According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a copula. Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence (Dasein)".

A starring role in Bolzano’s logical theory is played by the notion of variations: various logical relations are defined in terms of the changes in truth value that propositions incur when their non-logical parts are replaced by others. Logically analytical propositions, for instance, are those in which all the non-logical parts can be replaced without change of truth value. Two propositions are 'compatible' (vertraglich) with respect to one of their component parts x if there is at least one term that can be inserted that would make both true. A proposition Q is 'deducible' (ableitbar) from a proposition P, with respect to certain of their non-logical parts, if any replacement of those parts that makes P true also makes Q true. If a proposition is deducible from another with respect to all its non-logical parts, it is said to be 'logically deducible'. Besides the relation of deducibility, Bolzano also has a stricter relation of 'consequentiality' (Abfolge). This is an asymmetric relation that obtains between true propositions, when one of the propositions is not only deducible from, but also explained by the other.

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Mathematics

Bolzano made several original contributions to mathematics. In Parallelogram area theory he demonstrated that for similar rhombi, the ratio of the area of rhombus A to the area of rhombus B is equal to the square of the ratio of the width of A to the width of B. To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε-δ definition of a mathematical limit and the first purely analytic proof of the Intermediate Value Theorem (also known as Bolzano's theorem). Today he is mostly remembered for the Bolzano-Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered.

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Philosophical legacy

Due to the fact that Bolzano's most important work, the Wissenschaftslehre, could not be published during his lifetime, the impact of his thought on philosophy initially seemed destined to be slight. His work was rediscovered, however, by Edmund Husserl and Kazimierz Twardowski, both students of Franz Brentano. Through them, and through Gottlob Frege, also an admirer, Bolzano became a formative influence on both phenomenology and analytic philosophy.

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Writings in English

• Theory of science, attempt at a detailed and in the main novel exposition of logic with constant attention to earlier authors. (Edited and translated by Rolf George University of California Press, Berkeley and Los Angeles 1972)
• Theory of science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell - D. Reidel Publishing Company, Dordrecht and Boston 1973)
• Ewald, William B., ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press, 1996 contains the three following essays:
• 1810. Contributions to a better grounded presentation of mathematics, 174-224.
• 1817. Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation, 225-48.
• 1851. Paradoxes of the Infinite, 249-92 (excerpt).
• Paradoxes of the infinite - Translated from the German of the posthumous edition by Fr. Prihonský and furnished with a historical introduction by Donald A. Steele - Routledge & Kegan Paul, 1950.
• On the mathematical method and correspondence with Exner - Translated by Paul Rusnock and Rolf George - Amsterdam, Rodopi, 2004.
• The mathematical works of Bernard Bolzano - Edited by Steve Russ - Oxford, Oxford University Press, 2004.
• Selected Writings on Ethics and Politics - Translated by Paul Rusnock and Rolf George - Amsterdam, Rodopi, 2007.

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External links

• O'Connor, John J. & Robertson, Edmund F., “Bernard Bolzano”, MacTutor History of Mathematics archive 
• Biography of Bernard Bolzano
• Bernard Bolzano's Theory of Science
• Bernard Bolzano entry at the Stanford Encyclopedia of Philosophy by Edgar Morscher
• Bolzano's Logic entry at the Stanford Encyclopedia of Philosophy by Jan Sebestik

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References

Künne, Wolfgang. (1998). "Bolzano, Bernard". Routledge Encyclopedia of Philosophy 1: 823-827. London: Routledge. Retrieved on 2007-03-05

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