Throughout more than two millennia philosophers adhered massively to ideal standards of scientific rationality going back ultimately to Aristotle’s Analytica posteriora . These standards got progressively shaped by and adapted to new scientific needs and tendencies. Nevertheless, a core of conditions capturing the fundamentals of what a proper science should look like remained remarkably constant all along. Call this cluster of conditions the Classical Model of Science . In this paper we will do two things. First of all, we will (...) propose a general and systematized account of the Classical Model of Science. Secondly, we will offer an analysis of the philosophical significance of this model at different historical junctures by giving an overview of the connections it has had with a number of important topics. The latter include the analytic-synthetic distinction, the axiomatic method, the hierarchical order of sciences and the status of logic as a science. Our claim is that particularly fruitful insights are gained by seeing themes such as these against the background of the Classical Model of Science. In an appendix we deal with the historiographical background of this model by considering the systematizations of Aristotle’s theory of science offered by Heinrich Scholz, and in his footsteps by Evert W. Beth. (shrink)

This paper analyzes and evaluates Bolzano's remarks on the apagogic method of proof with reference to his juvenile booklet "Contributions to a better founded presentation of mathematics" of 1810 and to his ?Theory of science? (1837). I shall try to defend the following contentions: (1) Bolzanos vain attempt to transform all indirect proofs into direct proofs becomes comprehensible as soon as one recognizes the following facts: (1.1) his attitude towards indirect proofs with an affirmative conclusion differs from his stance to (...) indirect proofs with a negative conclusion; (1.2) by Bolzano's lights arguments via consequentia mirabilis only seem to be indirect. (2) Bolzano does not deny that indirect proofs can be perfect certifications (Gewissmachungen) of their conclusion; what he denies is rather that they can provide grounds for their conclusions. (2.1) They cannot do the latter, since they start from false premises and (2.2) since they make an unnecessary detour. (3) The far-reaching agreement between his early and late assessment of apagogical proofs (in the Beyträge of 1810 and the Wissenschaftslehre of 1837) is partly due to the fact that he develops his own position always against the background of Wolff's and Lambert's views. (shrink)

Contemporary historians of logic tend to credit Bernard Bolzano with the invention of the semantic notion, of consequence, a full century before Tarski. Nevertheless, Bolzano's work played no significant rôle in the genesis of modern logical semantics. The purpose of this paper is to point out three highly original, and still quite relevant themes in Bolzano's work, being a systematic study of possible types of inference, of consistency, as well as their meta-theory. There are certain analogies with Tarski's concerns here, (...) although the main thrust seems to be different, both philosophically and technically. Thus, if only obliquely, we also provide some additional historical perspective on Tarski's achievement. (shrink)

Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped (...) with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent. (shrink)

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.

A collection of material on Husserl's Logical Investigations, and specifically on Husserl's formal theory of parts, wholes and dependence and its influence in ontology, logic and psychology. Includes translations of classic works by Adolf Reinach and Eugenie Ginsberg, as well as original contributions by Wolfgang Künne, Kevin Mulligan, Gilbert Null, Barry Smith, Peter M. Simons, Roger A. Simons and Dallas Willard. Documents work on Husserl's ontology arising out of early meetings of the Seminar for Austro-German Philosophy.

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 ...

