Binding modalities

摘要

The standard first-order reading of modality does not bind individual variables, i.e. if x is free in F(x), then x remains free in square F(x). Accordingly, if square stands for 'provable in arithmetic,' for all x square F(x) states that F(n) is provable for any given value of n=0,1,2,...; this corresponds to a de re reading of modality. The other, de dicto meaning of square F(x), suggesting that F(x) is derivable as a formula with a free variable x, is not directly represented by a modality, though, semantically, it could be approximated by compound constructions, e.g. square for all xF(x).