Factor semantics for n-valued logics (Q800349)

The purpose of the article is to interpret truth-values of finite-valued logics in terms of the ''classical'' truth-values T (truth) and F (falsity) [see an earlier result by the author in Acta Philos. Fenn. 35, 7-22 (1982; Zbl 0525.03008)]. Assume that \(B=\{T,F\}\) and (B,\(\neg,\supset)\) is the two-element Boolean algebra. For any natural number \(s\geq 2\) the Descartes product of the algebra (B,\(\neg,\supset)\) is designated by \((B^ s,\neg,\supset)\). For any \(a\in B^ s\) we designate the number of occurrences of T by \(\eta\) (a); say that \(a\cong b\), if \(\eta (a)=\eta (b)\). The factor-set \(B^ s/\cong\) is provided with operations \(\sim\), \(\to\) as follows: \(\sim| a| =|\neg a|\) and \(| a|\to | b| =| a'\supset b'|\) where a'\(\in| a|\), b'\(\in| b|\) and a'Rb': \(<a'(1),...,a'(s)>R<b'(1),...,b'(s)>,\) iff either (1) \(\eta\) (a')\(\leq\eta (b')\) and \((\forall k\leq s)(a'(k)=T\Rightarrow b'(k)=T),\) or (2) \(\eta\) (a')\(>\eta (b')\) and \((\forall k\leq s)(b'(k)=T\Rightarrow a'(k)=T).\) Hence, the algebra \((B^ s/\cong,\sim,\to)\) is an adequate model of \((s+1)\)-valued \({\L}ukasiewicz's\) logic. Necessary conditions are established for a system of many-valued logics having semantics of this kind. It should be added that the author extends this semantics to infinite-valued \({\L}ukasiewicz's\) logic \({\L}_{\aleph_ 0}\) in the article ''Factor-semantics for infinite-valued \({\L}ukasiewicz's\) logic'' [Non-classical logic, Proc. Res. Semin. Inst. Philos. USSR (Russian) (Moscow, 1985)].