In his classic paper [5] Mal'cev applied certain methods of mathematical logic in order to prove local theorems for various classes of groups. In [1] Cleave simplified certain ideas of Mal'cev and proved the local theorems for the same classes of groups. In this paper we describe a metamathematical method similar to that of Mal'cev and Cleave. Using this method we study properties of various classes of finite and infinite groups expressed by second order sentences which will be called strongly boolean- (universal-existential).

We say that a class of groups 3 has the B-property if G1, G2 € J and G1 < G < G2 imply G € J, (B-property is the abbreviation of Betweenness property, meaning that whenever a group G is “between” two groups which belong to a class then necessarily G is in the class. As usual H <G means that H is a subgroup of G.) Then, an immediate consequence of Theorem 1 in Section I is that classes of groups defined by strongly boolean-(universal-existential) sentences have the B-property. Such sentences are closely related to crypto-universal sentences [1]. In particular, second order sentences of the form (QP) (Vv) Ф- where QP denotes a sequence of predicate quantifiers, Vv denotes a sequence of universal individual quantifiers and 4 is quantifier free – are called arypto-universa I. Properties defined from such sentences are hereditary.