Binding modalities

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 2F (x). Accordingly, if 2 stands for ‘provable in arithmetic,’ ∀x2F (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 2F (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., 2∀xF (x). We introduce the first-order logic FOS4∗ in which modalities can bind individual variables and, in particular, can directly represent both de re and de dicto modalities. FOS4∗ extends first-order S4 and is the natural forgetful projection of the first-order logic of proofs FOLP. The same method of introducing binding modalities obviously works for other modal logics as well.