1 Supposing, learning, and belief dynamics

What would be the case if P was the case? We deal with counterfactual questions all the time, in explanatory reasoning (‘Would the patient show such symptoms if it was smallpox?’), to ascertain legal and moral responsibilities (‘Would the pedestrian have been hit if the brakes had worked as expected?’), learn from mistakes (‘Would Johnson have scored if he had played centre forward?’), choose what to do next (‘Would I make more money if I invested in equities?’). Counterfactual thinking pervades intelligent life. How does it work?

It is often remarked that our primary way of assessing conditionals is by supposing the antecedent and checking the believability of the consequent on the supposition [Reference Williamson36]. The idea is at the core of ‘suppositional accounts’ of conditionals [Reference Adams1, Reference Bennett5, Reference Edgington11, Reference Evans and Over14]; however, these have been developed mainly in a probabilistic setting and for indicatives, rather than counterfactuals. The view is at least as old as a mythical footnote by Frank Ramsey:

This we now call the Ramsey Test. It makes our assessment of conditionals a sort of simulated belief revision. It takes our assessment of Q on the supposition of P as structurally similar to our assessing Q in the light of a revision of our beliefs upon learning that P. This makes sense because, just as imagining is a bit like seeing, supposing is a bit like learning. Supposing that Paul is in the kitchen, or imagining a situation where Paul is in the kitchen, we then adjust our beliefs to the supposition, checking what we take to be true in the imagined scenario. We can come to believe that in the imagined scenario, that is, within the supposition, it’s dinner time. This is a bit like seeing, and thus learning, that Paul is in the kitchen and then adjusting our beliefs accordingly, coming to believe that it’s dinner time.Footnote ¹ The idea sits well with simulationist accounts of imagination, which have it that imagination imitates counterpart non-imaginative mental states and activities [Reference Currie and Ravenscroft9, Reference Goldman18].

The formal literature on belief revision agrees that revision must be minimal [Reference Alchourrón, Gärdenfors and Makinson2, Reference Baltag, Smets, Bonanno, Hoek and Wooldridge4, Reference Van Benthem6, Reference Van Ditmarsch, van der Hoek and Kooi10, Reference Grove20]: one ought to change one’s beliefs as little as possible upon learning that P. Correspondingly, per the Ramsey Test, one should assess a conditional with antecedent P and consequent Q by assessing Q against a belief system that’s changed, in one’s mental simulation, as little as possible to accommodate the supposition P. Literature on the epistemic value of imagination agrees too [Reference Kind, Kind and Kung23–Reference Van Leeuwen, Kind and Kung25, Reference Williamson, Kind and Kung35]: imagination can have epistemic value in hypothetical thinking, to the extent that the development of an imaginative input is constrained by principles of plausibility and adherence to how one takes reality to be. What’s difficult is to fine-tune what ‘as little as possible’ exactly means.

Conditionals give us reasons to think that minimality comes in more than one fashion. For look at Lewis’s [Reference Lewis28] famous example (credited to Adams):

1. (1) If Oswald did not kill Kennedy, then someone else did.

2. (2) If Oswald had not killed Kennedy, then someone else would have.

(1) sounds very acceptable, and (2) doesn’t (if one believes in the verdict of the Warren commission). By the Ramsey Test, that’s because the consequent of (1) looks believable, the one of (2) doesn’t, on supposing the antecedent. But because it’s the same antecedent and consequent, the difference must be in how we suppose and, accordingly, revise beliefs. So the literature distinguishes indicative and subjunctive or counterfactual supposition [Reference Byrne8, Reference Eva, Shear and Fitelson13, Reference Joyce21, Reference Leitgeb and Segerberg27].

Beliefs can be all-or-nothing, or graded (then they’re often called ‘credences’). So belief revision can be phrased in a qualitative or quantitative fashion. This paper is the first of a two-part work, focusing on the qualitative case and using modal-epistemic logic to capture belief revision on a counterfactual supposition. Part II will deal with the quantitative case using probabilities.

