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)

Two hundred years ago Bernard Bolzano published a booklet on the philosophy of mathematics that is the first major step forward in this area since Pascal’s De l’esprit géométrique. Following Aristotelian lines Bolzano distinguishes in his opusculum two kinds of proofs, those that simply show that something is the case, and those that explain why something is the case. In his Wissenschaftslehre this contrast reappears as that between derivability and consecutivity . Husserl takes up some of Bolzano’s key concepts in (...) his Prolegomena to Pure Logic. In this paper I discuss, among other things, the question whether consecutivity can be regarded as a special case of derivability, as Husserl seems to think, and I contrast Bolzano’s rejection of any appeal to self-evidence with Husserl’s reliance on this notion.Genau vor zweihundert Jahren erschienen Bernard Bolzano Beyträge zu einer begründeteren Darstellung der Mathematik. Dieses Büchlein ist der erste Meilenstein in der Geschichte der Philosophie der Mathematik seit Pascals De l’esprit géométrique. Im Anschluss an Aristoteles unterscheidet Bolzano in seinem Opusculum zwei Arten von Beweisen: solche, die nur dartun, dass etwas der Fall ist, und solche, die erklären warum etwas der Fall ist. Im seiner Wissenschaftslehre erscheint dieser Kontrast als der zwischen Ableitbarkeit und Abfolge. Husserl übernimmt einige von Bolzanos Grundbegriffen in seinen Prolegomena zur reinen Logik. In diesem Aufsatz diskutiere ich u.a. die Frage, ob Abfolge ein Spezialfall von Ableitbarkeit ist, wie Husserl anzunehmen scheint, und ich kontrastiere Bolzanos Ablehnung jeder Berufung auf Evidenz mit Husserls Einstellung. (shrink)

The aim of the paper is to present and evaluate Bolzano's theory of grounding, that is, his theory of the concept expressed and the relation brought into play by 'because'. In the first part of the paper (Sections 1-4) the concept of grounding is distinguished from and related to three other concepts: the concept of an epistemic reason}, the concept of causality, and the concept of deducibility (i.e., logical consequence). In its second part (Sections 5-7) Bolzano's positive account of grounding (...) is reconstructed in axiomatic form and critically discussed. (shrink)

This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It (...) is argued that these two sorts of concepts must be kept strictly apart. (shrink)

This Paper is an Exploration of Bolzano’s views on causation, which have not been thoroughly examined yet. The paper reconstructs Bolzano’s position, with a focus on his analysis of the concept of causation, on its ontological presuppositions, and on how he relates causation to his theory of grounding.1 A comparison with standard positions from the contemporary debate on causation will prove his views to be quite original. Moreover, they are a valuable addition to the more recent debate on metaphysical grounding,2 (...) in which grounding is sometimes informally described as something like metaphysical causation with the exact connection of the two notions seldom being elaborated. Bolzano’s theory explicitly addresses .. (shrink)

In spite of its significance for everyday and philosophical discourse, the explanatory connective has not received much treatment in the philosophy of logic. The present paper develops a logic for based on systematic connections between and the truth-functional connectives.

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 is devoted to Bolzano’s theory of grounding (Abfolge) in his Wissenschaftslehre. Bolzanian grounding is an explanatory consequence relation that is frequently considered an ancestor of the notion of metaphysical grounding. The paper focuses on two principles that concern grounding in the realm of conceptual sciences and relate to traditionally widespread ideas on explanations: the principles, namely, that grounding orders conceptual truths from simple to more complex ones (Simplicity), and that it comes along with a certain theoretical economy among (...) them (Economy). Being spelled out on the basis of Bolzano’s notion of deducibility (Ableitbarkeit), these principles are revealing for the question to what extent grounding can be considered a formal relation. (shrink)

According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of (...) this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence. (shrink)

Bolzano's Sätze an sich and Frege's Gedanken are obviously close relatives. The paper underlines both similarities and dissimilarities between the psychological and semantical roles assigned to structured truth-evaluable contents in Bolzano's and Frege's theories. In particular, their different accounts of propositional identity are compared, and it is argued that Dummett's recent criticism of Frege's account is grist to Bolzano's mill.