1. Introducing the Diagonal

I am trying to keep the peace at a family reunion where everyone has different political views. The family goes to a flea market, where we come across a framed picture for sale. I inspect it and realize it is a rendering of actor Alec Baldwin lampooning Donald Trump on Saturday Night Live. I point to the picture and say:

1. (3) He makes me smile.

Three people overhear me: my conservative uncle, my cousin and my grandfather.

My conservative uncle looks at the picture and forms the belief that it is of Trump himself. My uncle also believes that I and everyone else in the family are big Trump fans. He concludes on this basis that what I said is true. Let ${w_1}$ be the world my uncle thinks we are in.

My cousin takes a look at the picture and forms the (true) belief that it is Baldwin in the picture. He believes that I enjoy watching Trump lampooned. So he agrees with my uncle that what I said was true – though he disagrees with my uncle about the content of what I said. (As a heuristic, we can think of the possible contents of sentences like (3) in terms of their disquoted analogues with the context-sensitive terms removed. So to my uncle, the content of what I said with (3) is Donald Trump makes MF smile; to my cousin, it is Alec Baldwin makes MF smile.) Let ${w_2}$ be the world my cousin thinks we are in.

Finally, my grandfather takes a look at the picture and forms the belief that it depicts Juan Trippe, the 1930s aviation pioneer. ^{Footnote 1} Knowing I have no particular knowledge or affection for that man, my grandfather concludes that what I said is false. His is yet a third nonequivalent hypothesis about the content of my utterance, viz., Juan Trippe makes MF smile. Let ${w_3}$ be the world my grandfather thinks we’re in.

As Stalnaker puts it, such examples are of interest because they highlight two ways the truth-value of my utterance depends on the way the world is (1987 [1999]: 78). The first way is a matter of reference: what (a particular token of) ‘he’ refers to. The second way is a matter of worldly facts. Here, those worldly facts concern: who in the world, independent of any facts about the in situ referents of actual and counterfactual pronouns, makes me smile.

Figure 1 depicts a two-dimensional semantic matrix for sentence (3), showing the different truth-values the sentence takes with respect to every pair $\langle {w_i},{w_j}\rangle$ of worlds drawn from $\left\{ {{w_1},{w_2},{w_3}} \right\}$ .

Figure 1. Matrix for (3).

In the matrix, worlds appear in the vertical axis in their role as world-as-actual, determining the content expressed by the sentence. At the level of the whole sentence, we can think of this as determining the proposition expressed by (3): as we saw, the candidates are, in order: {Donald Trump makes MF smile, Alec Baldwin makes MF smile, Juan Trippe makes MF smile}. Worlds in the matrix appear in the horizontal axis in their role as world of evaluation, where the truth of that proposition is interrogated. Whether a T (‘true’) or an F (‘false’) appears here often depends on counterfactual information of great conjectural interest to philosophers. ^{Footnote 2} For example, on the bottom left, at $\langle {w_3},{w_1}\rangle$ , we see an F: the claim here is that proposition Juan Trippe makes MF smile is false in the world my uncle thinks we’re in – which is to say: in a world where I and my family are all big Trump fans and Trump’s picture crops up at the flea market, I (still) don’t have a special place in my heart for figures in the history of commercial aviation. That I lack such affection in ${w_1}$ is not strictly entailed by the story I told at the beginning. It is a (reasonable) counterfactual stipulation.

Suppose we want to put all such counterfactual information aside, and summarize only the facts about whether my utterance is consistent with my family members’ views on what the actual world is like. The diagonal of the matrix (highlighted in Figure 2 left) brings out just this: my uncle thinks what I said was true (T), as does my cousin, but my grandfather thinks what I said was false (F). A third party genuinely uncertain about which of the worlds in $\left\{ {{w_1},{w_2},{w_3}} \right\}$ is actual is still in a position to see that my utterance is consistent with exactly the first and second of these worlds.

Figure 2. The matrix for (3) (left), and for $\dagger$ (3) (right).

We diagonalize the matrix when we apply the dagger ( $\dagger$ ) operator of 2D semantics to (3) (Figure 2 right), ‘projecting’ just this information across every row.

