Curry's paradox, generalized modus ponens axiom and depth relevance
Let a logical matrix \(M\) be a structure \((K, T, F, f_\rightarrow, f_\wedge, f_\vee, f_ \neg)\), where \(K\) is a set, \(T\) and \(F\) are non-empty subsets of \(K\) such that \(T \cup K = K\) and \(T \cap K = \emptyset\), \(f_\rightarrow, f_\wedge, f_\vee\) are binary function on \(K\), and \(f_ \neg\) is an unary function on \(K\). Then, \(M\) is said to be a weak relevant matrix (wr-matrix) iff there is \(a_F \in K\), \(K_1 \subseteq K\) and \(K_2 \subseteq K\), such that \(K_1\) and \(K_2\) are disjoint, and the following conditions are fulfilled: \(\forall x \forall y \in K_1(K_2) \, f_\sharp(x,y) \in K_1 (K_2) \text{ and } f_\neg\in K_1(K_2)\) (\(\sharp \in \{\rightarrow, \wedge,\vee\}\)); \(\forall x \in K_1 \forall y \in K_2 f_\rightarrow(x, y) = a_F\); \(\forall x \in K_1 \cup K_2 f_\rightarrow(x, a_F) = a_F\). Then, it can be shown that any logic veryfied by a wr-matrix has the variable-sharing property. A wr-model structure (wr-ms) \({\mathcal M}_M\) is defined as the set \(\{ M_0, M_1, \dots, M_n, \dots, M_\omega\}\), where \(M_0, M_1, \dots, M_n, \dots, M_\omega\) are all identical matrices to some wr-matrix \(M\). It can be shown that every logics veryfied by some wr-ms, satisfies the so-called depth relevance conditions in the sense of Brady. It can also be shown that the so-called ``deep relevant logics'' verified by a wr-ms are not subject to a generalized modus ponens used to derive Curry's paradox.
scientific article; zbMATH DE number 6305280
scientific article