Two types of indefinites: Hilbert & Russell

This paper compares Hilbert’s -terms and Russell’s approach to indefinite descriptions, Russell’s indefinites for short. Despite the fact that both accounts are usually taken to express indefinite descriptions, there is a number of dissimilarities. Specifically, it can be shown that Russell indefinites - expressed in terms of a logical ρ-operator - are not directly representable in terms of their corresponding -terms. Nevertheless, there are two possible translations of Russell indefinites into epsilon logic. The first one is given in a language with classical -terms. The second translation is based on a refined account of epsilon terms, namely indexed -terms. In what follows we briefly outline these approaches both syntactically and semantically and discuss their respective connections; in particular, we establish two equivalence results between the (indexed) epsilon calculus and the proposed ρ-term approach to Russell’s indefinites

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 paper, 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

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 paper, 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

The Epsilon-Reconstruction of Theories and Scientific Structuralism

Rudolf Carnap's mature work on the logical reconstruction of scientific theories consists of two components. The first is the elimination of the theoretical vocabulary of a theory in terms of its Ramsification. The second is the reintroduction of the theoretical terms through explicit definitions in a language containing an epsilon operator. This paper investigates Carnap's epsilon-reconstruction of theories in the context of pure mathematics. The main objective here is twofold: first, to specify the epsilon logic underlying his suggested definition of theoretical terms and a suitable choice semantics for it. Second, to analyze whether Carnap's approach is compatible with a structuralist conception of mathematics

Two types of indefinites: Hilbert & Russell

This paper compares Hilbert’s -terms and Russell’s approach to indefinite descriptions, Russell’s indefinites for short. Despite the fact that both accounts are usually taken to express indefinite descriptions, there is a number of dissimilarities. Specifically, it can be shown that Russell indefinites - expressed in terms of a logical ρ-operator - are not directly representable in terms of their corresponding -terms. Nevertheless, there are two possible translations of Russell indefinites into epsilon logic. The first one is given in a language with classical -terms. The second translation is based on a refined account of epsilon terms, namely indexed -terms. In what follows we briefly outline these approaches both syntactically and semantically and discuss their respective connections; in particular, we establish two equivalence results between the (indexed) epsilon calculus and the proposed ρ-term approach to Russell’s indefinites

Carnap's early semantics

In jüngerer Zeit hat sich ein verstärktes Interesse an den historischen und technischen Details von Carnaps Philosophie der Logik und Mathematik entwickelt. Meine Dissertation knüpft an diese Entwicklung an und untersucht dessen frühe und formative Beiträge aus den späten 1920er Jahren zu einer Theorie der formalen Semantik. Carnaps zu Lebzeiten unveröffentlichtes Manuskript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) beinhaltet ein Reihe von erstmals formal entwickelten Definitionen der Begriffe ‚Modell’, ‚Modellerweiterung’, und ‚logischer Folgerung’. Die vorliegende Dissertation entwickelt eine logische und philosophische Analyse dieser semantischen Begriffsbildungen. Darüber hinaus wird Carnaps frühe Semantik in ihrem historisch-intellektuellen Entwicklungskontext diskutiert. Der Fokus der Arbeit liegt in der Thematisierung einiger interpretatorischer Fragen zu dessen implizit gehaltenen Annahmen bezüglich der Variabilität des Diskursuniversums von Modellen sowie zur Interpretation seiner typen-theoretischen logischen Sprache. Mit Bezug auf eine Reihe von historischen Dokumenten aus Carnaps Nachlass, insbesondere zu dem geplanten zweiten Teil der Untersuchungen wird erstens gezeigt, dass dessen Verständnis von Modellen in wesentlichen Punkten heterodox gegenüber dem modernen Begriffsverständnis ist. Zweitens, dass Carnap von einer ‚nonstandard’ Interpretation der logischen Hintergrundtheorie für seine Axiomatik ausgeht. Die Konsequenzen dieser semantischen Annahmen für dessen Konzeptualisierung von metatheoretischen Begriffen werden näher diskutiert. Das erste Kapitel entwickelt eine kritische Analyse von Carnaps Versuch, die axiomatische Definition von Klassen von mathematischen Strukturen mittels des Begriffs von ‚Explizitbegriffen’ formal zu rekonstruieren. Im zweiten Kapitel werden die Implikationen von Carnaps frühem Modellbegriff für seine Theorie von Extremalaxiomen näher beleuchtet. Das letzte Kapitel bildet eine Diskussion der konkreten historischen Einflüsse, insbesondere durch den Mengentheoretiker Abraham Fraenkel, auf Carnaps formale Theorie von Minimalaxiomen.In recent years one was able to witness an intensified interest in the technical and historical details of Carnap’s philosophy of logic and mathematics. In my thesis I will take up this line and focus on his early, formative contributions to a theory of semantics around 1928. Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) includes some of the first formal definitions of the genuinely semantic concepts of a model, model extensions, and logical consequence. In the dissertation, I provide a detailed conceptual analysis of their technical details and contextualize Carnap’s results in their historic and intellectual environment. Certain interpretative issues related to his tacit assumptions concerning the domain of a model and the semantics of type theory will be addressed. By referring to unpublished material from Carnap’s Nachlass I will present archival evidence as well as more systematic arguments to the view that Carnap holds a heterodox conception of models and a nonstandard semantics for his type-theoretic logic. Given these semantic background assumptions, their impact on Carnap’s conceptualization of certain aspects of the metatheory of axiomatic theories will be evaluated. The first chapter critically discusses Carnap’s attempt to explicate one of the crucial semantic innovations of formal axiomatics, i.e. the definition of classes of structures, via his notion of ‘Explizitbegriffe’. The second chapter analyses the impact of Carnap’s early theory of model for his theory of extremal axioms. The final chapter reviews the mathematical influences, most importantly by the set theoretician Abraham Fraenkel on Carnap’s specific formalization of minimal axioms

What Are Structural Properties?†

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties