Begriffsschrift

Begriffsschrift
Title	Begriffsschrift
Author	Gottlob Frege
Language	German
Published	1879
Genre	logicalnotation
Publisher	Vierteljahrsschrift für wissenschaftliche Philosophie

Begriffsschrift Begriffsschrift is an 1879 work by Gottlob Frege that introduced a novel formal language for predicate logic and a rigorous treatment of inference, quantification, and identity. It presented a two-dimensional notation and proof theory that influenced later developments in mathematical logic, philosophy of language, and foundations of mathematics. The book laid groundwork later taken up by figures associated with logicism, analytic philosophy, and formalist and platonist debates in 19th-century philosophy and 20th-century philosophy.

Background and Context

Frege wrote his system in the intellectual milieu shaped by debates among proponents of Georg Cantor, Bernhard Riemann, Richard Dedekind, Leopold Kronecker, and contemporaries in German philosophy such as Immanuel Kant, G. W. F. Hegel, and Friedrich Schleiermacher. His project responds to mathematical problems exemplified in works by Carl Friedrich Gauss and the arithmetical foundational programs associated with David Hilbert. The Begriffsschrift emerged against discussions in periodicals like the Jahrbuch über die Fortschritte der Mathematik and institutions such as the University of Jena and University of Göttingen, and it engaged issues later central to debates involving Bertrand Russell, Alfred North Whitehead, Ludwig Wittgenstein, Rudolf Carnap, and Kurt Gödel.

Notation and Formal System

Frege's notation introduced a two-dimensional formula layout, combining concepts drawn from prior notational practices used by Augustus De Morgan and George Boole yet differing substantially from later linear systems developed by Peano, Giuseppe Peano, Charles Sanders Peirce, and Ernst Schröder. The Begriffsschrift formalism defined function-argument structure, quantifiers, and identity; it specified axioms and inference rules that anticipate aspects of predicate logic, propositional calculus, and substitutional principles used in systems by Alonzo Church and Emil Post. Frege employed distinct notation for the universal quantifier and conditional, and he formulated rules analogous to later sequent calculi associated with Gerhard Gentzen and natural deduction approaches by Jaśkowski and Dag Prawitz.

Main Results and Theorems

Begriffsschrift presented proofs aiming to reduce arithmetic to logic, offering derivations of basic arithmetical truths from logical axioms and definitions, a program later labeled logicism by commentators such as Russell and W. V. Quine. It introduced a formal treatment of identity and the identity of indiscernibles discussed by Gottfried Wilhelm Leibniz and addressed quantificational consequences that played a role in Russell's paradox and ensuing set-theoretic crises that concerned Ernest Zermelo, Thoralf Skolem, and Paul Bernays. Key theorems included compositionality results for functions and derivations of arithmetical laws resembling those later formalized in Principia Mathematica by Whitehead and Russell and in axiomatic systems studied by Kurt Gödel.

Reception and Influence

Initial reception among contemporaries such as Hermann Lotze and mathematicians at Humboldt University of Berlin was limited; later, figures like Bertrand Russell, Peano, Giuseppe Peano, and Ludwig Wittgenstein recognized Frege's innovations. The Begriffsschrift influenced the rise of analytic philosophy through its impact on G. E. Moore, Rudolf Carnap, and the Vienna Circle including Moritz Schlick and Otto Neurath, and it informed technical developments by logicians at Princeton University and University of Göttingen. Its ideas filtered into work by Alonzo Church, Kurt Gödel, Gerhard Gentzen, and Alan Turing, and they contributed to later symbolic and model-theoretic traditions associated with Alfred Tarski and Jerzy Łoś.

Editions and Translations

The original 1879 German edition was published in the journal of Bruno Bauer's circle; subsequent reprints and scholarly editions have been produced by editors in Berlin and academic presses in Oxford and Cambridge. Translations into English and other languages were prepared by scholars working on Frege's collected works alongside editors responsible for editions such as those published in volumes associated with J. van Heijenoort and critical commentaries by Michael Dummett, Hans Hermes, G. J. Thompson, and Geoffrey Warnock. Modern critical editions pair the original text with commentary linking Begriffsschrift to archival materials held at repositories like the University of Jena and the Humboldt University of Berlin.

Legacy in Logic and Philosophy

Begriffsschrift's formal innovations underpin later theoretical work in model theory, proof theory, and the philosophy of mathematics pursued by Kurt Gödel, Alfred Tarski, Hartry Field, Michael Dummett, and Hilary Putnam. Its influence extends to computational theories developed by Alan Turing, Alonzo Church, and Stephen Kleene, and to contemporary debates in metaphysics and semantics engaged by Donald Davidson, David Kaplan, and Saul Kripke. The conceptual resources introduced in Begriffsschrift continue to inform research in philosophy of language, formal semantics, and foundations treated in programs at institutions such as Princeton University, Oxford University, and the University of Cambridge.

Category:Works by Gottlob Frege