Ways of Branching Quantifers* Generalizing from the Existence of Skolem Normal Forms

Branching quantifiers were first introduced by L. Henkin in his 1959 paper 'Some Remarks on Infinitely Long Formulas'. By 'branching quantifiers' Henkin meant a new, non-linearly structured quantifier-prefix whose discovery was triggered by the problem of interpreting infinitistic formulas of a certain form. 1 The branching (or partially-ordered) quantifier-prefix is, however, not essentially infinitistic, and the issues it raises have largely been discussed in the literature in the context of finitistic logic, as they will be here. Our discussion transcends, however, the resources of standard lst-order languages and we will consider the new form in the context of 1st-order logic with 1and 2-place 'Mostowskian' generalized quantifiers. 2 Eventually we would like to know whether branching quantification is a genuine logical form. But today we find ourselves in an interesting