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Title: Axiomatization via Translation
Subtitle: Hiż's Warning for Predicate Logic
Author(s): BADIA, Guillermo , CROSSLEY, John N. , HUMBERSTONE, Lloyd
Journal: Logique et Analyse
Volume: 257    Date: 2022   
Pages: 39-56
DOI: 10.2143/LEA.257.0.3291070

Abstract :
The problems of logical translation of axiomatizations and the choice of primitive operators have surfaced several times over the years. An early issue was raised by H. Hiż in the 1950s on the incompleteness of translated calculi. Further pertinent work, some of it touched on here, was done in the 1970s by W. Frank and S. Shapiro, as well as by others in subsequent decades. As we shall see, overlooking such possibilities has led to incorrect claims of completeness being made (e.g. by J.L. Bell and A.B. Slomson as well as J.N. Crossley) for axiomatizations of classical predicate logic obtained by translation from axiomatizations suited to differently chosen logical primitives. In this note we begin by discussing some problematic aspects of an early article by W. Frank on the difficulties of obtaining completeness theorems for translated calculi. Shapiro had established the incompleteness of Crossley’s axiomatization by exhibiting a propositional tautology that was not provable. In contrast, to deal with Bell and Slomson’s system which is complete for propositional tautologies, we go on to show that taking a formal system for classical does not guarantee completeness of the resulting system.

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