By Jonathan Barnes

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Predicates: For simplicity, I reduce the list of predicates to just one binary predicate ‘R2’. Formulas Taking ‘variable’ to mean ‘unstarred variable’, ‘individual constant’ to mean ‘unstarred individual constant’, and ‘individual symbol’ to mean ‘unstarred individual symbol’, the definition of ‘formula’ is to remain as before with only the following changes: replace the definition of ‘atomic formula’ with clause (a) given below and add to the definition of ‘formula’ clause (b) given below: (a) Atomic formula: If α is a starred individual symbol and β is an individual symbol, then (b) is an atomic formula.

The starred individual symbols are to be thought of as referring only to the things in the actual world. K*-Interpretations A K*-interpretation = differs from a K-interpretation in the following way: 1. 10U*(0, k) is the set of things in world k that the starred variables range over, and U*(1, k) is the set of things in world k that the unstarred variables range over. )11 2. I* assigns to each individual constant some element of Z = the union of all the U*(1, w), where w is any element of W*; I* assigns to each starred individual constant some element of U*(0, a*); and I* assigns to ‘R2’ a subset of the Cartesian product of U*(0, a*) with Z.

Gottlieb suggests that we can regard these sentences as asserting [3] ◇ (∃x)(x is an S-sentence that is true). Now as [1] is intended, it is true just in case there are finitely many 38 THE CONSTRUCTIBILITY THEORY stars. But [3] is true even if, in fact, there are infinitely many stars. For no matter how many stars may in fact exist, it is possible that only finitely many stars exist. Strangely, what Gottlieb concludes from this is not that it is improper to replace by ‘(Cx)’ with ‘◇ (∃x)’, but rather that the constructibility interpretation of the existential quantifier that I gave does not “assign truth values in an acceptable manner” (Economy, p.

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