It was Putnam who started the idea of refuting the brain-in-vat hypothesis, with sematic externalism, in this paper. The money quote:

By what was just said, when the brain in a vat (in the world where every sentient being is and always was a brain in a vat) thinks 'There is a tree in front of me', his thought does not refer to actual trees. On some theories that we shall discuss it might refer to trees in the image, or to the electronic impulses that cause tree experiences, or to the features of the program that are responsible for those electronic impulses. These theories are not ruled out by what was just said, for there is a close causal connection between the use of the word 'tree' in vat-English and the presence of trees in the image, the presence of electronic impulses of a certain kind, and the presence of certain features in the machine's program. On these theories the brain is right, not wrong in thinking 'There is a tree in front of me.' Given what 'tree' refers to in vat-English and what 'in front of' refers to, assuming one of these theories is correct, then the truth conditions for 'There is a tree in front of me' when it occurs in vat-English are simply that a tree in the image be 'in front of' the 'me' in question — in the image — or, perhaps, that the kind of electronic impulse that normally produces this experience be coming from the automatic machinery, or, perhaps, that the feature of the machinery that is supposed to produce the 'tree in front of one' experience be operating. And these truth conditions are certainly fulfilled.

By the same argument, 'vat' refers to vats in the image in vat-English, or something related (electronic impulses or program features), but certainly not to real vats, since the use of 'vat' in vat-English has no causal connection to real vats (apart from the connection that the brains in a vat wouldn't be able to use the word 'vat', if it were not for the presence of one particular vat — the vat they are in; but this connection obtains between the use of every word in vat-English and that one particular vat; it is not a special connection between the use of the particular word 'vat' and vats). Similarly, 'nutrient fluid' refers to a liquid in the image in vat-English, or something related (electronic impulses or program features). It follows that if their 'possible world' is really the actual one, and we are really the brains in a vat, then what we now mean by 'we are brains in a vat' is that we are brains in a vat in the image or something of that kind (if we mean any thing at all). But part of the hypothesis that we are brains in a vat is that we aren't brains in a vat in the image (i.e. what we are 'hallucinating' isn't that we are brains in a vat). So, if we are brains in a vat, then the sentence 'We are brains in a vat' says something false (if it says anything). In short, if we are brains in a vat, then 'We are brains in a vat' is false. So it is (necessarily) false.

And a nice counterargument from Nagel's The View From Nowhere:

If I accept the argument, I must conclude that a brain in a vat can't think truly that it is a brain in a vat, even though others can think this about it. What follows? Only that I can't express my skepticism by saying "Perhaps I'm a brain in a vat." Instead I must say: "Perhaps I can't even think the truth about what I am, because I lack the necessary concepts and my circumstances make it impossible for me to acquire them!" If this doesn't qualify as skepticism, I don't know what does.