The presentist seems to claim that:
(*) Only things that are present exist.
This threatens triviality or implausibility. Either the underlined occurrence of "exist" is to be read in the present tense, as "exist now", or it is to be read in some tenseless sense not tied to the present. On the tensed reading, (*) says that only things that presently exist exist now, which no eternalist will dispute. On the untensed reading, the claim is implausible—it contradicts the fact that in the tenseless sense we are quite willing to say that among historical events there exists the Battle of Waterloo.
Let's slow down.
Take first the untensed horn of the dilemma. One way to go for this horn is crazy. The tenseless "exist" is some disjunction like: "existed, exists now, will exist or exists outside of time" (maybe we omit the fourth disjunct, maybe we don't). But on this view, (*) is looney: it commmits one to the claim that the Battle of Waterloo not only doesn't exist, but never existed. No sane presentist wants to be an irrealist about past-tensed statements.
The other way to defend the untensed horn is more interesting. On this approach, the presentist needs to defend the idea of a basic, tenseless existential quantifier (or basic, tenseless existence property), and then claim that, as a matter of (necessary?) fact, all the things that this quantifier quantifies over are presently existent. In doing this, the presentist will depart from common sense, for she will be using the tenseless "exists" while denying the common-sensical claim that if there is a tenseless "exists", then past things "exist" in that sense. But that is far from a decisive objection.
Here is a more worrisome objection: How do we motivate the claim that all and only the things that exist now exist in the tenseless sense? Consider a rival view: All and only the things that existed exist in the tenseless sense. Or: All and only the things that existed exactly 17 seconds ago exist. Why think that the present is special ontologically?
One answer that won't work: We perceive present things, so that we'd be as bad as the sceptics if we were pastists. For in fact the things we exist are always also past things—there is always a delay in the perceptual process.[note 1] Indeed, to be strictly correct, we can be more sure that what we perceive existed than that what we perceive exists. But even if this weren't true, we could parallel the claim that we perceive present things with the claim that we perceived past things.
A better answer might be that the present has a special iterativeness property. If an event E is present, then the being-present of the event is also present. Being 17 seconds in the past doesn't: if E is 17 seconds in the past, then E's being 17 seconds in the past is present, not 17 seconds in the past. However, if E is in the past, and the past is dense, then E's being past is also past. But this does depend on the density of time, and one can argue against density. Another response to the iterativeness response is that it is a mistake to take iterativeness as decisive: it is to be balanced against stability—while what is present won't ever be present again, what is past will always be past.
Here is another approach. Focus on the iterativeness. A proposition holds if and only if it holds at present. Thus, it seems natural to say that present-truth is ontologically prior to past-truth and future-truth. After all, when p will be true, then it is presently true that p will be true.
Perhaps, then, the presentist should take the first horn of the dilemma: the underlined "exist" in (*) is present tensed. Then, (*) is trivial. But the presentist can still make other claims that will be non-trivial: she can, for instance, hold that there is no single tenseless "exists"—all there is is the disjunction of "existed", "exists" and "will exist" (and maybe of "exists outside of time"). However, if she says this, then it becomes somewhat mysterious why she calls herself a "presentist". The thesis now is that there are three existence predicates: "existed", "exists" and "will exist" (or three quantifiers or three properties or whatever). But why is that any more a presentist claim than a pastist or a futurist one?
Maybe the presentist can claim that the present has a special priority. But what kind of priority? The future has a teleological priority. The past has a causal and epistemic priority (the events we observe are always at least in part past).
So perhaps the presentist should just become a tensist, recognizing three co-primitive existential quantifiers, say. But actually it's worse than that. For why should we lump together "existed five minutes ago" and "existed ten minutes ago" as a single kind of quantifier? Shouldn't we replace the single "existed" with "existed t seconds ago", and the single "will exist" with "will exist in t seconds". If so, then the view comes to be that there is a sequence of quantifiers, one for each place in the A-series.
But it is simpler just to accept a single quantifier and be a B-theorist.
I am not entirely convinced by all these arguments.