Here’s one further important feature of such mental simulation activity, as recognized in the literature: it is episodic [see, e.g., Reference Badura, Badura and Kind3, Reference Wansing33]. In an episode of hypothetical thought, we suppose that P deliberately, wondering what follows. We usually have specific Qs to focus on. We carry out the exercise, revising our beliefs under the supposition. The exercise unfolds in time. We stop after a (long or short) while, when we reach a stable verdict on the acceptability of Q on the supposition P. (Or, when the real world gets in the way and interrupts our mental exercise!)

Thus, simulated belief revision under supposition has its dynamics. Now the dynamics of belief revision under the indicative supposition of P must be very different from that of belief revision under the counterfactual supposition of P. And while the former has been extensively investigated in the formal literature on belief revision and in particular in dynamic epistemic logic (DEL) [see, e.g., Reference Baltag, Smets, Bonanno, Hoek and Wooldridge4, Reference Van Benthem6, Reference Van Ditmarsch, van der Hoek and Kooi10], the latter is less well-understood and studied, one notable exception being the wonderful work of Leitgeb and Segerberg [Reference Leitgeb and Segerberg27] which, however, rather than DEL, employs the framework of Segerberg’s dynamic doxastic logic (DDL). We get back to the Leitgeb–Segerberg work later on.

2 Features of supposition

We start by listing four general features of the kind of hypothetical thinking we are after:

1. (Certainty) One who supposes P believes P on the supposition.

2. (Minimality) If P is already believed, one does not revise one’s beliefs on the supposition of P.

3. (Consistency) If Q is believed on the supposition of P, and P is believed on the supposition of Q, then revising by P and by Q must lead to the same result.

4. (Possibility) If P is possible (and one’s beliefs are consistent), then one’s beliefs on the supposition of P will (still) be consistent.

These surface here and there in the literature and there’s some agreement on their holding for both indicative and counterfactual supposition, both for all-or-nothing belief and for credences [Reference Joyce21, Reference Lewis28, Reference Lewis29, Reference Skyrms31].

Certainty is endorsed, e.g., by Williamson: ‘trivially, one accepts A on the supposition A’. ‘Anything is to be accepted on the supposition of itself. Therefore, by the Suppositional Rule [i.e. the Ramsey Test], “If A, A” is always to be accepted’ [Reference Williamson36, pp. 18 and 53]. One finds the idea in formal accounts of belief revision under the label of the ‘Success’ principle: after updating by P, one believes P [Reference Alchourrón, Gärdenfors and Makinson2, Reference Van Benthem, Smets, Ditmarsch, Halpern, Hoek and Kooi7]. For supposition, it may not hold unrestrictedly, however: there may be such a thing as partial supposition, whereby one treats the supposed content as having an increased degree of plausibility or likelihood but stops short of endorsing it as true within the supposition. This is best dealt with in a probabilistic setting, so we postpone the discussion to our Paper II. The idea is investigated in [Reference Eva and Hartmann12].

Minimality follows from the idea that revision must be, well, minimal. It says: ‘no revision when none is needed’ [Reference Lewis29, p. 150]. If you are already convinced that Paris is in France, and you’re asked to suppose that Paris is in France, you’ll feel there is nothing new for your belief system to adjust to on the supposition.

We gave Consistency its name following Lewis, who says that ‘the method [of belief revision] must be consistent in its comparisons’ [Reference Lewis29, ibid.]: if it failed, revising by P would be both more and less of a revision than revising by Q.

Possibility encodes the idea that when you suppose a possible P, you’ll only look at possible scenarios—unless your beliefs were inconsistent in themselves already, before the supposition, and then it’s their fault if your revised beliefs could not be true in any possible scenario.

We consider it a desideratum for a good account of the dynamics of counterfactual supposition that it satisfies these conditions. We mentioned that these are (taken as) shared by indicative supposition. We’ll come later to a key feature pulling the two kinds apart.

3 Making things precise

Start with a nonempty set of possible worlds W and take a proposition $P \in \mathcal {P}(W)$ as represented, in traditional fashion, by the set of worlds where it holds: P is true at w iff $w \in P$ .

