History and Philosophy of Logic 35 (1):1-26 (2014)

María Manzano
Universidad de Salamanca
This paper focuses on the evolution of the notion of completeness in contemporary logic. We discuss the differences between the notions of completeness of a theory, the completeness of a calculus, and the completeness of a logic in the light of Gödel's and Tarski's crucial contributions.We place special emphasis on understanding the differences in how these concepts were used then and now, as well as on the role they play in logic. Nevertheless, we can still observe a certain ambiguity in the use of the close notions of completeness of a calculus and completeness of a logic. We analyze the state of the art under which Gödel's proof of completeness was developed, particularly when dealing with the decision problem for first-order logic. We believe that Gödel had to face the following dilemma: either semantics is decidable, in which case the completeness of the logic is trivial or, completeness is a critical property but in this case it cannot be obtained as a corollary of a previous decidability result. As far as first-order logic is concerned, our thesis is that the contemporary understanding of completeness of a calculus was born as a generalization of the concept of completeness of a theory. The last part of this study is devoted to Henkin's work concerning the generalization of his completeness proof to any logic from his initial work in type theory
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DOI 10.1080/01445340.2013.816555
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References found in this work BETA

Introduction to Mathematical Logic.Alonzo Church - 1944 - London: Oxford University PRess.
The Concept of Truth in Formalized Languages.Alfred Tarski - 1936 - In A. Tarski (ed.), Logic, Semantics, Metamathematics. Oxford University Press. pp. 152--278.
The Concept of Logical Consequence.John Etchemendy - 1990 - Harvard University Press.
Completeness in the Theory of Types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.

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Citations of this work BETA

Completeness: From Husserl to Carnap.Víctor Aranda - 2022 - Logica Universalis 16 (1):57-83.
Visions of Henkin.María Manzano & Enrique Alonso - 2015 - Synthese 192 (7):2123-2138.
Weak and Post completeness in the Hilbert school.Víctor Aranda - 2019 - Humanities Journal of Valparaiso 14:449-466.

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