Time and World Quantifiers

Abstract

The example sentences used in Chapter 2 to argue that natural language quantifies over times introduced them by explicitly using the noun ‘time’, and referred back to them by explicitly anaphoric phrases like ‘the first’, ‘the second’, and so on. So the next task is to integrate this way of referring to times and worlds with the multiply-indexed languages introduced in Chapters 3 and 13. The language ℒ, introduced in Chapter 13 is in fact a two-sorted language with individual variables and world variables, but it is standard that a many-sorted classical predicate language can be reduced to a one-sorted language if for every sort s, ℒ contains a one-place predicate S, and every wff of the form ∀ϰα where ϰ is a variable of sort s is replaced by ∀ϰ(Sϰ⊃ α). These quantifiers are called restricted quantifiers and the standard result shews that sorted quantifiers can be replaced by unsorted ones provided they are restricted. Now the point about a language like ℒ with generalized quantifiers is that all its quantifiers are restricted. So if ℒ contains a predicate time, as the discussion in Chapter 2 suggested it should, then an expression like ∀tα can be replaced by

$$every\left( {\lambda x{\text{ }}time{\text{ }}x} \right)\left( {\lambda x\alpha \left[ {x/{\text{t}}} \right]} \right)$$

((1))