Thursday, February 8, 2018

Ersatz objects and presentism

Let Q be a set of all relevant unary predicates (relative to some set of concerns). Let PQ be the powerset of Q. Let T be the set of abstract ersatz times (e.g., real numbers or maximal tensed propositions). Then an ersatz pre-object is a partial function f from a non-empty subset of T to PQ. Let b be a function from the set of ersatz pre-objects to T such that b(f) is a time in the domain of f (this uses the Axiom of Choice; I think the use of it is probably eliminable, but it simplifies the presentation). For any ersatz pre-object f, let n(f) be the number of objects o that did, do or will exist at b(f) and that are such that:

  1. o did, does or will exist at every time in the domain of f

  2. o did not, does not and will not exist at any time not in the domain of f

  3. for every time t in the domain of f and every predicate F in Q, o did, does or will satisfy F at t if and only if F ∈ f(t).

Then let the set of all ersatz objects relative to Q be:

  • OQ = { (i,f) : i < n(f) },

where i ranges over ordinals and f over ersatz pre-objects. We then say that an ersatz object (i, f) ersatz-satisfies a predicate F at a time t if and only if F ∈ f(t).

The presentist can then do with ersatz objects anything that the eternalist can do with non-ersatz objects, as long as we stick to unary predicates. In particular, she can do cross-time counting, being able to say things like: “There were more dogs than cats born in the 18th century.”

Extending this construction to non-unary predicates is a challenging project, however.

1 comment:

Alexander R Pruss said...

This construction also generates, for any set of worlds, ersatz trans-world objects.