For $P \subseteq W$ , let $\leqslant _{w}^{P}$ be the restriction of $\leqslant _w$ to P and let $min_{P} \leqslant _w$ be the set of minimal elements of P with respect to $\leqslant _w$ :

$$ \begin{align*}min_{P} \leqslant_w=\{w' \in P \, \vert\, \forall w"\in P\ , \, w"\not<_w w' \},\end{align*} $$

where $w" <_w w'$ if and only if $w" \leqslant _w w'$ and $w' \not \leqslant _w w"$ . We write $min \leqslant _w$ for $min_{W} \leqslant _w$ .

Plausibility will be at work in the characterization of belief and (dispositions to) belief revision. The idea will be, as is common in the literature (see, e.g., [Reference Van Benthem, Smets, Ditmarsch, Halpern, Hoek and Kooi7, Reference Grove20]), that what one believes is what holds at the worlds one finds most plausible given one’s total information. One could call these, the agent’s belief-worlds.

That plausibility is a preorder, not an order, means there can be ties in the competition for plausibility. That it’s total means we can always compare worlds for plausibility. That it’s a well-preorder means that there will never be infinite descending chains of worlds, getting more and more plausible, forever, i.e., $min_{P} \leqslant _w \neq \emptyset $ when $P \neq \emptyset $ .

( $w" \prec _w w'$ if and only if $w" \preccurlyeq _w w'$ and $w' \not \preccurlyeq _w w"$ .) For $P \subseteq W$ , let $\preccurlyeq _{w}^{P}$ be the restriction of $\preccurlyeq _{w}$ to P and let $min_{P} \preccurlyeq _{w}$ be the set of minimal elements of P with respect to $\preccurlyeq _{w}$ . In words: $min_{P} \preccurlyeq _w$ are the closest to w (least divergent) worlds in P. Condition (ii) ensures that in general $min_{P} \preccurlyeq _w \neq \emptyset $ .

Divergence is the hallmark of counterfactual supposition. One can take $min_{P} \preccurlyeq _{w}$ as the worlds one reaches, starting from w and applying, in one’s counterfactual imagination, the minimal changes required to make P true. One who counterfactually supposes that Paul is in the kitchen will imagine a situation that’s changed as little as possible from the viewpoint of w, compatibly with making it the case that Paul is in the kitchen.

This becomes obvious as soon as one phrases the idea in terms of selection functions.

Each divergence relation on W imposes a selection function: when $\{\preccurlyeq _w\}_{w\in W}$ is a divergence relation on W, then $f: W \times \mathcal {P}(W) \to \mathcal {P}(W)$ , defined as $f(w, P)= min_{P} \preccurlyeq _w$ , is a selection function.

Selection functions are familiar from the literature on conditional logics [Reference Lewis28, Reference Stalnaker and Rescher32], where one gives the truth conditions for a ceteris paribus conditional, say $\varphi> \psi $ , by claiming that it’s true at w iff the consequent $\psi $ is true at the selected antecedent-worlds, i.e., iff $f(w, ||\varphi ||) \subseteq ||\psi ||$ ( $||\varphi ||$ being the proposition expressed by $\varphi $ : the set of worlds where $\varphi $ is true).

Now give ‘ $\!\! f(w,P)$ ’ a suppositional reading: take it as picking the worlds one looks at, from the viewpoint of w, on the counterfactual supposition of P; that is, the worlds one reaches by implementing, in imagination, the least drastic change required to make P true (an insight already in [Reference Stalnaker and Rescher32]). Then our conditions (1)–(4) mirror exactly our four conditions on suppositional thought. (1) maps to Certainty: all the worlds one looks at on supposing P must make P true. (2) maps to Minimality: when P is already true at w, the single closest world to w is w itself, i.e., one moves nowhere away from w (this is often called (Strong) Centering in the literature on conditional logics). (3) maps to Consistency (in Comparisons): if the least divergent P-worlds make true Q and vice versa, one looks at the same worlds when supposing P and when supposing Q (this is sometimes called Uniformity). (4) maps to Possibility: if P is possible, one must look at some possible scenario when supposing it.

