What Does "I" Really Mean? The Two-Dimensional Puzzle
Who Does “I” Pick Out?
Imagine you are at summer camp. Someone shouts, “I’m hungry!” If it’s your friend Mia, the word “I” means Mia. If Jamal says it, “I” suddenly means Jamal. The same word races around the room and lands on a different person every time it is spoken. This is not a trick — it’s just how language works. Yet it poses a real puzzle: how can a word have a meaning if what it points to changes from moment to moment?
Philosophers call expressions like “I,” “here,” “now,” and “this” context-sensitive terms. Their job is to stand for different things depending on who says them, where, and when. A simple dictionary definition won’t capture that. In the 1970s, the philosopher David Kaplan (1933–2023) invented a powerful tool to explain such words: a two-dimensional semantic framework. His key idea was that a word like “I” has two layers of meaning, not one. The first layer is character — a rule that tells you how the word behaves in any context. The second layer is content — the particular person, place, or thing the word picks out once a context is fixed.
To see the two layers, Kaplan used a chart, or 2D matrix. He lined up possible situations where someone might speak along one side of the chart, and possible ways the world could be along the other. Each cell shows what the word picks out in that combination. For “I,” the rule is simple: it always picks out the person who is speaking in that context, no matter what happens in the rest of the world. So if Mia says “I,” the word rigidly points to Mia in every possible circumstance, even in situations where she is not hungry or where Jamal is the one who shouts.
This idea — that “I” is a rigid designator — is one of the big insights of two-dimensional semantics. It means that once you fix who is talking, “I” latches onto that person for good. The same holds for “here” and “now.” They are like arrows that point to a location in space and time and stay there, no matter how you spin the rest of the story.
”I Am Here” Is Always True … But Not a Deep Truth
Try this: close your eyes and say “I am here now.” The sentence cannot be false when you utter it. Wherever you are, whenever you speak, you are indeed there at that moment. Kaplan showed that such sentences are logically valid: they are true in every possible context of use. But that doesn’t mean they are deep truths about the world. In fact, the content of “I am here now” is usually a very ordinary, contingent fact — you could easily have been at the beach or in another city. The sentence is guaranteed to be true only because of the rules that govern “I,” “here,” and “now,” not because the world has to be a certain way.
The matrix reveals this neatly. Suppose we consider three different contexts: Mia says “I am here” at camp, Jamal says it at home, and a girl named Rosa says it at the dentist. In each row — each context — the sentence comes out true at the world where it is spoken, but false in many other worlds where the speaker ends up somewhere else. The diagonal of the matrix, from top left to bottom right, records what happens when the world of the context and the world of evaluation are the same. That diagonal is all “true.” Kaplan called this the diagonal intension. When the diagonal is necessarily true, the sentence is valid in his special sense — it is a truth of language, not a revelation about reality.
This discovery was a big deal for philosophy. It showed that some sentences can be bulletproof in conversation yet still be about ordinary, changeable facts. In other words, necessity can come in two flavours: there’s the necessity of logic and definitions, and then there’s the deeper necessity of how things must be, no matter what. The 2D framework gave philosophers a clean way to separate them.
The “Actually” Twist: When Hidden Necessity Sneaks In
Another puzzle arises with the word “actually.” Consider this sentence: “It is possible that everything that is actually red is shiny.” You might think you are saying that there is a possible world where all the red things there are also shiny there. But the word “actually” forces you to look back at the real world: it says that the things that are red in our actual world could all be shiny in some other world. Standard one-dimensional semantics couldn’t capture this, so logicians introduced a second dimension again. They designated one world to play the role of the actual world and added the operator Actually, written ( \mathcal{A} ).
