# math history – Understanding the notation in Lowenheim’s 1915 proof of decidability of monadic predicate logic

One of the results of Lowenheim’s (1915) is the decidability of the monadic fragment of first-order predicate logic. As I understand it, the method of the proof is to push quantifiers inwards, converting formulas into boolean expansions (sums/products) of atomic formulae consisting of a single quantifier (hence a single quantified variable given the monadic language).

However, the conversion and the resulting “normal form” is opaque to me because I don’t understand Lowenheim’s notation and terminology. For example, I don’t understand how he singles out a finite number of “distinguished classes” (predicates) and corresponding “subscripts” (variables). I also don’t understand what it means for the remaining classes to be “symmetric with respect to all subscripts”, especially since this is monadic logic so there are no binary relations.

As examples, see the two passages below, p. 243 and 245 of From Frege to Godel. Does anyone know a reference for an explanation of the framework Lowenheim is working in?  References:

LÖWENHEIM, LEOPOLD

(1915) On Possibilities in the Calculus of Relatives, in van Heijenoort (ed.) (1967), pp. 228–251.

VAN HEIJENOORT, JEAN (ED.)

(1967) From Frege to Gödel; a source book in mathematical logic, 1879-1931. Cambridge, Harvard University Press.

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