Yes, once again I feel I really should have made the key right away. Lots of good questions coming in over email. Here’s the response to another one.

There is no explicit definition of the interpretation of Present tense on the handout. This is mainly because Past tense seemed more exciting, so I always used that as an example.

This is what it looked like:

[[ PAST S ]]^M,g,w,t = ∃t′[t′ < t ∧ [[S]]^M,g,w,t′ ]

(Where “t′ < t” is suppose to mean that t′ is before t.) That is: “There’s a time t′ before now, such that S is true at t′.”

There is a sort of obvious thing you can do to create the present tense version of this, which is just to change “<” to “=”. However, if you do this, you wind up saying something kind of overly complicated:

[[ PRES S ]]^M,g,w,t = ∃t′[t′ = t ∧ [[S]]^M,g,w,t′ ]

That is: “There’s a time t′, which is now, such that S is true at t′.”

But that’s really pretty much just the same thing as saying “S is true now.” So, really, the simplest version of this (and still just as accurate, and still requiring that Tense raises) would be simply to assume that the rule for present tense is:

[[ PRES S ]]^M,g,w,t = [[S]]^M,g,w,t

That is, PRES doesn’t really contribute anything to the meaning. You can use either of the interpretation rules for [PRES S] I gave above, but the second one is of course the simpler of the two.