Resolving Radzki’s issues with Łukasiewicz logics’ axiomatics via correspondence analysis
Yaroslav Petrukhin, Vasilyi Shangin · Journal of Applied Non-Classical Logics · 2025
This paper examines a series of works by Radzki, who addresses the problem of axiomatizing Łukasiewicz’s groundbreaking three-valued logic Ł3T (Ł3 with Słupecki’s operator T) and n-valued logic Łn for n⩾3. According to Radzki, the solution presented in Słupecki’s textbook proof for the case n = 3 is flawed. Furthermore, Radzki demonstrates that the textbook solutions provided by Rosser and Turquette, as well as by Grigolia, for the case n>3 are also inadequate. As a result of Radzki’s studies, the only indisputable solution is attributed to Tuziak. However, Radzki highlights certain shortcomings in this solution: specifically, Tuziak’s completeness proof is indirect and relies on non-logical algebraic methods. This paper is inspired by the open question posed by Radzki, namely: ‘How exactly we can construct, e.g. the Kalmár-type completeness proof for the Tuziak axiom systems for Łn’? We adopt a proof-theoretic method known as correspondence analysis and present natural deduction systems for Łukasiewicz’s three-valued logic Ł3T and n-valued (n⩾3) logic Łn. The completeness of these systems is established in a unified manner using the one hundred-percent logical Henkin method. Additionally, we provide formal proofs for the formulas Radzki uses to demonstrate the failure of the aforementioned solutions. Finally, we address another issue raised by Radzki: while applying correspondence analysis, we propose the first proof system for Radzki’s logic of atomic transactions AtL, which he defines semantically using the interjunction connective introduced by Blamey.