This operation captures Stalnaker’s answer to the question we began with: how a rational agent should update in cases where the referents of context-sensitive terms (here, he) differ across epistemically possible worlds. ^{Footnote 3} His conjecture is that no matter which world is actual, the rational thing to do is keep in play exactly the worlds consistent with the diagonalized matrix. If a bystander who starts out having no idea whether we’re in world ${w_1}$ , ${w_2}$ , or ${w_3}$ hears me utter sentence (3), and believes I am honest, he should ‘throw out’ ${w_3}$ , but keep ${w_1}$ and ${w_2}$ .

2. 2D Thirding in Sleeping Beauty

Let’s apply diagonalization to a puzzle of credal self-location: the Sleeping Beauty problem discussed by Elga (Reference Elga2000). On Sunday, Beauty signs up for a laboratory experiment. First, she’ll be put to sleep, and a fair coin will be tossed. Whether the coin comes up heads or tails, Beauty will be awakened briefly on Monday. Moreover, if the coin comes up heads, she’ll sleep through Tuesday; otherwise, she’ll be woken again briefly on Tuesday (Figure 3). Each time she awakens, a drug will erase her memory of any previous wakings.

Figure 3. Sleeping Beauty Experiment.

The next thing Beauty knows, she wakes up in the lab. The experimenter asks her a question:

In Sleeping Beauty, there are two intuitively rational answers to (SB). The first is 1/2: the coin is fair. The second answer is 1/3. This answer is suggested by Figure 3: of the three cells of the table consistent with Beauty’s being awake, only one of them coincides with the coin’s landing heads. The challenge for this answer is to articulate a reply to a subsequent question: given that Beauty started out with credence 1/2 in the coin’s landing heads, what proposition $p$ did she learn upon waking up, such that conditioning her prior on $p$ results in a posterior of 1/3?

Diagonalization can help, by increasing the expressive capacity of the language relative to Beauty’s epistemically possible worlds. Begin with a sentence Beauty can truly utter when she wakes up:

1. (4) I am awake today.

As we saw in the Trump-Baldwin example, sentence (4), which contains the indexical expression ‘today’, expresses different propositions at different worlds in Beauty’s information state (Figure 4). ^{Footnote 4}

Figure 4. Candidate propositions for (4).

Figure 5. Heads is true at 1/3 of $\dagger$ (4)-worlds along the diagonal.

In some worlds ( ${w_1}$ and ${w_2}$ ), (4) expresses the a priori proposition Beauty awakens on Monday – a priori in the sense that Beauty knew all along that she would wake up on Monday. But in worlds ${w_3}$ and ${w_4}$ , (4) expresses the previously-uncertain, chance-0.5 proposition Beauty awakens on Tuesday – chance 0.5 because this was slated to happen iff the coin came up tails (viz., as per Figure 3, only in worlds ${w_2}$ and ${w_3}$ ).

It is just this predicament, for Stalnaker, that rationally calls for diagonalization. Figure 5 shows the semantic matrix for (4), with the four possible worlds in Figure 4 arrayed along each axis. The diagonal is highlighted.

Figure 6. Value matrices for a sentence $\phi$ .

Leveraging the assumption that, upon waking, Beauty’s credence function is conditioned on the proposition expressed by the diagonal of the matrix – viz., on $\dagger$ (4) – we can model the third answer: $Cr$ (heads $|\dagger$ (4)) = $1/3$ . The total package involves an appeal to Conditionalization enriched with a two-dimensional bridge principle; we can call the combined norm ‘Sentential Conditionalization’:

The Equivalence Lemma of two-dimensional logic – which roughly states that diagonalization is inert for sentences which do not contain ‘disputed’ context-sensitive terms ^{Footnote 5} – allows us to write the new Conditionalization norm with daggers scoped over both the evidence-sentences and hypothesis sentences:

So in Beauty’s case: $Cr$ (heads $|\dagger$ (4)) = $Cr$ ( $\dagger$ heads $|\dagger$ (4)) = $1/3$ .