Eine vollständige Darstellung von Edmund Husserls Verhältnis zu Gottlob Frege steht noch aus, so dass es nicht verwundert, einige Missverständnisse, dieses Verhältnis betreffend, im Umlauf zu finden. Selbst scheinbar längst überwundene systematische Dogmen tauchen wieder auf, so z.B. die Auffassung, dass Husserl nicht nur entscheidend von Gottlob Frege beeinflusst wurde, sondern darüber hinaus auch seine schärfste Frege-Kritik 1891 zurückgenommen habe. Mein Beitrag enthält eine überwiegend historisch vorgehende Entgegnung auf solche fälschlich vertretenen Ansichten wie sie sich auch in dem neu erschienenen (...) und mit vielen dokumentarischen Belegen versehenen Frege-Kommentarbuch von W. Künne finden. Die wichtigsten Argumente, die in diesem Buch gegen Husserl gerichtet werden, stützen sich auf Stellen in Husserls Frühwerk, die zwar sehr aufschlussreich für die Beurteilung des philosophischen und menschlichen Habitus Husserls in den 1890er Jahre sind, nicht aber Künnes Thesen stützen oder gar beweisen können. Die als zurückgenommene Frege-Kritik verstandene Selbstkritik Husserls betrifft jedoch nach meinen Darlegungen gerade nicht die Kritik, die er an Freges Buch Grundlagen der Arithmetik geübt hatte, sondern seine eigene systematische Stellungnahme hinsichtlich der Begründung der Arithmetik. Ausgehend von dieser Einsicht werde ich auf ein wichtiges Spezifikum von Husserls Erstlingswerk (Philosophie der Arithmetik) aufmerksam machen, das voreiligen Interpretationen einen Riegel vorschiebt: Dieses Buch ist ein Mosaik aus Einsichten verschiedener Arbeits- und Denkperioden des Verfassers, die sich vor allem auch in dem zweifachen Begriff der Äquivalenz zeigen. Dies knüpft an einen langen geschichtlichen Entwicklungsprozess an, der in eine von Bolzano, Lotze, Brentano und Carl Stumpf geprägte Tradition zurückführt und das Problem der Unterscheidung zwischen Sinn und Gegenstand eines Aktes betrifft. (shrink)

In this paper I argue that Bolzano's concept of deducibility and Tarski's concept of logical consequence differ with respect to their philosophical intent. I distinguish between epistemic and ontic approaches to logic, and argue that Bolzano's deducibility presupposes an epistemic approach, while Tarski's logical consequence presupposes an ontic approach.

This compelling reevaluation of the relationship between logic and knowledge affirms the key role that the notion of judgement must play in such a review. The commentary repatriates the concept of judgement in the discussion, banished in recent times by the logical positivism of Wittgenstein, Hilbert and Schlick, and the Platonism of Bolzano. The volume commences with the insights of Swedish philosopher Per Martin-Löf, the father of constructive type theory, for whom logic is a demonstrative science in which judgement is (...) a settled feature of the landscape. His paper opens the first of four sections that examine, in turn, historical philosophical assessments of judgement and reason; their place in early modern philosophy; the notion of judgement and logical theory in Wolff, Kant and Neo-Kantians like Windelband; their development in the Husserlian phenomenological paradigm; and the work of Bolzano, Russell and Frege. The papers, whose authors include Per Martin-Löf, Göran Sundholm, Michael Della Rocca and Robin Rollinger, represent a finely judged editorial selection highlighting work on philosophers exercised by the question of whether or not an epistemic notion of judgement has a role to play in logic. The volume will be of profound interest to students and academicians for its application of historical developments in philosophy to the solution of vexatious contemporary issues in the foundation of logic. ​. (shrink)

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less (...) numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding. (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)

This paper concentrates on some aspects of the history of the analyticsynthetic distinction from Kant to Bolzano and Frege. This history evinces considerable continuity but also some important discontinuities. The analytic-synthetic distinction has to be seen in the first place in relation to a science, i.e. an ordered system of cognition. Looking especially to the place and role of logic it will be argued that Kant, Bolzano and Frege each developed the analytic-synthetic distinction within the same conception of scientific rationality, (...) that is, within the Classical Model of Science: scientific knowledge as cognitio exprincipiis. But as we will see, the way the distinction between analytic and synthetic judgments or propositions functions within this model turns out to differ considerably between them. (shrink)

volumes of his work, in his discussions of what underlay a Wissenschaftslehre or theory of science in the sense of his conception; he did so with such purity and scientific strictness, and with such a rich store of original, scientifically confirmed and fruitful thoughts, that we must count him as one of the greatest logicians of all time. He must be placed historically in fairly close proximity to Leibniz, with whom he shares important thoughts and fundamental conceptions, and to whom (...) he is also philosophically akin in other respects.". (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.