Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

This paper is aimed at understanding one central aspect of Bolzano's views on deductive knowledge: what it means for a proposition and for a term to be known a priori. I argue that, for Bolzano, a priori knowledge is knowledge by virtue of meaning and that Bolzano has substantial views about meaning and what it is to know the latter. In particular, Bolzano believes that meaning is determined by implicit definition, i.e. the fundamental propositions in a deductive system. I go (...) into some detail in presenting and discussing Bolzano's views on grounding, a priori knowledge and implicit definition. I explain why other aspects of Bolzano's theory and, in particular, his peculiar understanding of analyticity and the related notion of Ableitbarkeit might, as it has invariably in the past, mislead one to believe that Bolzano lacks a significant account oï a priori knowledge. Throughout the paper, I point out to the ways in which, in this respect, Bolzano's antagonistic relationship to Kant directly shaped his own views. (shrink)

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

Bernard Bolzano of Prague was one of the few thinkers of his time who combined real talent in mathematics and philosophy. He was especially drawn to the common ground between these fields, interested in questions of method and what would today be called foundations . Interestingly, he was neither a professional mathematician nor a professional philosopher. As a young man, he had decided that his first priority must be to work for the reform and improvement of society. This led him, (...) after much reflection, to become a priest and to occupy a newly-created chair in religious science at the Charles University in 1805. In this post, he was widely known as what would later be called a dissident, denouncing abuses of power and the ills that were the consequence of the still largely feudal institutions and attitudes of the Austrian Empire, and inspiring a generation of students to transform these institutions from within. While his activities brought him a large and committed following, they also gained him powerful enemies in Vienna and Rome, who eventually convinced the Emperor to remove him from his post in December of 1819. Amazingly, during these years of energetic public life, in which he suffered horribly from lung disease and a variety of other ailments, Bolzano remained quite active as a mathematician. As his notebooks attest, he read the latest mathematical literature voraciously and systematically.1 He also made a number of important discoveries, especially in the foundations of real analysis, and published four mathematical works between 1810 and 1817 [Bolzano, 1810; 1816; 1817a; 1817b].2 After his dismissal, he continued his work, pursuing his earlier inquiries with greater depth and precision, …. (shrink)

G. F. Stout is famous as an early twentieth century proselyte for abstract particulars, or tropes as they are now often called. He advanced his version of trope theory to avoid the excesses of nominalism on the one hand and realism on the other. But his arguments for tropes have been widely misconceived as metaphysical, e.g. by Armstrong. In this paper, I argue that Stout’s fundamental arguments for tropes were ideological and epistemological rather than metaphysical. He moulded his scheme to (...) fit what is actually given to us in perception, arguing that our epistemic practices would break down in an environment where only universals were given to us. (shrink)

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...) with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

This thesis is on the history and philosophy of logic and semantics. Logic can be described as the ‘science of reasoning’, as it deals primarily with correct patterns of reasoning. However, logic as a discipline has undergone dramatic changes in the last two centuries: while for ancient and medieval philosophers it belonged essentially to the realm of language studies, it has currently become a sub-branch of mathematics. This thesis attempts to establish a dialogue between the modern and the medieval traditions (...) in logic, by means of ‘translations’ of the medieval logical theories into the modern framework of symbolic logic, i.e. formalizations. One of its conclusions is that, when properly understood within their own framework, the interest of medieval logical theories for modern investigations go beyond mere historical interest, but that a thorough conceptual analysis of such theories must be undertaken in order to avoid conceptual misprojections. While such translations of medieval into modern logic have been attempted before, the approach presented here is innovative in that attention is paid to the similarities as well as to the dissimilarities between the two traditions, and to what can be learned from the medieval masters for modern investigations in logic and semantics. (shrink)

In a series of publications beginning in the 1980s, John Etchemendy has argued that the standard semantical account of logical consequence, due in its essentials to Alfred Tarski, is fundamentally mistaken. He argues that, while Tarski's definition requires us to classify the terms of a language as logical or non-logical, no such division is guaranteed to deliver the correct extension of our pre-theoretical or intuitive consequence relation. In addition, and perhaps more importantly, Tarski's account is claimed to be incapable of (...) explaining an essential modal/epistemological feature of consequence, namely, its necessity and apriority. Bernard Bolzano (1781-1848) is widely recognized as having anticipated Tarski's definition in his Wissenschaftslehre (or Theory of Science ) of 1837. Because of the similarities between his account and Tarski's, Etchemendy's arguments have also been extended to cover Bolzano. The purpose of this article is to consider Bolzano's theory in the light of these criticisms. We argue that, due to important differences between Bolzano's and Tarski's theories, Etchemendy's objections do not apply immediately to Bolzano's account of consequence. Moreover, Bolzano's writings contain the elements of a detailed philosophical response to Etchemendy. (shrink)

In Poggiolesi we have introduced a rigorous definition of the notion of complete and immediate formal grounding; in the present paper our aim is to construct a logic for the notion of complete and immediate formal grounding based on that definition. Our logic will have the form of a calculus of natural deduction, will be proved to be sound and complete and will allow us to have fine-grained grounding principles.

The aim of this paper is to provide a definition of the the notion of complete and immediate formal grounding through the concepts of derivability and complexity. It will be shown that this definition yields a subtle and precise analysis of the concept of grounding in several paradigmatic cases.

