Meaning-Preserving Translations of Non-classical Logics into Classical Logic: Between Pluralism and Monism

Abstract

In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value (sec. 1). Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language (sec. 2–5). Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic.

Optimality justifications and the optimality principle: New tools for foundation‐theoretic epistemology

“…The optimality program has been applied to the domain of logics (Schurz, 2021b). This application is based on two observations:…”

“…The optimality program has been applied to the domain of logics (Schurz, 2021b). This application is based on two observations: The most fundamental goal of logical frameworks is providing the necessary logical means of describing arbitrary ‘manifolds’ or worlds, and non‐classical logics can be translated into classical logic. …”

“…Following Schurz (2021b), a meaning‐preserving translation of non‐classical into classical logics is possible by introducing suitable intensional operators into classical logic. In what follows, L i stands for systems of (propositional) logic and L i for the corresponding languages.…”

“…For any L 3 ‐assertion A, its translation into L 2.Luk is defined as Τ 3 (A), i.e., trans(A) = T 3 (A). Based on this translation function the following translation theorem is proved (Schurz, 2021b, theorem (4)); “|==” for “logical inference”: (7) Theorem . For all Γ ⊆ L 3 and A∈ L 3 (where T 3 (Γ) = def {T 3 (B): B∈Γ}): (7.1)For every L 3 ‐valuation v 3 there exists exactly one L 2.Luk ‐valuation v 2 such that: v 3 (A) = o 3 iff v 2 (O 3 (A)) = t 2 , where O 3 = T 3 /U 3 /F 3 iff o 3 = t 3 /u 3 /f 3 . (7.2)Γ |== L3 A iff Ax Luk ∪ T 3 (Γ) |== L2 T 3 (A). …”

Improving the efficiency of using multivalued logic tools: application of algebraic rings

“…Multivalued logic has been developing quite actively in recent decades 1 , 2 . Various varieties of multivalued logics have been developed, in particular, paralogics 3 , including paracomplete 4 and paraconsistent 5 logic.…”