Invariant transformations of formulas (Q1065781)

The paper is concerned with the invariance of some transformations of formulas in predicate logic with respect to classes of models. These transformations, called \(\zeta\)-transformations, are based on a general modus ponens rule. An existential quantifier (\(\exists y)\) is defined to be essential with respect to a class of models K if \(Q_ 1(\exists y)Q_ 2(F(y)\to G(y))\) is valid in all models of K, while \(Q_ 1(\forall x)Q_ 2(F(x)\to G(x))\) is false in all models of K. It is proved, that \(\zeta_ A\)- for A of the form \(Q(\exists y)(F(\bar x)\to G(\bar x,\bar y))\)- is invariant iff A has no essential \(\exists\)- quantifiers \((\zeta_ A\) is a transformation of formulas which depends on A and has the form \(Q(F(\bar x)\to G(\bar x)).\) Moreover variants \(\zeta\), \(\zeta^*\) of the transformation \(\zeta\) are defined, which differ with respect to the quantifier prefix of the transformation. The second theorem establishes a criterion for the invariance of \(\zeta^*\) under restriction on the signature of the models.