To model the appeal of the Stalnakerian analysis, it is helpful to reflect on what happens if we don’t diagonalize. Suppose one maintains that, when one receives testimony $S$ at world ${w_i}$ , one is rationally required to update on the proposition $S$ expresses at ${w_i}$ , rather than on $\dagger S$ (which, as we noted, expresses the same proposition relative to any world in ${w_j}$ in the support of $Cr$ ). Such insistence is vulnerable to an objection from stringency: un-diagonalized norms generate unreasonable demands on rational agents. Suppose, unbeknownst to her, that Beauty has in fact woken up on Tuesday. We can see from Figure 4 that she is therefore in ${w_3}$ . Should she be rationally required to adopt $Cr$ (heads) = 0? Intuitively, no, although this is $Cr$ (heads $|$ (4)) at the context she occupies. ^{Footnote 6} The diagonalizer’s answer $Cr$ (heads $|$ $\dagger$ (4)) is much more natural.

Stepping back, it’s now possible to be a bit more precise about how natural language is hypothesized to relate to the guiding question of Mahtani’s book. Diagonalization lets us reply to the question:

with aspects of both the traditional answers (‘sentences!’, ‘propositions!’). An agent’s credences can be modeled as distributed over a set of worlds she considers epistemically possible. Linguistic competence converts sentences to matrix semantic values, where these worlds to populate the vertical and horizontal axes. The diagonals of these matrices are propositions. It is those diagonal propositions which, at a first pass, are the objects of credence and update.

3. Old Norms in New Bottles

Now that diagonalization is up and running, we can consider diagonalized versions of other rational norms. A recent one, due to Dmitri Gallow, two-dimensionalizes Lewis’s famous ‘Principal Principle’. According to the Principal Principle, one’s credence in a proposition $H$ should defer to the objective chance ( $Ch$ ) of $H$ (Lewis Reference Lewis1971; Mahtani Reference Mahtani2024: Ch. 5.3). Gallow’s sentential version, adapted to the present notation, is:

• Sentential PP. A rational agent’s credence in $\phi$ should defer to her expectation of the chance of $\dagger \phi$ (Gallow Reference Gallow2023: §3).

$$Cr(\dagger \phi |Ch\left( {\dagger \phi } \right) = n) = n$$

To Sentential PP and Sentential Conditionalization, I’d propose to add a third. The rough idea is:

• Sentential Utility. The value of $\phi$ , for a rational agent, is the value of $\dagger \phi$ . ^{Footnote 7}

In the next section, I’ll discuss Sentential Utility in the context of Mahtani’s finite version of the Two-Envelope Paradox (Ch. 6). First, a warm-up example should help us move from matrix semantic values (with range {T, F} – Figure 6 left) to matrix values with range $\mathbb{N}$ – Figure 6 right.

At first pass, it may seem silly to worry about functions from sentences to numbers of this kind. But – as I think Mahtani’s Two Envelopes show – it isn’t. Agents who do not use language cannot, for better or worse, enter into a wide range of betting arrangements, for the simple reason that the payoff conditions of such arrangements cannot plausibly be described to them. The question of how linguistic competence enters into the calculation of expected utility, when the outcomes are described with all the richness of natural language, is important and far from trivial.

Our warm-up will consider this sentence:

1. (5) I have as many Tamagotchi as I actually have.

What is – not the probability, but – the value of (5)? ^{Footnote 8}

We consider the question relative to an information state with four possible worlds, ${w_1}$ through ${w_4}$ . As in the Trump-Baldwin story, we need to fill in the nature of ${w_1}$ through ${w_4}$ in a away that makes vivid how the value of the outcome described by (5) depends both on the nonlinguistic facts and on the semantic facts about indexicals (here, ‘actually’).

Here’s a story that fits the bill. Say I have inherited a large treasure-chest from my grandmother, the Dread Pirate Roberta. I am digging through the coins and goblets inside when I find a Tamagotchi. How much is it worth? I do not know, and different worlds in my information state will represent different resolutions of my ignorance. Let’s say there are four price possibilities: in ${w_1}$ , a Tamagotchi is worth $10; in ${w_2}$ , a Tamagotchi is worth $20; in ${w_3}$ , $30; and in ${w_4}$ , $40. All these worlds (and these alone) are epistemic possibilities for me.

