Two theorems on many-valued logics (Q1106833)

A consequence operation on a propositional language \(L=<| L|,\{f_ k\}_{k\in K}>\) is a function C on the set of subsets of L such that for X,Y\(\subseteq | L|\) [X\(\subseteq C(X);X\subseteq Y\Rightarrow C(X)\subseteq C(Y);C(C(X))\subseteq C(X)]\). If \({\mathcal A}\) is an algebra similar to L, then \({\mathcal M}=<{\mathcal A},D,H>\) with \(D\subseteq | {\mathcal A}|\) and \(H\subseteq Hom(L,{\mathcal A})\) is called a matrix for L. Then \(C_{{\mathcal M}}=\cap_{h\in H,h(x)\subseteq D}h^{(-1)}(D)\) is called a matrix consequence operation. For a given logic \({\mathcal L}=<L,C>\), a class \({\mathcal K}\) of matrices for L is said to be a semantics for \({\mathcal L}\) iff \(C(X)=\cap \{C_{{\mathcal M}}(X):{\mathcal M}\in {\mathcal K}\}\) for all \(X\subseteq | L|\). If for each \({\mathcal M}\in {\mathcal K}\), card(\({\mathcal M})=\kappa\) then \({\mathcal K}\) is a \(\kappa\)-valued semantics for \({\mathcal L}\). \({\mathcal L}\) is called strongly \(\kappa\)-valued iff there is a one-element \(\kappa\)-valued semantics for \({\mathcal L}\). On the other hand \({\mathcal L}\) is said to have the ``Extensionality Property'' (EP) iff C has a basis \({\mathcal B}\) such that for every \(X\in {\mathcal B}\) the relation \((\Phi_ X:\) \(\alpha \Phi_ X\beta\) iff \(\alpha\in X\Leftrightarrow \beta \in X)\) is a congruence of L. \({\mathcal L}\) has the ``Strong Extensionality Property'' (SEP) iff there is a basis \({\mathcal B}\) of C such that for every \(X,Y\in {\mathcal B}\) if for \(i\in n\), \(\alpha_ i\in X\Leftrightarrow \beta_ i\in Y\) then \(f(\alpha_ 0...\alpha_{n- 1})\in X\Leftrightarrow f(\beta_ 0...\beta_{n-1})\in Y\). Now it is shown that \({\mathcal L}\) is 2-valued iff \({\mathcal L}\) has EP and \({\mathcal L}\) is strongly 2-valued iff \({\mathcal L}\) has SEP.