In the course of the last few decades, Bolzano has emerged as an important player in accounts of the history of philosophy. This should be no surprise. Few authors stand at a more central junction in the development of modern thought. Bolzano's contributions to logic and the theory of knowledge alone straddle three of the most important philosophical traditions of the 19th and 20th centuries: the Kantian school, the early phenomenological movement and what has come to be known as analytical (...) philosophy. This paper identifies three Bolzanian theoretical innovations that warrant his inclusion in the analytical tradition: the commitment to ‘logical realism’, the adoption of a substitutional procedure for the purpose of defining logical properties and a new theory of a priori cognition that presents itself as an alternative to Kant's. All three innovations concur to deliver what counts as the most important development of logic and its philosophy between Aristotle and Frege. In the final part of the paper, I defend Bolzano against a common objection and explain that these theoretical innovations are also supported by views on syntax, which though marginal are both workable and philosophically interesting. (shrink)

Alongside his groundbreaking work in logic, Bernard Bolzano (1781?1848) made important contributions to ontology, notably with his theory of collections. Recent work has done much to elucidate Bolzano's conceptions, but his notion of a sum has proved stubbornly resistant to complete understanding. This paper offers a new interpretation of Bolzano's concept of a sum. I argue that, although Bolzano's presentation is defective, his conception is unexceptionable, and has important applications, notably in his work on the foundations of arithmetic.

Here I revisit Bolzano's criticisms of Kant on the nature of logic. I argue that while Bolzano is correct in taking Kant to conceive of the traditional logic as a science of the activity of thinking rather than the content of thought, he is wrong to charge Kant with a failure to identify and examine this content itself within logic as such. This neglects Kant's own insistence that traditional logic does not exhaust logic as such, since it must be supplemented (...) by a transcendental logic that will in fact study nothing other than thought's content. Once this feature of Kant's views is brought to light, a much deeper accord emerges between the two thinkers than has hitherto been appreciated, on both the nature of the content that is at issue in logic and the sense of logic's generality and formality. (shrink)

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary (...) mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao. (shrink)

El artículo tiene por objetivo analizar ciertos pasajes fundamentales de la Wissenschaftslehre y de las Paradoxien des Unendlichen de Bernard Bolzano en cuanto al análisis conjuntista se refiere. En dichos pasajes, Bolzano desarrolla conceptos fundamentales tales como multitud, colección e infinito que anticipan el carácter conjuntista y del análisis matemático moderno. Asimismo, se presentará un breve estudio de las Contribuciones a una más fundada exposición de la matemática y el apéndice, Sobre la teoría kantiana de la construcción de conceptos a (...) través de intuiciones, textos donde Bolzano muestra su rechazo por Kant.The article addresses the treatment of certain fundamental passages from Bolzano’s Wissenschaftslehre and Paradoxien des Unendlichen. In these passages, Bolzano describes some of his key concepts such “Multitude”, “Collections” and “Infinite” in the context of mathematical analysis and Set theory. Therefore, I discuss at length the unknown Contributions to a Better-Grounded Presentation of Mathematics and the appendix On the Kantian Theory of the Construction of Concepts through Intuitions, texts where Bolzano shows his rejection of Kant. (shrink)

The article addresses the works of three 19th-century philosophers: Herbart, Bolzano and Lotze. Despite their differences, i will analyze two essential features they share: their refusal to psychologism as the radicalization of psychology as the ultimate source of philosophy , and the positing of an ideal ‘reality’, independent from both sensibility and the mental and linguistic dimensions, upon which the refusal to psychologism is founded . As a conclusion, i will show how, according to these authors, this platonic, ideal background (...) constitutes the very foundation of every meaningful thought. (shrink)

This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent (...) research in metaphysics on the pair metaphysical grounding–metaphysical explanation, and especially with the literature in philosophy of science on the pair causality-causal explanation. (shrink)

El artículo aborda ciertos pasajes fundamentales de la Wissenschaftslehre de Bolzano, analizando algunos de sus conceptos claves tales como los de «proposición en sí» o «representación en sí». Haciendo especial hincapié en el estatus ontológico peculiar de estas «entidades en sí» que presenta Bolzano, mostraremos que su obra desarrolla una auténtica teoría semántica de la dimensión del sentido y de lo pensable, que no sólo no depende de la ontología, sino que desborda a la misma.

The paper discusses Bernard Bolzano’s epistemological approach to believing and knowing with regard to the epistemic requirements of an axiomatic model of science. It relates Bolzano’s notions of believing, knowing and evaluation to notions of infallibility, immediacy and foundational truth. If axiomatic systems require their foundational truths to be infallibly known, this knowledge involves both evaluation of the infallibility of the asserted truth and evaluation of its being foundational. The twofold attempt to examine one’s assertions and to do so by (...) searching for the objective grounds of the truths asserted lies at the heart of Bolzano’s notion of knowledge. However, the explanatory task of searching for grounds requires methods that cannot warrant infallibility. Hence, its constitutive role in a conception of knowledge seems to imply the fallibility of such knowledge. I argue that the explanatory task contained in Bolzanian knowing involves a high degree of epistemic virtues, and that it is only through some salient virtue that the credit of infallibility can distinguish Bolzanian knowing from a high degree of Bolzanian believing. (shrink)