By J N Crossley, M A E Dummett

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HERMES AND D. ), Germany 1. A class T of formulae with unsolvable decision problem for validity The method is based on an appropriate description of a Semi-Thuesystem in the language of the restricted lower predicate calculus. No effort has been made to optimize the resulting reduction type. We hope that our method will yield better results in the future. We start with the well-known notion of a Semi-Thue-system S. , aN' The relations W =;Os W' (W immediately produces W') and W ~s W' (W produces W') are introduced in the usual way by the use of a finite system (L k , R k ) (k = 1, ...

L) and from the free individual symbols ai' a2' ... , ai. > a 2, ... , ak) (i = 1,2, ... ) be all those atomic formulae so defined which contain at least one occurrence of ak' Then we have, trivially, DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC (4) II, A;(a1' ... , a k - i = 1 = IIs i = 1 A;(a 1, ... , ak - 1) & l' 53 ak) = IIt B;(a 1, ... l)-(P. 3), it is advisable to define a closely related kind of formula which will be called an attributive constituent (in short, an a-constituent) with the same parameters.

Memoirs of the American Math. Soc. 44 (1963). DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC 49 parallel normal forms are so different as to make a separate treatment advisable. 2. Special cases The distributive normal forms of first-order logic are generalizations of the well-known "complete" normal forms of propositional logic and of monadic first-order logic. The notation and terminology that will be employed in this paper can be conveniently explained in terms of these more restricted normal forms.