We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. (...) This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature. (shrink)

This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) in his theory of axiomatics is presented. Specifically, it is shown that Carnap’s early model theory is based on a convention to simulate domain variation that is not identical but logically comparable to the modern account. (shrink)

This paper considers the question whether the notion of the relative apriori, central to Michael Friedman’s transcendentalist programme for philosophy of science, is available also to philosophers who reject appeals to a synthetic a priori. After tracing the rediscovery of the relative a priori and delineating its potential, the question is considered whether Friedman’s arguments against Quinean naturalism and Carnap’s attenuated logicism tell against a conception of philosophy as scientific metatheory that combines logical and empirical inquiries. Finding an opening here (...) it is argued that this conception can also help itself to the notion of a relative a priori which, it is agreed, is central to historically informed theorising about science. (shrink)

Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.

En este artículo que dedicamos a las concepciones epistemológicas de Jean Cavaillès sobre el pensamiento formal y la teoría de la ciencia, analizaremos, en un primer tiempo, cómo la filosofía de las matemáticas de Cavaillès se presenta como una severa censura del logicismo y, en particular, de la empresa universalista de Carnap cuya sintaxis lógica es el blanco de las reprobaciones del filósofo francés, tanto en lo que atañe a la cuestión del formalismo como a la que estudia la relación (...) que mantiene la mathesis formalis con la física en la epistemología de Carnap. En un segundo tiempo, dado las simpatías de Cavaillès por la matemática intuicionista y una cierta resonancia de su epistemología con la de Brouwer, procuraremos subrayar, a partir del análisis crítico realizado por el propio Cavaillès, las dificultades a las que se expone el intuicionismo. Lo veremos, por una parte, en la indeterminación de la matemática a la física y, por otra parte, en el peligro de encontrarse la ciencia absorbida por la matemática si se admite, con el matemático holandés, que ésta es pensamiento racional del mundo. En la tercera y última parte del artículo, esbozaremos el programa de trabajo de Cavaillès, su última hoja de ruta, a favor de una epistemología constructiva y refinada de las matemáticas, entendidas a la vez en su esencia más pura y en su relación praxeológica con las ciencias de la naturaleza. En definitiva, se tratará de aproximarnos al proyecto de Cavaillès, entendido como una epistemología conceptual, fundada en una aproximación rigurosamente bolzaniana de lo lógico, una teoría pura de los encadenamientos racionales, que contempla a la vez la determinación de las posibilidades de los objetos y las posibilidades de determinación de éstos. Tal empresa, lo veremos, se apoya sobre una noción fuerte de necesidad cuyo carácter imprevisible escapa a las mallas de la lógica y atraviesa toda la praxis matemática en su temporalidad propia. (shrink)

Intentionality may be dealt with in two different ways: either ontologically, as an ordinary relation to some extraordinary objects, or epistemologically, as an extraordinary relation to some ordinary objects. This paper endorses the epistemological view in order to provide a model of semiotic intentionality defined as the meaning-and-cognizing process that constitutes to power of the mind to be about something on the basis of a semiotic system. After a short introduction that presents the components of semiotic intentionality (viz. sign, act, (...) content, referent) along with their division into an intending and a fulfilling side (Sect. 1), the first main part of the paper analyzes semiotic intentionality at its primary level (a.k.a. 'concrete intentionality') as a real and subjective relation between meaning-intending and meaning-fulfilling acts grounded in the manipulation of some semiotic system (Sect. 2). Then, building on such concrete intentionality, the second main part of the paper analyzes semiotic intentionality at its secondary level (a.k.a. 'abstract intentionality') as an ideal and objective relation between intentional and fulfillment contents, which in turn: (i) proceed from an abstraction performed on the intending and fulfilling acts, respectively, and (ii) retroactively categorize the intending and fulfilling acts, respectively (Sect. 3). Finally, from this combination of an act-based conception of content with a presentationalist account of intentionality, the conclusion of this paper produces the intended model of semiotic intentionality in such a way that knowledge and truth are then respectively defined in it as the subjective correspondence between the two acts of concrete intentionality and as the objective correspondence between the two contents of abstract intentionality (Sect. 4). (shrink)

In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, 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. (shrink)

Carnap’s work was instrumental to the liberalization of empiricism in the 1930s that transformed the logical positivism of the Vienna Circle to what came to be known as logical empiricism. A central feature of this liberalization was the deployment of the Principle of Tolerance, originally introduced in logic, but now invoked in an epistemological context in “Testability and Meaning”. Immediately afterwards, starting with Foundations of Logic and Mathematics, Carnap embraced semantics and turned to interpretation to guide the choice of a (...) theoretical language for science. The first thesis of this paper is that recourse to an intended interpretation led to a partial retrenchment of the conventionalism implied by the Principle of Tolerance. It required that the choice of a language be based on abstraction from a context; this procedure later became a component of the process of explication that was distinctive to Carnap’s mature views. The interpretive origin of formal systems also ensured their likely syntactic consistency, an issue on which Carnap was strongly criticized by figures such as Beth and Gödel. The second thesis of this paper is that this reliance on an intended interpretation enabled constructed formal systems to be relevant to the development of empirical science. (shrink)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, 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. (shrink)

In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style ofPrincipia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of (...) axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory. (shrink)

This paper consists of three sections. In the first one, some of the main developments in the philosophy of science through the xx century up to the present will be pointed out, and inserted them in the frame of some more general philosophical transformations, such as the so-called “linguistic turn” and “pragmatic turn”, respectively. In the second one, the established connection will be nuanced, from a revision of the work of a “classical” author such as Carnap. Finally, it will be (...) intended a kind of “balance and future perspectives”. (shrink)