It will be useful to compare indicative and counterfactual supposition by mapping the former to belief revision as captured in DEL. The following essentially re-purposes what one can find in the lovely introduction to DEL by Van Benthem and Smets [Reference Van Benthem, Smets, Ditmarsch, Halpern, Hoek and Kooi7].

In dynamic logics in general, one has in the language operators $[a]$ (‘after action a has been executed’) which work as model-transformers: one claims that $[a]\varphi $ holds in a model when $\varphi $ holds in the model(s) obtained by transforming the initial one as per the instructions encoded in $[a]$ . In DEL, the focus is on epistemic actions such as learning.

Learning a true P with certainty should reduce the set of possibilities W to those consistent with P. This is often called ‘hard information’: information that irrevocably changes the set of epistemic possibilities for an agent. However, not all information has such drastic effects. For instance, learning information from not fully reliable sources, or that is uncertain in itself, does not necessarily reduce the set of possibilities. What such information does change is what is believed locally, at a world w. Let’s unpack.

So P is believed when it’s true at all of one’s belief-worlds, i.e., at the worlds one finds most plausible. Q is believed on the indicative supposition of P when it’s true at all of one’s belief-worlds where P is true, i.e., at the most plausible P-worlds.

Soft information can prompt a dynamics of belief revision by changing the plausibility ordering between worlds. Two popular revision policies are the radical (aka lexicographic) and conservative upgrades [see Reference Van Benthem, Smets, Ditmarsch, Halpern, Hoek and Kooi7, pp. 334–336]—in terms of supposition.

Definition 6. Q is believed on the radical supposition of P, $^{\Uparrow P}Bel(Q)$ , at w, iff ${min \leqslant ^{\prime }_w \subseteq Q}$ where (with $\bar {P}$ the complement of P):

(1) $$ \begin{align} w' \leqslant^{\prime}_w w" \iff \begin{cases} w' \in P \,\, \& \,\, w" \in \bar{P} \\ w', w" \in P\,\, \& \,\, w' \leqslant_w w"\\ w', w" \in \bar{P} \,\, \& \,\, w' \leqslant_w w". \end{cases} \end{align} $$

Definition 7. Q is believed on the conservative supposition of P, $^{\uparrow P}Bel(Q)$ , at w, iff $min \leqslant ^{\prime }_w \subseteq Q$ where

(2) $$ \begin{align} w' \leqslant^{\prime}_w w" \iff \begin{cases} w' \in min_{P} \leqslant_w \\ w' \leqslant_w w". \end{cases} \end{align} $$

So the radical setting makes any P-world more plausible than any not-P world while leaving the old plausibility ordering in place within the two groups. The conservative setting replaces the current ordering with one that puts the most plausible P-worlds ahead, leaving the rest unchanged.

Other formats have been studied, but this should suffice to get an idea. What matters for us is that the counterfactual supposition of P is quite different from its indicative counterpart.

Definition 8. Q is believed on the counterfactual supposition of P, $Bel_P(Q)$ , at w, iff

$$ \begin{align*}\bigcup_{w' \in \min \leqslant_w} min_{P} \preccurlyeq_{w'}\subseteq Q.\end{align*} $$

So Q is believed on the counterfactual supposition of P when it’s true at all of the P-worlds least divergent from each of one’s belief-worlds. Thus, notice the difference: on the indicative supposition of P, we look at the most plausible P-worlds; on the counterfactual supposition of P, we look at the P-worlds closest to the most plausible worlds.

When we suppose that P indicatively, we pretend to be getting new information about actuality. Hence, only plausibility is at work: there is no departure from here to check how things would be if P, i.e., to check how things are there, i.e., at the minimally divergent world(s) where P. So we work globally: we take all of our belief-worlds, representing how we take reality to be, and we have them interact with P by intersecting them with the P-worlds, or by variously re-ordering their plausibility in the light of P.

When we suppose that P counterfactually, instead, we depart from how we believe reality to be so that we can make nontrivial comparisons with it from within the supposition: divergence enters into play. We work locally: for each of our belief-worlds, we switch to its closest P-world(s), i.e., to what we reach by minimal change imposed, in our imagination, in order to make P true; and we lump those together.

