems as well as his later contributions to formal semantics and to modal logic. Regarding Carnap’s philosophy of logic, several talks at the conference have focused on his ‘principle of tolerance’—first expressed in 1934 in his Logical Syntax of Language—and on the logical pluralism expressed in it. I hope that the present selection of talks from the conference will provide its viewers with an overview over Carnap’s wide-ranging contributions to logic as well as over the present state of Carnap scholarship.

From Intuition to Tolerance

In his talk “From Intuition to Tolerance in Carnap’s Philosophy of Mathematics”, Michael Friedman gives a general study of the evolution of Carnap’s views on arithmetic and geometry from his disseration Der Raum, through the Aufbau period, to Logical Syntax and the semantic period. The talk focuses on Carnap’s changing understanding of the role of intuition in mathematics. Geometry, in particular in Der Raum of 1922, is explicitly tied to spatial intuition, and then, even though it remains distinguished from arithmetic, its ties to spatial intuition become empirical rather than a priori. As Friedman points out, the character of Carnap’s generally tolerant attitude towards philosophical disputes in the foundations of mathematics thereby changes as well. Carnap’s attitude towards intuition in the arithmetical case becomes explicit in his exchange with E.W. Beth in the Schilpp volume, and this clarifies his application of the principle of tolerance in this case as well.

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Invariance and Logical Truth

Steve Awodey’s talk “On the Invariance of Logical Truth” focuses on a general idea underlying Carnap’s definition of logical truth in Logical Syntax which is later called the invariance conception of logical truth. As Awodey points out, Carnap’s specific approach in 1934 to demarcate between the logical and descriptive vocabulary of a language reflects a more general understanding of logicality in terms of invariance conditions that is present also in other work by Carnap and Tarski, as well as in more recent contributions in the philosophy of logic. In Carnap’s framework this is the idea that the truth values of statements are invariant under reinterpretations of the descriptive or empirical terms of the language in question. In his talk, Awodey defends the claim that the invariance conception of logical truth is a valid idea that reoccurs in different versions in Carnap’s work on logic. In particular, there is a pre-Syntax version of this idea, formulated in Carnap’s general axiomatics project from the late 1920s where the notion of ‘formal’ or ‘structural’ properties of models of axiomatic theories is specified in terms of their invariance under isomorphic transformations.

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Carnap as a Logician

Richard Zach’s talk “Carnap as a Logician” concentrates on Carnap’s early work on the logical metatheory of axiomatic theories. Based on a general overview of Carnap’s contributions to logic, the talk focuses on Carnap’s ‘general axiomatics’ project developed in Vienna in the late 1920s. More specifically, Zach gives a modern reconstruction and philosophical discussion of Carnap’s logical explication of several metalogical concepts presented in his unpublished manuscript Investigations into General Axiomatics from 1928. Concepts introduced here include the notions of logical consequence and truth in a structure, as well as several notions of completeness—roughly, in modern terminology, the categoricity, semantic completeness, and syntactic completeness—of axiomatic theories. A specific focus in Zach’s talk is put on Carnap’s failed proof of the so-called ‘Gabelbarkeitssatz’, a metatheorem stating the equivalence of these three metatheoretic properties of completeness for theories expressed in higher-order logic.

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The Limits of Tolerance

Florian Steinberger, in his talk “The limits of tolerance? Carnap on the normativity of logic”, discusses Carnap’s understanding of the normative status of logical laws in light of his principle of tolerance. The talk starts with a discussion of two important issues concerning Frege’s views on the normativity of logic, namely first why logic is normative according to him and, secondly, how it is normative. Regarding the second question, Steinberger introduces a distinction between a regulative and a constitutive understanding of norms and argues that, at least for Frege, logical norms are constitutive for reasoning. The second part of the talk then focuses on Carnap’s account of the normative status of logic. As Steinberger points out, Carnap rejects both Frege’s logical absolutism and his logical realism, i.e. the idea that there exists one true logic that describes reality. Instead, the principle of tolerance stated in Logical Syntax introduces a logical pluralism according to which any logical system can be adopted for the analysis of mathematics and the natural sciences. The justification of such systems is not to be based, as in Frege’s understanding, on criteria of external correctness, but rather based on purely pragmatic considerations. As a consequence of this, Steinberger argues, Carnap’s conception of logic is best described as a voluntarist account of logical norms: by choosing one logical system over another, we deliberately adopt norms of reasoning that will guide our thinking.

