The quick argument for double-indexing

Consider the following logical truths (or indexical validities) of English:

(1) It rains if and only if it rains now.

(2) It rains now if and only if it always rains now.

Assume (to reach a contradiction) a singly-indexed semantics. Given that (1) is a logical truth, then the semantic clause for the indexical sentential operator `Now' (or `It is now the case that') must be as follows:

[[Now(φ)]]^t =1 iff [[φ]]^t =1

But then (2) is not valid (as long as there are times in structures where it rains and fails to rain). But (2) is valid. Contradiction.

[This is basically the argument given in Kamp (1971), "The formal properties of `now'"]