In addition to my ignorance regarding the resale price of Tamagotchi, I am ignorant of how many Tamagotchi I have inherited. Perhaps there is an additional Tamagotchi in the box, and perhaps not; thus I do not know what $n \in \left\{ {1,2} \right\}$ the noun phrase ‘the actual number of Tamagotchi I have’ denotes.

Finally, let us stipulate that, for some reason, these aspects of the world – the price of Tamagotchi, and the number of them I inherited – are not independent. Perhaps I know, given her hoarding habits, that the Dread Pirate would only have stashed a second Tamagotchi in the chest if each was individually quite valuable.

This example would then give us the left matrix in Figure 6 as the utility value (rather than the semantic value) of sentence (5). Once again, we have a case of horizontal mismatch; $Val$ ( $\dagger$ 5) is not equivalent to any un-diagonalized candidate for $Val$ (5). Rather, the diagonal represents a kind of value polarization: in worlds where I have fewer Tamagotchi, they are each individually worth less; in worlds where I have more (viz., two) Tamagotchi, each is individually worth more. Sentential Utility says: the value of the state of affairs expressed by (5) is the function from worlds to utilities expressed by the diagonal of the matrix. ^{Footnote 9} Action-guiding norms that employ $Val$ -expressions, such as expected utility:

(1) $$EU\left( A \right) = \mathop \sum \limits_E \,Cr\left( E \right)Val\left( {A \wedge E} \right)$$

are targets for diagonalization because $Val$ appears in the weighted sum. Sentential Utility encourages us to replace Equation (1) with Equation (2):

(2) $$EU\left( A \right) = \mathop \sum \limits_E Cr\left( {\dagger E} \right)Val\left( {\dagger \left( {A \wedge E} \right)} \right)$$

4. Making Value 2D in the Finite Two-Envelope Paradox

So much for the warm-up exercise. We turn to Mahtani’s example, the Finite Two-Envelope Paradox. ^{Footnote 10}

Mahtani continues:

As with (5), a two-dimensionalist will insert a neon “actually” in the descriptions M and N in Mahtani’s vignette. We should start by glossing $M$ and $N$ as full sentences.

1. (6) (Stick): I get the envelope I actually have.

   $M$

2. (7) (Switch): I get the envelope I don’t actually have.

   $N$

There are also two states:

1. (8) I have the lesser envelope.

   ${E_1}$

2. (9) I have the greater envelope.

   ${E_2}$

If we let $x$ be the amount in the lesser envelope, we can rewrite Mahtani’s Table 7 (with some convenient world-labels) as follows:

When we generate the matrices for $M$ and $N$ with the four worlds generated by $\left\{ {M,N} \right\} \times \left\{ {{E_1},{E_2}} \right\}$ (Figure 8), we see that the two acts denote different constant functions $W \to \mathbb{R}$ in different worlds ( ${w_1}$ and ${w_2}$ are relevant for $M$ ; ${w_3}$ and ${w_4}$ for $N$ ).

We can, therefore, tap the diagonalization strategy. It is easy to show, using Equation (2) rather than Equation (1), that the expected values of the two acts are in fact the same:

$$\eqalign{ EU\left( M \right) & = \mathop \sum \limits_E Cr\left( E \right)Val\left( {\dagger \left( {M \wedge E} \right)} \right) \cr & = Cr\left( {{E_1}} \right)Val\dagger \left( {M \wedge {E_1}} \right)) + Cr\left( {{E_2}} \right)Val\left( {\dagger \left( {M \wedge {E_2}} \right)} \right) \cr & = .5 \cdot Val\left( {\left\{ {{w_1}} \right\}} \right) + .5 \cdot Val\left( {\left\{ {{w_2}} \right\}} \right) \cr & = .5 \cdot x + .5 \cdot 2x \cr & = 1.5 \cdot x \cr} $$

$$\eqalign{ EU\left( N \right) & = \mathop \sum \limits_E Cr(E|N)Val\left( {\dagger \left( {N \wedge E} \right)} \right) \cr & = Cr\left( {{E_1}} \right)Val\left( {\dagger \left( {N \wedge {E_1}} \right)} \right) + Cr\left( {{E_2}} \right)Val\left( {\dagger \left( {N \wedge {E_2}} \right)} \right) \cr & = .5 \cdot Val\left( {\left\{ {{w_3}} \right\}} \right) + .5 \cdot Val\left( {\left\{ {{w_4}} \right\}} \right) \cr & = .5 \cdot x + .5 \cdot 2x \cr & = 1.5 \cdot x\; \cr} $$

