Georg Schiemer

Carnap on extremal axioms, completeness of the models, and categoricity

This paper provides a historically sensitive discussion of Carnaps theory of extremal axioms developed first in the late 1920s. The main focus is set on the unpublished documents of the projected second part of his manuscript Untersuchungen zur allgemeinen Axiomatik (RC 081-01-01 to 081-01-33). Carnaps theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnaps understanding of the relationship between the completeness of the models and other metatheoretic notions investigated by him at the time, most notably that of categoricity. The paper surveys Carnaps different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.

Carnap’s Untersuchungen: Logicism, Formal Axiomatics, and Metatheory

This paper discusses Carnap’s attempts in the late 1920s to provide a formal reconstruction of modern axiomatics.1 One interpretive theme addressed in recent scholarly literature concerns Carnap’s underlying logicism in his philosophy of mathematics from that time, more specifically, his attempt to “reconcile” the logicist approach of reducing mathematics to logic with the formal axiomatic method. For instance, Awodey & Carus (2007) characterize Carnap’s manuscript Untersuchungen zur allgemeinen Axiomatik from 1928 as a “large-scale project to reconcile axiomatic definitions with logicism, and transform implicit into explicit definitions.” (ibid., 29) It is argued that Carnap’s central idea was to balance a Fregean (or Russellian) foundational stance with the modern model-theoretic viewpoint introduced in Hilbert’s Grundlagen der Geometrie (see (Reck 2004)). It was also shown in recent literature that Carnap’s attempt to provide a logicist reconstruction of axiomatics is limited in several ways.2 No closer attention, however, has so far been dedicated to some of the details of his proposed reconciliation.

The Structuralist Thesis Reconsidered

Øystein Linnebo and Richard Pettigrew ([2014]) have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of positions are structural. In this article, we argue that the structuralist thesis, even when restricted to fundamental properties, does not follow from the theory of structures that Linnebo and Pettigrew have developed. To make their account work, we propose a formal framework in terms of Kripke models that makes structural abstraction precise. The formal framework allows us to articulate a revised definition of fundamental properties, understood as intensional properties. Based on this revised definition, we show that the restricted version of the structuralist thesis holds. 1. Introduction2. The Structuralist Thesis3. LP-Structuralism4. Purity5. A Formal Framework for Structural Abstraction6. Fundamental Relations7. The Structuralist Thesis Vindicated8. Conclusion Introduction The Structuralist Thesis LP-Structuralism Purity A Formal Framework for Structural Abstraction Fundamental Relations The Structuralist Thesis Vindicated Conclusion

Carnap on logic and rationality

Rudolf Carnap is undoubtedly one of the central figures in twentieth-century scientific philosophy. While his work covers a wide spectrum of topics in philosophy of logic and mathematics, philosophy of language, formal philosophy of science, and probability theory (among other fields), a particular method of doing philosophy can be identified in all of his contributions. This is the use of formal logic in his approach to philosophical problems. Logic plays an essential role in Carnap’s work throughout his intellectual career, from his early type-theoretic formulation of a constitution theory of knowledge in Der logische Aufbau der Welt (Carnap 1928) to his mature work on a theory of probability, e.g. (Carnap 1950).

Carnap's Early Metatheory: Scope and Limits

In Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap’s contributions to the development of modern logic.

A choice-semantical approach to theoretical truth

A central topic in the logic of science concerns the proper semantic analysis of theoretical sentences, that is sentences containing theoretical terms. In this paper, we present a novel choice-semantical account of theoretical truth based on the epsilon-term definition of theoretical terms. Specifically, we develop two ways of specifying the truth conditions of theoretical statements in a choice functional semantics, each giving rise to a corresponding logic of such statements. In order to investigate the inferential strength of these logical systems, we provide a translation of each truth definition into a modal definition of theoretical truth. Based on this, we show that the stronger notion of choice-semantical truth captures more adequately our informal semantic understanding of scientific statements.