Here’s the twist: if you take any true sentence ( S ) and put “actually” in front of it, the result ( \mathcal{A}S ) becomes necessarily true in a certain formal sense. For example, “Jamal actually arrived first” — if that’s true at the designated actual world, then in every possible world in the model it is true that Jamal arrived first at the designated actual world. So the sentence is necessary even though, intuitively, it seems contingent — Jamal could have been second. Logicians Martin Davies and Lloyd Humberstone introduced another operator, Fixedly Actually, to capture the deeper sense of necessity we were reaching for. A sentence is fixedly actually true if it is true no matter which world we pick as the actual one.
This machinery may sound abstract, but it turned out to be philosophically explosive. It offered a way to understand some famous puzzles from Saul Kripke (1940–2022). Kripke had argued that there are truths that are necessary but can only be known by looking at the world — like “water is H₂O” — and truths that can be known just by thinking but are contingent — like “the standard metre bar in Paris is one metre long.” Davies and Humberstone showed how the two-dimensional logic of “actually” could model such cases. Using the operator ( \mathcal{A} ), one could construct sentences that were logically necessary yet aposteriori (knowable only through experience) and sentences that were logically contingent yet apriori (knowable without experience). The 2D framework seemed to hold the key to a deep mystery about how our minds hook onto the world.
Water on Twin Earth: One Word, Two Worlds
Now take the word “water.” Most English speakers know it refers to the clear, drinkable liquid that falls as rain and fills lakes. But chemists tell us that water is H₂O. Could you have known that just by thinking about your idea of water? It seems not — you had to go out and discover it. Yet many philosophers, including David Chalmers (born 1966) and Frank Jackson (born 1943), argue that your ordinary understanding of “water” still contains an implicit two-dimensional rule: water is whatever chemical kind actually explains the clear, potable stuff around here. This is the A-intension — a function from worlds considered as actual to an extension. The C-intension — from worlds considered as counterfactual — then captures what the word picks out once you know the actual chemical kind. In our world, the C-intension maps every possible world to H₂O.
To see this, imagine a planet called Twin Earth. It is exactly like Earth except that the liquid in its rivers and taps is not H₂O but a complex substance we can call XYZ. XYZ looks, tastes, and behaves like water. On Twin Earth, when someone says “water,” which substance do they refer to? Many people, following the philosopher Hilary Putnam (1926–2016), feel that Twin Earthlings refer to XYZ — because that is the stuff that fills the water-role in their actual environment. On the 2D empiricist picture, this shows that the meaning of “water” has a hidden indexical rule: it picks out different things depending on which world is actual. A 2D matrix with Earth and Twin Earth as rows makes this vivid. Each row gives the C-intension for that world considered as actual, and the diagonal records the A-intension — the pre-scientific rule “the clear, potable stuff around here.” That rule, Jackson says, is what every competent speaker grasps and shares, even before they know any chemistry.
This view is called generalized two-dimensional semantics because it applies the 2D framework to all expressions, not just words like “I.” The hope is to save the old philosophical idea that we can figure out the deepest rules of our words just by armchair reflection — by running thought experiments in our heads.
The Rationalist Dream and the Armchair Explorer
David Chalmers offers a bolder, more rationalist version of this idea. He believes there is a “golden triangle” of tight connections between meaning, apriority (knowability by pure thought), and possibility. In this picture, if two words have the same meaning, then you can know that they pick out the same thing just by reflection. And if something is necessary, then ideally rational reflection can reveal it. The 2D matrix is supposed to be the tool that makes this triangle work: the diagonal intension — what Chalmers calls the 1-intension — captures the aspect of meaning that is accessible from your armchair.
This approach faces a powerful objection. How do you assign an extension to a sentence like “Language exists” when you consider a possible world where there are no speakers, no words, and no thoughts? If you are supposed to imagine yourself inside that world using the word, you can’t — there’s nothing there to do the imagining. So the standard 2D method breaks down. Chalmers’ solution is to reinterpret the possible worlds as scenarios — maximally specific hypotheses about the way the actual world could be, described in a neutral, fundamental vocabulary (like microphysics, plus the facts of consciousness). You then ask: given that this scenario is actual, is my sentence true? This “epistemic” interpretation lets you assign a 1-intension even to worlds without language, because you are using your own words right now to evaluate the scenario, not imagining someone speaking inside it.
