A note on the principle of predication

Let A be a well-formed formula of first-order modal logic whose only free variable is x. We shall use the following abbreviations: Mat(A) for (x)[OA * 0~A] Form(A) for (x)[DA vD-i] Ban(A) for ((x)DA) v ((x)U~A) Pred(A) for Form(A) v Mat(A). We read Mat(A), FormjA), and Ban(A) respectively as: "A is material", "A is formal", and "A is banal"; Pred(A) is the assertion of the Principle of Predica-tion for A. We prove that if F and M are formulas whose only free variable is x such that BanjF), Form(F), and Mat(M) are true in any suitable T-model, then PredjF A M) and PredjF v M) are not acceptable as axioms. Theorem The formulas: (1) ~Ban{F) A Form(F) A MoUiM) D ~Pred(M A F) (2) ~Ban(F) A Form(F) A 7to(Af) D ~Pred(M v F) ore r-vfl/W. be a T-model ([1], p. 171) and w t e W. If V(~Ban(F) A Form(F) A Mat(M), w () = 1 then V(~Ban(F), w () = 1 and there exist a, Z? e Z); such that: (3) F*(O~F, w/) = 1 and V b (OF, w t) = 1 where V a and F & are just like V except for assigning a and b, respectively, tox.