4 Counterfactual dynamics recaptures KGM

The celebrated AGM theory of belief revision by Alchourrón et al. [Reference Alchourrón, Gärdenfors and Makinson2] provides the standard framework when the revision corresponds to learning new information; it is naturally mapped to indicative supposition. For counterfactual supposition, the theory to look at is the KGM account (so-named in [Reference Leitgeb and Segerberg27] after the seminal paper by Katsuno and Mendelzon [Reference Katsuno, Mendelzon and Gärdenfors22], complemented by the work of Grahne, e.g., [Reference Grahne19]). Let’s explore why this is so.

The kind of revision captured by KGM is often called ‘belief update’, with ‘belief revision’ reserved to AGM in contexts of comparison between the two—but we need not quibble about the terminology. Katsuno and Mendelzon marked the difference with AGM by claiming that KGM models how one changes one’s beliefs in response to relevant worldly change, AGM models how one does it in response to new information on a relevantly unchanged world, but this way of capturing their difference is disputed (see, e.g., [Reference Friedman and Halpern16]).

What matters for us is that KGM is good for counterfactual supposition, a point investigated in [Reference Eva, Shear and Fitelson13]. We will now show this, by revisiting in an abstract setting the dynamics we introduced in §3 for counterfactual supposition and showing that it satisfies the KGM belief update postulates one can find in [Reference Katsuno, Mendelzon and Gärdenfors22].

The KGM postulates include principles that match our four general features of supposition, singled out in §3: Certainty, Minimality, Consistency (in Comparisons), and Possibility. By recapturing KGM, our dynamics will therefore comply with them as well. In §4, we will then see that the dynamics of counterfactual supposition fails a principle which is, instead, satisfied by indicative supposition and its dynamics. This failure marks their crucial difference.

Notice that the belief and counterfactual belief sets characterise the set of beliefs and the set of P-counterfactual beliefs, in the sense that they define the strongest belief and P-counterfactual belief at w. In particular, $Bel(P)$ and $Bel_{P}(Q)$ hold at w iff $Bel(w) \subseteq P$ and $Bel_{P}(w) \subseteq Q,$ respectively.

Counterfactual dynamics transforms a proposition (a set of worlds) into another.

Definition 10. The P-counterfactual dynamics is a mapping $(\cdot )_{P}: \mathcal {P}(W) \to \mathcal {P}(W)$ defined for all $V \subseteq W$ as

$$ \begin{align*}V_{P} = \underset{v \in V} \bigcup f(v, P).\end{align*} $$

This essentially rephrases in our abstract setting the idea originally formulated by Katsuno and Mendelzon in terms of ‘possible models’: one updates V on P by (a) moving from each point/world/model $v \in V$ to those closest to v where P holds and (b) collecting all of these points/worlds/models together (see [Reference Katsuno, Mendelzon and Gärdenfors22, p. 188] and compare [Reference Grahne19, p. 94]).

In particular, the P-counterfactual dynamics transforms a belief set $Bel(w) = \min \leqslant _w$ to the belief set $Bel_P(w)= \underset {v \in \min \leqslant _w} \bigcup f(v, P) = \underset {v \in \min \leqslant _w} \bigcup \min _P \preccurlyeq _v$ .

5 Preservation of certainties

It is not difficult to show that (counterpart formulations of) Certainty, Minimality, Consistency in Comparisons, and Possibility hold for AGM and indicative supposition. So while it’s good for our counterfactual dynamics to satisfy them as well, insofar as they’re conditions on the adequacy of an account of supposition in general, this tells us little on how counterfactual supposition differs from its indicative cousin. Let’s delve into this by starting with what we take to be a mistaken way of drawing the distinction.

It is a piece of common wisdom that we only suppose that P indicatively when we are not certain that P is false. When we are so certain, this is the territory of counterfactual supposition. For instance, glossing on the Ramsey Test for indicatives, Bennett says:

(Bennett makes the point for degrees of belief represented by probabilities, but of course it carries over to all-or-nothing belief.)

