Classical Logic with n Truth Values as a Symmetric Many-Valued Logic
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Classical Logic with n Truth Values as a Symmetric Many‑Valued Logic A. Salibra1 · A. Bucciarelli2 · A. Ledda3 · F. Paoli3
© The Author(s) 2020
Abstract We introduce Boolean-like algebras of dimension n ( nBA s) having n constants 𝖾1 , … , 𝖾n , and an (n + 1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The nBA s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, nCL , and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in nCL , and, via the embeddings, in all the finite tabular logics. Keywords Many-valued logics · Classical logic with n truth values · Boolean-like algebras · Equipollence · Rewriting systems Mathematics Subject Classification 03B50 · 08B05 · 08A70
1 Introduction This paper means to kick off a general programme aimed at bridging several different areas of logic, algebra and computation—the algebraic analysis of conditional statements in programming languages, the theory of factorisations of algebras, the theory of Boolean vector * A. Salibra [email protected] A. Bucciarelli [email protected] A. Ledda [email protected] F. Paoli [email protected] 1
Università Ca’Foscari Venezia, Venice, Italy
2
IRIF, CNRS and Université de Paris, Paris, France
3
Università di Cagliari, Cagliari, Italy
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spaces, the investigation into generalisations of classical logic—most of which somehow revolve around the main concept that lies at the crossroads of the three disciplines: the notion of a Boolean algebra. There is a thriving literature on abstract treatments of the if-then-else construct of computer science, starting with McCarthy’s seminal investigations (McCarthy 1963). On the algebraic side, one of the most influential approaches originated with the work of Bergman (1991). Bergman modelled the if-then-else by considering Boolean algebras acting on sets: if the Boolean algebra of actions is the 2-element algebra, one simply puts 1(a, b) = a and 0(a, b) = b . In Salibra et al. (2013), on the other hand, some of the present authors took their cue from Dicker’s axiomatisation of Boolean algebras in the language with the ifthen-else as primitive (Dicker 1963). Accordingly, this construct was treated as a proper algebraic operation q𝐀 on algebras 𝐀 whose type contains, besides the ternary term q, two constants 0 and 1, and having the property that for every a, b ∈ A , q𝐀 (1𝐀 , a, b) = a and q𝐀 (0𝐀 , a, b) = b . Such algebras, called Church algebras in Salibra et al. (2013), will be termed here Church algebras of dimension 2. The reason for this denomination is as follows. At the root of the most i
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