Many secular materialist are puzzled by Sam Harris's frequent assertion that science can bridge Hume's is–ought gap. Indeed, bafflement abounds on both sides whenever he debates his "bridge" with other materialists. Both sides are unable to understand how the other can fail to grasp elementary and undeniable points. This podcast conversation with the physicist Sean Carroll provides a vivid yet amicable demonstration.
I believe that this mutual confusion is a consequence of two distinct but unspoken ways of thinking about idealized moral argumentation. I'll call these two ways logical and dialectical.
Roughly, logical argumentation is focused on logical proofs of statements. Dialectical argumentation is geared towards rational persuasion of agents. These two different approaches lead to very different conclusions about what kinds of statements are necessary in rigorous moral arguments. In particular, the is–ought gap is unavoidable when you take the logical point of view. But this gap evaporates when you take the dialectical point of view.
I won't be arguing for one of these views over the other. My goal is rather to dissolve disagreement. I believe that properly understanding these two views will render a lot of arguments unnecessary.
Logical moral argumentation
Logical argumentation, in the sense in which I'm using the term here, is focused on finding rigorous logical proofs of moral statements. The reasoning proceeds by logical inference from premises to conclusion. The ideal model is something like a theory in mathematical logic, with all conclusions proved from a basic set of axioms using just the rules of logic.
People who undertake moral argumentation with this ideal in mind envision a theory that can express "is" statements, but which also contains an "ought" symbol. Under suitable circumstances, the theory proves "is" statements like "You are pulling the switch that diverts the trolley," and "If you pull the switch, the trolley will be diverted." But what makes the theory moral is that it can also prove "ought" statements like "You ought to pull the switch that diverts the trolley."
Now, this "ought" symbol could appear in the ideal formal theory in one of only two ways: Either the "ought" symbol is an undefined symbol appearing among the axioms, or the "ought" symbol is subsequently defined in terms of the more-primitive "is" symbols used to express the axioms.
When Harris claims to be able to bridge the is–ought gap in purely scientific terms, many listeners think that he's claiming to do so from this "logical argumentation" point of view. In that case, such a bridge would be successful only if every possible scientifically competent agent would accept the axioms of the theory used. In particular, "accepting" a statement that includes the "ought" symbol would mean something like "actually being motivated to do what the statement says that one 'ought' to do, at least in the limit of ideal reflection".
But, on these terms, the is–ought gap is unavoidable: No moral theory can be purely scientific in this sense. For, however "ought" is defined by a particular sequence of "is" symbols, there is always a possible scientifically competent agent who is not motivated by "ought" so defined.
Thus, from this point of view, no moral theory can bridge the is–ought gap by scientific means alone. Moral argumentation must always include an "ought" symbol, but the use of this symbol cannot be justified on purely scientific grounds. This doesn't mean that moral arguments can't be successful at all. It doesn't even mean that they can't be objectively right or wrong. But it does mean that their justification must rest on premises that go beyond the purely scientific.
This is Sean Carroll's point of view in the podcast conversation with Harris linked above. But Harris, I claim, could not understand Carroll's argument, and Carroll in turn could not understand Harris's, because Harris is coming from the dialectical point of view.
Dialectical moral argumentation
Dialectical moral argumentation is not modeled on logical proof. Rather, it is modeled on rational persuasion. The ideal context envisioned here is a conversation between rational agents in which one of the agents is persuading the other to do something. The persuader proceeds from assertion to assertion until the listener is persuaded to act.
But here is the point: Such arguments shouldn't include an "ought" symbol at all!—At least, not ideally.
By way of analogy, suppose that you're trying to convince me to eat some ice cream. (This is not a moral argument, which is why this is only an analogy.) Then obviously you can't use "You should eat ice cream" as an axiom, because that would be circular. But, more to the point, you wouldn't even have to use that statement in the course of your argument. Instead, ideally, your argument would just be a bunch of "is" facts about the ice cream (cold, creamy, sweet, and so on). If the ice cream has chocolate chips, and you know that I like chocolate chips, you will tell me facts about the chocolate chips (high in quantity and quality, etc.). But there's no need to add, "And you should eat chocolate chips."
