, Volume 79, Issue 2, pp 405–429 | Cite as

Submodels in Carnap’s Early Axiomatics Revisited

  • Iris Loeb
Original Article


G. Schiemer has recently ascribed to Carnap the so-called domains-as-fields conception of models, which he subsequently used to defend Carnap’s treatment of extremal axioms against J. Hintikka’s criticism that the number of tuples in a relation, and not the domain of discourse, is optimised in Carnap’s treatment. We will argue by a careful textual analysis, however, that this domains-as-fields conception cannot be applied to Carnap’s early semantics, because it includes a notion of submodel and subrelation that is not only absent from Carnap’s work at that time, but even contradicts it. As a consequence, Schiemer’s defense of Carnap’s extremal axioms against Hintikka’s criticism fails. We will reconcile Carnap’s treatment of extremal axioms and Hintikka’s observation by taking into account the practice of axiomatics in the early twentieth century. If one realises that, in Carnap’s time, a predicate for the domain of discourse was often introduced in the formal theory, and that Carnap defined such predicates from the basic relations of an axiom system, the apparent disagreement between optimising relations and optimising domains disappears.


Object Language Basic Relation Axiom System Propositional Function Completeness Axiom 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author thanks Georg Schiemer, for sending her his recent papers, for discussions, and for his comments on an earlier version of this paper. She thanks Stefan Roski for many interesting discussions on Carnap and Carnap’s notion of model. She also thanks Arianna Betti, Hein van den Berg, Rik Peels, Jeroen de Ridder, Stefan Roski, Jeroen Smid, and René van Woudenberg for comments on an earlier version of this paper. Finally she thanks both referees for their insightful remarks. Work on this paper was made possible by ERC Starting Grant TRANH 203194. All passages from the Rudolf Carnap Papers are quoted by permission of the University of Pittsburgh. All rights reserved.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Faculty of Philosophy and Network InstituteVU University AmsterdamAmsterdamThe Netherlands

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