It seems that quite a few philosophers have gotten an erroneous idea about the Many Worlds Interpretation (MWI) of quantum mechanics, namely that it is a deterministic interpretation. Even the Internet Encyclopedia of Philosophy makes this incorrect claim. So, I feel I ought to set the record straight.
The classic definition of determinism is that, given the way the world is at some time, plus the laws of nature, the future is completely fixed. Now, in one sense, this definition is clearly satisfied by the MWI. If, by "the way the world is at some time," we mean the quantum state of the universe at that time, and by "the future" we mean the quantum state of the universe at some later time, then the MWI is indeed deterministic. For the MWI takes the quantum state to be all that there is, and the unitary evolution of the quantum state to be the complete dynamics of the state. And that evolution is deterministic: there is one and only one state at a future time, and it is completely determined by the quantum state at the earlier time. (Here I am omitting the very serious difficulties of this approach, including the dubious nature of such an entity as "the quantum state of the universe" and the even more dubious assertion that that state evolves in a unitary fashion.)
But this sense of "determinism" is completely useless for normal philosophical discussion.
The problem is very easy to see in the famous example of Schroedinger's cat. Let's assume that some quantum process yields a 50-50 chance of outcomes A and B, and we rig up some equipment so that if outcome A happens then a vial of poison is smashed and the cat dies, but if outcome B happens, then the cat lives. And all of this takes place inside a perfectly impenetrable box, so that we have no way of determining the outcome until we open the box. Then, according to the rules of quantum mechanics, the cat will be in a superposition of alive and dead states until the box is opened.
In the MWI, an experiment like this has a determinate outcome: that the quantum state evolves into two branches, one with a live cat and one with a dead cat. But in the real world we don't see such superposition states. We see a live cat or a dead one. The MWI solves this by invoking decoherence, yada, yada, yada, so that the two branches are effectively split into different worlds that never interfere thereafter.
But now you see the problem: the future, in the normal sense of the word, involves the cat being either alive or dead, not both. And the MWI does not determine which future we will see. Indeed, from the point of view of the MWI, there is not even any sense in asking which outcome actually occurred, because both actually occurred, in different "worlds."
So in the case of the cat, the future (in the normal everyday sense) is undetermined. But it's much worse than this: anything that occurs because of quantum effects has an indeterminate outcome. But in the MWI, everything that occurs at all occurs because of quantum effects. So (almost) everything is indeterminate.
(The sole exceptions would be those events that occur with 100% probability: outcomes that are represented by operators for which the quantum state of the universe is an eigenvector. But, given that the wave function splits every time a quantum event occurs, we can be sure that none of the events of ordinary interest will have the state of the universe as an eigenvector.)
The thing is, even though the MWI has restored determinism of a sort to the description of the universe, it completely fails to predict what we will see in our particular future. John Earman says that the MWI exhibits ontological determinism, but at the price of "radical epistemic indeterminism."
Clearly, though, it is the indeterministic aspect that is relevant for most philosophical concerns. For example, in discussions of free will the question is often asked, "Could I have done otherwise?" And the MWI gives the emphatic answer, "Yes!" Because not only could I have done otherwise, but in some other "world" I did do otherwise. Austin missed his putt, but in some other world where the quantum events in his brain fired the neurons in a different way he made the putt.
Maybe it is possible to continue to do philosophy while keeping in mind the ontological viewpoint of the MWI - though I have to say I doubt it. But it would at least be a very different philosophy than what we have now, and all of the classic discussions would have to be revisited in light of that ontology.