Instead, you will just give me all of those "is" facts about the ice cream, maybe draw some "is" implications, and then rely on my internal motivational drives to find those facts compelling. If the "is" facts alone aren't motivating me, then something has gone wrong with the conversation. Either you as the persuader failed to pick facts that will motivate me, or I as the listener failed to understand properly the facts that you picked.
Now, practically speaking, when you attempt to persuade me to X, you might find it helpful to say things like "You ought to X". But, ideally, this usage of "ought" should serve just as a sort of signpost to help me to follow the argument, not as an essential part of the argument itself. Nonetheless, you might use "ought" as a framing device: "I'm about to convince you that you ought to X." Or: "Remember, I already convinced you that you ought to X. Now I'm going to convince you that doing X requires doing Y."
But an ideal argument wouldn't need any such signposts. You would just convince me of certain facts about the world, and then you'd leave it to my internal motivational drives to do the rest—to induce me to act as you desired on the basis of the facts that you showed me.
Put another way, if you're trying to persuade me to X, then you shouldn't have to tell me explicitly that doing X would be good. If you have to say that, then the "is" facts about X must not actually be motivating to me. But, in that case, just telling me that doing X would be good isn't going to convince me, so your argument has failed.
Likewise, if the statement "Doing X would cause Y" isn't already motivating, then the statement "Doing X would cause Y, and Y would be good" shouldn't be motivating either, at least not ideally. If you're doing your job right, you've already picked an is-statement Y that motivates me directly, or which entails another is-statement that will motivate me directly. So adding "and Y would be good" shouldn't be telling me anything useful. It would be at best a rhetorical flourish, and so not a part of ideal argumentation.
Thus, from this point of view, there really is a sense in which "ought" reduces to "is". The is–ought gap vanishes! Wherever "ought" appears, it will be found, on closer inspection, to be unnecessary. All of the rational work in the argument is done purely by "is". Of course, crucial work is also done by my internal motivational structure. Without that, your "is" statements couldn't have the desired effect. But that structure isn't part of your argument. In the argument itself, it's all just "is... is... is...", all the way down.
This, I take it, is Sam Harris's implicit point of view. Or, at least, it should be.
 I am not saying that these views of moral argumentation exhaust all of the possibilities. I'm not even saying that they are mutually exclusive in practice. Normally, people slide among such views as the needs of the situation require. But I think that some people tend to get needlessly locked into one view when they try to think abstractly about what counts as a valid and rigorous moral argument.
 From the logical point of view, all moral argumentation is first and foremost about the assertions that we can prove. Argumentation is not directly about action. There is only an indirect connection to action in the sense that arguments can prove assertions about actions, like "X ought to be done". Furthermore, this "ought" must be explicit. Otherwise, you're just proving "is" statements.
 Analogously, you can get the symbol in a theory of arithmetic in two ways. On the one hand, in first-order Peano arithmetic, the symbol is undefined, but it appears in the axioms, which govern its behavior. On the other hand, in the original second-order Peano axioms, there was no symbol. Instead, there was only a successor-of symbol. But one may subsequently introduce by defining it in terms of the successor-of symbol using second-order logic.
 Some might dispute this, especially regarding the meaning of the phrase "possible scientifically competent agent". But this is not the crux of the disagreement that I'm trying to dissolve. [ETA: See this comment.]
 Here I mean "persuaded" in the sense of "choosing to act out of a sense of moral conviction", rather than out of considerations of taste or whatever.
 ETA: The phrases "internal motivational drives" and "internal motivational structure" do not refer to statements, for example to statements about what is good that I happen to believe. Those phrases refer instead to how I act upon beliefs, to the ways in which different beliefs have different motivational effects on me.
The point is: This unspoken "internal" work is not being done by still more statements, and certainly not by "ought" statements. Rather, it's being done by the manner in which I am constituted so as to do particular things once I accept certain statements.