What they show is that contrary to current orthodoxy in the philosophy of mind, intentionality is a two-stage connection to the world rather than a one-stage connection.
The term "intentionality" refers to the phenomenon that mental states/attitudes/concepts are about, are directed on, or represent other things. "Intentionality" is this rather vaguely characterized notion of "aboutness". There is something that I affirm, deny, interpret, understand, believe, hope, wish, like, love, hate, (and according to some, perceive or sense), and so forth. The modern use of the term was initiated by Brentano towards the end of the 19th Century. He claimed that every mental state has intentionality and is directed towards an "intentional object". In modern philosophy of mind, is it hotly debated whether this notion of "aboutness" is a necessary and / or sufficient condition for things mental. Some claim that only things mental can have this "aboutness." Others claim that non-mental (i.e. physical) things can have the same sort of "aboutness". Regardless of this debate, "intentionality" is generally taken to be an indicator of things mental. It may not be a necessary and sufficient mark. But it is a close enough approximation for most discussions not relying on precision. It will do for the purposes of this essay,
"Frege cases" are examples of multiple words/names/expressions all referring to one referent-in-the-world. The example that Frege used was the case of "Hesperus", "Phosphorus", and "Venus". These are three names used to refer to the same object. Frege noted that when used within intentional idioms, co-referring names are not substitutable salva veritate. If Ann believes that Hesperus is now visible, that does not entail that she believes that Phosphorus is now visible. Ann may not know they co-refer. Frege cases are used to argue that the meaning of a word/name/expression is not simply the referent-in-the-word (as is claimed by Millian semantics). Frege argued that a name must have more than just the referent as its meaning. He proposed that names also have "senses". To Frege, however, senses were not subjective entities, but rather objective entities in the public domain. They must be external to the minds that use them, argued Frege, if two people are ever to mean the same thing by a word. But if they are external, goes the "externalist" orthodoxy, then the nature of intentionality cannot be explained just by the physical properties of the brain.
"Twin Earth cases" reverse the many-to-one relationship of Frege cases - one word/name/expression having multiple referents-in-the-world. These thought experiments were initially proposed by Hillary Putnam. Putnam's example is the case of Oscar on the Earth, and Toscar on Twin-Earth both using the word "water" to talk/think about the clear, odourless liquid that comes out of taps, falls as rain, and fills lakes and streams. Putnam argued that since Oscar and Toscar are ex hypothesi physically identical, there is nothing physical in their brains to distinguish the fact that when Oscar thinks about "water" he is thinking about "H2O", and when Toscar thinks about "water" he is thinking about "XYZ". Putnam famously concludes that "meaning ain't just in the head". Again if the intentional relation is defined by something external, then the nature of intentionality cannot be explained just by the physical properties of the brain.
The "externalist" about intentionality employs these cases to show that intentionality (the "aboutness" of mental states) derives not just from stuff within the brain (so-called "narrow content" in the words of Fodor, Millikan and Dennett), but requires stuff in the world as well (so-called "broad content" in the words of Putnam, Horgan, Dretske, and Burge). When Ann affirms that Hesperus is visible, what her affirmation is about involves something outside of Ann -- the Fregean objective sense of "Hesperus". When Toscar wants some "water", what his want is about involves something outside of Toscar -- the fact that to Toscar, what he calls "water" is not made of H2O, but actually made of XYZ. The externalist understands these cases through the premise that the intentional relationship involved between a name/word/expression and the referent-in-the-world is single-stage. Frege's puzzle about the non-interchangability of supposedly co-referring names in intentional idioms is based on the assumption that the referent-in-the-world of those names is actually the same. Putnam's puzzle over the meaning of "water" is based on the assumption that Oscar's "water" is a relationship to the referent-in-the-world of H2O while Toscar's "water" is a relationship to the referent-in-the-world of XYZ.
But a better interpretation of these cases is that the relationship involved is two-stage. The referent of a name like "Hesperus" or of a word like "water" is actually a mental existent (a "concept"). This is the approach offered by John Searle's cluster theory of identifying descriptions. Searle proposed that what picks out the referent-in-the-world of a name (and it works just as well for any word) is a whole collection of descriptions and other contextually relevant information, internal to the mind using that information. This cluster of descriptions is not something that grants an objective sense of meaning to the name, as argued by Frege and Kripke. What ensures that we both pick out the same referent-in-the-world (if and when we do) is the conventional nature of language (and word) acquisition.
In Ann's mind, "Hesperus" refers not to Venus, but to a subjective collection of information that in this actual world just happens to pick out a certain "star" visible in the evening. And "Phosphorus" refers to a subjective collection of information that just happens to pick out a certain "star" visible in the morning. The two clusters of information in Ann's mind are not the same, so there is no co-referring taking place. Frege's puzzle dissolves. And in Oscar's mind the word "water" refers to a subjective collection of information that in this actual world just happens to pick out H2O. But in Toscar's mind, the word "water" refers to exactly the same (ex hypothesi) subjective collection of information that in his world just happens to pick out XYZ.
