This paper defends the Fregean view that logical forms don't contain variables (No Variables!). That is, the logical form of a statement like "Everybody needs somebody", which we'd normally formalize as ∀x∃yR(x,y), contains only the quantifiers ∀ and ∃ and the predicate R, but no variables x and y.

This view is motivated by the so-called Antinomy of the Variable, which Fine (2003) attributes to Russell (1903).

We defend No Variables! by developing a 100% variable-free predicate calculus, which has a fully compositional Fregean semantics, where terms denote objects, predicates denote value-ranges, and sentences denote truth-values. This calculus is formulated using de Bruijn indices, which are common in automated theorem proving, but rather uncommon in philosophy.

We argue that the calculus provides a better guide to logical form than standard Tarskian or Fregean predicate calculi.