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Empirical Significance

In his talk “On Carnap on Empirical Significance”, Sebastian Lutz presents a particular case study of Carnapian explication, namely a study of different proposals in Carnap’s work on the logic of science to formalize a criterion of empirical significance for scientific statements and terms. Lutz argues that while all of Carnap’s early attempts to provide a precise account of empirical significance for terms ultimately fail, Carnap did in fact develop successful criteria for the significance of statements in his mature philosophy of science from the 1950s. More specifically, it is shown that Carnap’s different proposals to explicate the concept in terms of the technical notions of ‘translatability’, ‘verifiability’, ‘falsifiability’, and ‘non-trivial content’ turn out to be successful if his mature views on logical theory reconstruction are taken into account. Moreover, given Carnap’s understanding of the Ramsey sentence and the Carnap sentence as expressing the theory’s synthetic and analytic content, these criteria for empirical significance in fact turn out to be equivalent in a sense specified in the talk.

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Carnapian Explication

Catarina Dutilh Novaes’ talk “Two conceptions of formalization: Carnapian explication, and formalisms as cognitive tools” focuses on a key concept in Carnap’s methodology of formal philosophy, namely logical explication. Dutilh Novaes gives a comparison between Carnap’s account of explication as a way to formalize informal or pre-theoretical concepts in the sciences and her own account of “formalisms as cognitive tools” developed in her recent book Formal Languages in Logic of 2012. The talk starts with a closer discussion of Carnap’s specific understanding of the four adequacy criteria of a successful explication—namely similarity, fruitfulness, exactness, and simplicity—as well as his philosophical motivations for each. Dutilh Novaes then further illustrates Carnap’s adequacy conditions by turning to so-called statistical prediction rules, that is, mathematical algorithms formulated on the basis of given statistical data. As she shows, these formal tools often outperform expert intuitive judgments or intuition-based predictions since cognitive biases or by other non-relevant factors are ruled out in them. How are these examples of statistical prediction rules related to Carnapian explication? While the formulation of SPRs based on statistical data are not generally cases of explication since no concept formation is involved here, there is an important link to Carnap’s method. In particular, as Dutilh Novaes shows, two of Carnap’s adequacy conditions, namely exactness and fruitfulness, also play central role in the formulation of such rules.

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Logic and the Analytic

In his talk “The Logical and the Analytic”, Richard Creath analyzes Carnap’s definition of logical expressions in Logical Syntax as well as several objections raised against it in the subsequent literature. The talk starts with a detailed discussion of Carnap’s formal definition and then turns to three objections formulated by Saunders Mac Lane in a review of 1938. Mac Lane’s objections are based on counter-examples of expressions that should intuitively count as logical but turn out as descriptive according to Carnap’s definition. Two further objections are mentioned that were raised against Carnap’s criterion in Quine’s article “Carnap and Logical Truth” from 1963. Quine’s first objection concerns the question whether the addition of non-logical terms such as ‘is heavier than’ to a class of determinate statements will qualify as logical according to Carnap’s definition. Creath shows, based on a number of examples, that this is not the case. He then goes on to discuss two more serious objections against Carnap’s syntactic approach to specify logical expressions. In the second and more philosophical part of the talk, Creath then analyzes what impact the lack of a principled definition of logical expressions has for Carnap’s more general project of clarifying the notion of analyticity. The main point made here is that this lack of a demarcation principle between logical and non-logical (or descriptive) terms is not as fatal to Carnap’s subsequent semantic program as one would expect.

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For further details on the “Carnap on Logic” conference and its participants, I refer to www.carnaponlogic2013.philosophie.uni-muenchen.de and to the other recordings on the MCMP iTunes U channel. I hope you like the collection — comments or thoughts? Send me a message! Georg