I am reading Chalmers' "Two dimensional semantics" and "Two dimensional argument against materialism" and a point is unclear: As per Kripke (1980), "Water is not H2O" is false in all possible worlds. I.e., water and H2O have the same extension in all possible worlds and so have the same intension. Now consider a chemical XYZ with properties identical to the H2O of our actual world. Let Twin earth (T) be a possible world which is identical to our actual world, except that H2O is replaced by XYZ everywhere. (For example, oceans in P contain XYZ and no H2O).

The extension of water in T is XYZ and the extension of H2O in T is the empty set. So,"Water is not H2O" is true in T, contradicting Kripke's claim. Can someone clarify this point?

As per Chalmers' papers, if a person P were to imagine that T is actual, then P will hold that "Water is XYZ", and hence that "Water is not H2O" (In his words: T verifies "Water is not H2O" or "Water is not H2O" is positively primarily conceivable or the primary intension of water in T is XYZ).

I am under the impression that to evaluate the truth of "Water is not H2O" in T, one considers the primary intension of water in T and checks if it is the same as the primary intension of H2O in T (I.e., one imagines that T is actual and then evaluates the truth of "Water is not H2O" conditioned on T being actual) But if this were the case, then "Water is not H2O" is true in T.

Can someone please explain?

  • SEP has a detailed explanation of how intension is modified in 2D semantics, including the difference between actual and counterfactual intensions in the Twin Earth experiment.
    – Conifold
    Commented May 29, 2023 at 4:34
  • "water and H2O have the same extension in all possible worlds and so have the same intension" Does Kripke say that? It's a non sequitur according to the traditional meaning of intension, but it may fit the possible worlds definition of intension. Commented May 29, 2023 at 8:25
  • @David Gudeman. Notation is from the above SEP article: I am thinking of intension as a function from circumstance of evaluation to extensions. In the 2D matrix of 2D semantics, each context of occurrence defines a row and hence an intension. If the context is Earth, then S="Water is H2O" is true in all circumstances. Thinking of circumstances as possible worlds, S has the same extension (namely 'true') in all possible worlds, when the context is earth. If the context were twin earth, then S has the same extension (namely 'false') in all possible worlds. Commented May 29, 2023 at 9:03
  • @Conifold. The link you mentioned was very helpful Commented May 29, 2023 at 9:04
  • Yes, I assumed it was something like that. I've always been a bit confused when people bring up intensions in the context of possible worlds semantics, because it never seemed to fit and they never defined it. It looks like a rather extensional attempt to define intensions. Commented May 29, 2023 at 10:37

1 Answer 1


The de re/de dicto distinction is in play, here. It's not enough to say the word "water," we have to differentiate between, "Use of the word 'water' on Earth," and, "Use of the word 'water' on Twearth." Then:

  1. The word "water" as used on Earth refers only to H2O in all possible worlds.
  2. The word "water" as used on Twearth refers only to XYZ in all possible worlds.


  1. WaterEarthspeak is H2O in all possible worlds.
  2. WaterTwearthspeak is XYZ in all possible worlds.

At least, this is the kind of thing that Kripkeans are proposing. Imagining a situation where an Earthling ends up talking with Twearthlings and uses the word "water" does complicate matters, although supposing that possible worlds are closed off from each other might make the thought experiment into a non-starter.

Caveat: the above presupposes that the doctrine of possible analytic aposteriority is roughly correct. I myself am not educated enough to be confident that such a doctrine is correct/incorrect specifically, as things like Chalmers' semantic theory make me wonder if words like "water" are fully semantically indexed to/per possible worlds. Moreover, assuming that H2O has a specific geometrical/topological character, I am hard-pressed to imagine that a world with XYZ differing topologically from H2O would have its chemicals assemble into macroscopic objects indistinguishable from Earth-water. But so if Earth-water and Twearth-water are topologically equivalent, I don't know that we'd have grounds for saying that H2O ≠ XYZ in the first place.

To illustrate my own confusion:

You are on a trolley with your pet kittens Awe and Wonder. Like Luna and Artemis from Sailor Moon, your cats have symbols on their heads, a ? and a !. Suddenly, the trolley is derailed; your cats put their heads together, and a ?! forms and opens a portal to Twearth, upon which the trolley crashes with the passengers otherwise unharmed. While stuck on Twearth, you start to get thirsty, and go up to a pond. You're about to drink when Awe and Wonder spring into action again. "You can't drink that!" Awe says. "That's not H2O, that's XYZ!" Wonder explains. "Your body can't absorb it right!"

Meanwhile, there's no oxygen on Twearth, either, but you seem to be breathing just fine...

EDIT: iterated modality: in standard modal logic, iterating the operators is superfluous. For example, if something is possibly possible, it's just possible; if something is possibly necessary, it's just necessary; etc. However, perhaps the standard is wrong, or perhaps qualitative compounding is nontrivial. E.g.:

  1. It is epistemically possible that it is not metaphysically necessary that water is H2O. Or (slightly differently): it is epistemically possible that water is not H2O even if it is metaphysically necessary that water is H2O.

So, "The word 'water' on Earth refers only to H2O in all metaphysically possible worlds, but there are epistemically possible worlds where Earth's 'water' refers to XYZ," etc.

  • 1
    Is this correct?: Let P be the set of possible worlds of Earth (E) and let P' be the set of possible worlds of Twearth (T). Let w be in P and let w != E. So, water in w will necessarily mean H2O since water_earthspeak refers to H2O and water on w must be the same as water_earthspeak since w is in P. Similarly, if w' is in P', then water in w' will mean water_twearthspeak = XYZ. In particular, P and P' are disjoint. It seems that the physics of w is identical to that of Earth. Then, how does w differ from E? Is it that the events on w differ from E's? Commented May 29, 2023 at 4:53
  • By "possible worlds of Earth/Twearth" do you mean "possible worlds accessible from Earth/Twearth"? Commented May 29, 2023 at 4:58
  • Yes. In the "2D semantics" paper by Chalmers, each row is an actual world. With respect to this actual world, one can talk of possible (counterfactual) worlds. By P, I mean the set of counterfactual possible worlds which can be imagined from Earth (these will lie on the row corresponding to Earth). But not all authors are explicit about using a reference actual world when talking about a possible world. This point is at the root of my confusion. When one defines a "possible world", does one also have to first define a reference with respect to which the possible world is defined? Commented May 29, 2023 at 5:24
  • @TejasBhojraj I myself am unsure about "where" truths about modal logic are supposed to be true when different modal logics seem possible. Some time ago, Conifold directed me to an essay on "the modal aether" which might not have really resolved my confusions, but did show me how to look at them from a better POV. Commented May 29, 2023 at 5:35
  • Thanks. The SEP article pointed to by Conifold above is very clarifying too. Commented May 29, 2023 at 8:49

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