This refined version, epistemic two-dimensional semantics, promises that you can know, without ever leaving your chair, exactly what it would take for “water” to refer to XYZ or for “consciousness” to be physical. Critics worry that the required ideal reflection is too hard for real humans, but Chalmers sees it as an extension of ordinary conceptual analysis — the kind you use when you decide whether a weird liquid in a thought experiment counts as water. The prize is enormous: a rational method for discovering the boundaries of possibility itself.
The Outsider’s View: Just a Tool for Conversations
Not everyone thinks the diagonal intension is a genuine kind of meaning. Robert Stalnaker (born 1940) pioneered the use of 2D matrices as early as the 1970s but insists on a much more modest interpretation. He calls his approach metasemantic, not semantic. For Stalnaker, the only real meaning of a word like “water” is its ordinary horizontal intension — the function that picks out H₂O in every possible world. The 2D matrix does not reveal a hidden layer of meaning; it simply tracks our ignorance about what that single meaning is.
Think of it this way: suppose you meet a kid named Lloyd at a party, and later someone says, “Lloyd is I.L. Humberstone.” You don’t know that I.L. Humberstone is the author of a famous article, and you don’t yet know whether the boy in front of you is that author. Your epistemic state can be captured by a tiny 2D matrix over two scenarios: one where Lloyd is the author, and one where he isn’t. When the speaker asserts the identity, what you learn is not a new layer of meaning — you learn the contingent fact that the actual world is the first scenario. The diagonal intension is just a convenient way to summarise the update to your beliefs. It doesn’t belong to the dictionary entry for “Lloyd.”
On this picture, diagonal intensions are ad hoc tools that we cook up when the normal meaning of a sentence is a necessary truth or falsehood and therefore uninformative. They aren’t stable aspects of linguistic competence, nor are they knowable a priori in any deep sense. Stalnaker’s externalism — the view that meaning is fixed by facts outside your head — remains fully intact. The 2D matrix is simply a method for modelling how we talk when we don’t quite know what we’re talking about.
This debate is still wide open. Some philosophers find the metasemantic view realistic and economical; others worry that it gives up on explaining how rational thought works. The argument is, at heart, about whether the rules inside your head are enough to determine what your words refer to — or whether the world always has the last word.
Why It Still Matters When You Learn a New Word
You encounter new words all the time. Maybe it’s “algorithm” in computer class, “schizophrenia” in a book, or “fair” when your friend complains about a rule. In each case, you seem to start with a rough idea — a reference-fixing rule — that guides your early guesses. As you learn more, that initial rule might need to be revised or even replaced. The two-dimensional framework puts its finger on exactly this dance between what you grasp from the armchair and what you discover by looking.
Whether you side with the rationalists or the externalists, the question matters. If Jackson and Chalmers are right, then careful thought about imaginary cases can reveal genuine necessities — perhaps even truths about the mind or morality. If Stalnaker is right, your initial “meaning” is mostly a placeholder, and the world fills in the blank. In that case, philosophy from the armchair is a riskier business.
Next time you say “I,” notice how effortlessly it picks out the right person. Then try the same trick with “water” or “game” or “kindness.” The two-dimensional puzzle is still at work, tracking the double life of every word you speak.
Think about it
- Imagine you have never seen a platypus, but a friend describes it as “a furry animal that lays eggs and has a duck-like bill.” If you later discover a real platypus, did your original description already pick out exactly that animal? Or did you only truly know what “platypus” meant after you saw one?
- If you call a tomato a “vegetable,” but a scientist says it’s a fruit, does the meaning of “vegetable” for you change? Does it make sense to say one of you is wrong, or could you both be right in different dimensions of meaning?
- Suppose a perfect computer could predict every word you will ever say. Would that mean the words had no real meaning until they were spoken, or would it mean the meanings were already there, waiting in a two-dimensional grid?
