Frege's Begriffsschrift is Indeed First-Order Complete

History and Philosophy of Logic 38 (4):342-344 (2017)
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It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete.

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Yang Liu
Cambridge University

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References found in this work

Begriffsschrift, a Formula Language, Modeled upon that of Arithmetic, for Pure Thought [1879].Gottlob Frege - 1879 - From Frege to Gödel: A Source Book in Mathematical Logic 1931:1--82.
Introduction to Mathematical Logic.John Corcoran - 1964 - Journal of Symbolic Logic 54 (2):618-619.
Introduction to Mathematical Logic.Dirk van Dalen - 1964 - Journal of Symbolic Logic 34 (1):110-111.

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