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On Boolean modus ponens. (English) Zbl 0931.03075

The inequality \(x\wedge(x\to y)\leq y\) is valid in an arbitrary Boolean algebra \((B,\vee, \wedge,',0,1)\), and it is a Boolean analogue of modus ponens. E. Trillas and S. Cubillo determined [Mathware Soft Comput. 3, No. 1-2, 105-112 (1996; Zbl 0862.03037)] other Boolean variants of modus ponens by replacing conjunction and implication by other truth functions \(f,g: B^2\to B\) that can be written in the form \(f(x,y)= axy\vee bxy' \vee cx'y \vee dx'y'\) and \(g(x,y)= pxy\vee qxy'\vee rx'y \vee sx'y'\), where \(a, \dots, d,p, \dots,s\in \{0,1\} \subseteq B\) and which satisfy the identity \(f(x, g(x,y))\leq y\). In the present paper the author uses the general theory of Boolean equations to obtain a generalization by allowing the coefficients \(a, \dots,s\) to be any constants from \(B\).

MSC:

03G05 Logical aspects of Boolean algebras
06E30 Boolean functions

Citations:

Zbl 0862.03037
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