Accepting the intentional relation as existing not between a mental state and a referent-in-the-world, but between a mental state and a mental "concept" also solves a couple of classic "paradoxes of intentionality". The first is that the intentional object need not exist. If Billy believes that Santa Claus is coming, then "Santa Claus" is a real mental object that just happens to have no referent-in-the-world. The second is that a mental state can bear an intentional relation to something general, without their being any particular referent-in-the-world that the state bears that relationship to. If Betty wants a cat, but has no particular cat in mind, then Betty's mental state of wanting bears an intentional relationship to her concept of a cat without having to pick out a particular cat.
What Frege cases demonstrate is that Fregean "senses" are not objective as Frege claimed. A and B are not the same intentional object, even though when those descriptions are projected on the world they happen to pick out the same referent-in-the-world. What Twin-Earth cases demonstrate is that in different scenarios, the (ex hypothesi) identical collection of information can pick out different referents-in-the-world. Intentionality is a two-stage relation to the world, not a one-stage relation to the world.
Yes, it does. At least it does if one adopts physicalism. It doesn't, of course, if one adopts anti-physicalism. In other words, the answer one provides will depend on one's prior attitude towards physicalism.
The term "intentionality" refers to the phenomenon that mental states/attitudes/concepts are about, are directed on, or represent other things. There is something that I affirm, deny, interpret, understand, believe, hope, wish, like, love, hate, (and according to some, perceive or sense), and so forth. "Intentionality" is this rather vaguely characterized notion of "aboutness". The modern use of the term was initiated by Brentano towards the end of the 19th Century. He claimed that every mental state has intentionality and is directed towards an "intentional object". In modern philosophy of mind, is it hotly debated whether this notion of "aboutness" is a necessary and / or a sufficient condition for things mental. Some claim that only things mental can have this "aboutness." Others claim that non-mental (i.e. physical) things can have the same sort of "aboutness". The division between views matches the division between physicalism and anti-physicalism.
There are, of course, many purely physical things that have an "aboutness". Maps, photographs, sentences of a language, and so forth, are all about something. But the anti-physicalist draws a distinction between "intrinsic" versus "derivative" intentionality. Intrinsic intentionality is intentionality that is not derivative. An object has derivative intentionality when its "aboutness" relies on the interpretation of something outside that object. So a sentence of a language, for example, is about something only because something outside of it (a mind) provides that aboutness. The anti-physicalist argument is that only mental states have intrinsic intentionality. Philosophers arguing in this direction include Jerry Fodor, John Searle, Fred Dretske, Tyler Burge, Saul Kripke, among others.
In contrast to this view, consider the Martian Spirit rover. It has a panel of photocells fixed to its back. To angle its panel of cells to maximize power acquisition, the drivers sitting at JPL in Pasadena had to drive the rover up the side of a hill. Suppose JPL builds a Spirit2 rover that has two advances. One is a panel of solar cells that the rover can tilt and swivel on its own -- it has a program module specifically programed to do so when provided with a position to point to. And the second is a skycam (with its associated analysis software module) that tells that solar panel program module where the sun (the brightest spot in the camera view) is located. Now Spirit2 can maximize its power acquisition without the involvement of the drivers at JPL. The message from the skycam analysis program module to the solar panel manager program module is clearly "about" the power maximization coordinates. The intentionality of this "aboutness" does not rely on the interpretation of something outside of it. The drivers at JPL need have no inkling about the messages passed between the two program modules. If this counts as "intrinsic" intentionality, then the physicalist case is made.
But the general reply of the anti-physicalist is that the entire Spirit2 rover is a device designed by a mind, and hence derives any intentionality of its parts from the intrinsic intentionality of the designing mind. But this response runs into the counter argument, offered by the likes of Ruth Millikan and Daniel Dummett, among others. We human beings are but a device designed by "Mother Nature" -- otherwise known as the blind watchmaker, or the processes of evolution. According to Dawkins, we (including our minds) are the result of the purely physical processes of evolutionary selection, "designed" for the purposes of ensuring the survival and proliferation of our genes. And no one suggests that "Mother Nature" (in the guise of evolution or our genes) has a mind, or intrinsic intentionality. If we are the results of a purely physical process of evolution, and it is admitted that we have intentionality (intrinsic or derivative), then the physicalist case is made.
If this is the case (and a Designing Intelligence is not smuggled in), then where is the principled difference between us, and our "intrinsic" intentionality, and the Spirit2 rover and its "derived" intentionality? If we (our minds), are an artifact designed by the physical processes of evolution, and do have intentionality, then intentionality obviously admits of a naturalistic explanation. If intentionality does not admit of a naturalistic explanation, then we (at least our minds) are not a result of the physical processes of evolution. The anti-physicalist must provide something extra.
Either way, the answer to the title question will depend on one's prior commitment or opposition to physicalism.