5. The Big Picture

I’ve argued that the two-dimensionalist’s diagonalizing strategy has something to contribute to (at least) two of the puzzles studied by Mahtani. Here I briefly compare her own analysis. Two discussions in the book are particularly relevant. First, in the Two-Envelope case, Mahtani endorses a refinement of a strategy suggested by Horgan (Reference Horgan2002):

Because Mahtani’s decision table in Figure 7 violates Sameness of expectation in value, arguments based on it (like the argument that one should switch) are ‘defective’ (127), and should be rejected.

Figure 7. From Mahtani (Reference Mahtani2024: 117).

Figure 8. Value of $M$ (left) and $N$ (right).

Second and more generally, in Chapter 7 Mahtani critically discusses David Chalmers’s two-dimensional approach to understanding credal reports (Chalmers Reference Chalmers2011a, b). In Chalmers’s terminology – which is importantly related to Stalnaker’s, though it connotes a different array of philosophical commitments – the objects of credence should be interpreted as primary, rather than secondary, intensions. That distinction maps roughly onto our claim that the objects of credence (and learning! and value!) are diagonal, rather than horizontal, propositions.

I begin with Horgan, whose recommendation is that outcomes must be stated in terms of ‘epistemically rigid designators’ (quoted in Mahtani: Reference Mahtani2024 127). What this comes to, I think, is that terms like $M$ shouldn’t be used in the formulation of decision problems. My worry about the strategy is how restrictive it is. For example, in Sleeping Beauty, ‘today’ is not an epistemically rigid designator, because when Beauty wakes up, she doesn’t know what day it is (ergo: does not know which day ‘today’ denotes). Should we therefore prohibit Beauty from thinking about her situation using the concept (that she would articulate with) ‘today’? This move seems to deprive Beauty of any way of thinking about where she is located in time – to say nothing of precluding the attractive thirder strategy I sketched above.

When it comes to Chalmers’s two-dimensionalist view, Mahtani worries about the destructive effects of a wholesale move from secondary to primary intensions. In ${\rm{\S}}$ 7.6, she lists three important ones: (i) thinking-alike, (ii) deference, and (iii) conditionalization:

$$P{r_{{\rm{Tom}}}}(P|P{r_{{\rm{Tess}}}}\left( P \right) = v) = v$$

Without providing a full defense of two-dimensionalism from these concerns, I think that what we have covered so far gives grounds for optimism. We can consider Mahtani’s concerns in reverse order. For worry (iii) – concerning Conditionalization – I argued above that we need diagonalization for a proper version of the norm. For worry (ii) – deference to experts – I have appealed to others who have argued similarly: Gallow, cited above, claims for independent reasons that the proper way to operationalize the Principal Principle, which has exactly the same form as Mahtani’s Tom-Tess deference norm, will insert diagonalizing operators into the scope of both probability functions.

Finally, for (i) – attribution of sameness (and difference) of credal content, we can tap Sleeping Beauty again. Suppose Beauty is a thirder, and it is now Sunday. Consider these two true statements:

1. (10) Present Beauty’s credence that [today is not Tuesday] is 1.

2. (11) Future Beauty’s credence that [today is not Tuesday] is 2/3.

From these two statements, a simple thinking-alike norm of attribution would lead us to infer that Beauty is going to violate conditionalization: there is some $P$ such that her credence in $P$ is going to drop from 1 to something lower. The more complex norm of credal attribution the two-dimensionalist would endorse blocks this inference. But that’s a good thing. In the envisioned scenario, Beauty does not update irrationally.