Common wisdom is wrong, we think. One can non-trivially reason with indicatives whose antecedent one disbelieves: one can assess Indicative Oswald also when one is sure that Oswald killed Kennedy [Reference Gillies17]. Thinkers who don’t deem themselves infallible can suppose in the indicative mood that P also when they utterly disbelieve P, and assess whether Q is the case under that supposition:

One example is given by Williamson [Reference Williamson36]: you can imagine that you don’t exist and reasonably assess the indicative ‘If I do not exist now, then thinking occurs without a thinker’ although, presumably, you have Cartesian certainty in the falsity of the antecedent. One doesn’t need to get so philosophical anyway, for mathematics provides a plethora of cases. The maths teacher in the business of a proof by reductio can felicitously assert: ‘If there is a largest prime, then some number is both prime and composite’.Footnote ³

If the difference between indicative and counterfactual supposition is not what common wisdom claims it is, then what is it? We think, following Joyce [Reference Joyce21], that it has to do with the difference in the background beliefs that can be preserved, or lost, in the respective dynamics. And this will be the right way to mark the crucial aspect whereby indicative and counterfactual supposition differ.

Start with another definition.

So a belief revision dynamics preserves certainties when, if one already believes that Q (and doesn’t disbelieve that P), one will retain one’s belief in Q on supposing that P.

Indicative supposition preserves certainties—look at Indicative Oswald again: we are certain that Q, that Kennedy has been killed. So when we wonder, What if P?, What if Oswald didn’t kill Kennedy?, we retain our belief in Q. Since Kennedy has been killed, it must have been someone else. Hence we believe the consequent on indicatively supposing the antecedent.

Counterfactual supposition, instead, can make us lose certainties—look at Counterfactual Oswald again: when we wonder, What if it had been the case that P?, What if Oswald hadn’t killed Kennedy?, we look, for each of our belief-worlds w, at the minimally divergent scenarios where P, i.e., we switch to $f(w, P)$ , and lump them together. If we are Warrenite enough, these may include worlds with no killing. Hence we don’t believe the consequent on counterfactually supposing the antecedent.

We think the opposition between preserving and losing certainties marks the fundamental difference between indicative and counterfactual supposition (thanks to an anonymous referee of this Journal for pressing us to make this explicit). The reason is that, in the indicative way of supposing that P, we simulate getting new information about the actual world, or the actual world as unchanged in the relevant respects. So things taken as facts, i.e., that we are certain of as actually holding, such as that Kennedy has been killed, are left untouched. In the counterfactual way of supposing that P, we simulate that the world has been changed in the relevant respects, no more than needed to make it true. So we may twist (things taken as) facts in our counterfactual mental simulation. For a different example used (in a quantitative, degrees-of-belief setting) by Hannes Leitgeb, let P = The banana is not in the basket. Then the difference is between ‘whether the evidence is meant to express some additional information about an unchanged world (the banana is not in the basket, i.e., it has not been there) or rather information about how the world has changed (the banana is not in the basket, i.e., it was removed if it had been there at all)’ [Reference Leitgeb26, p. 474].

Our sticking to actuality in the former case, not in the latter, is mirrored in the famously different behaviour of the two kinds of conditionals when actuality operators (‘Actually P’ is true at world w iff P is true at the actual world) are embedded in their consequent. The indicative (‘If Midori is working, she is actually working’) is trivially true; the counterfactual (‘If Midori was working, she would actually be working’) is not.

So it would be good for our counterfactual dynamics to deserve its name by not preserving certainties. And indeed, it doesn’t.

So we have shown both how counterfactual dynamics satisfies the four general features of suppositional thinking, i.e., Certainty, Minimality, Consistency, and Possibility, and how it diverges from indicative supposition by failing Preservation of Certainties. We now put our counterfactual dynamics to work in a full-fledged logical system. We record here our debt to Leitgeb and Segerberg [Reference Leitgeb and Segerberg27] which presents what is, to the best of our knowledge, the closest antecedent (pun intended) to our logic. We will comment on some similarities and differences at the end of §6.

6 Dynamic logic of counterfactual supposition

6.1 Syntax and semantics

Given a set of propositional variables $\mathfrak {P}$ , the sentences of dynamic logic of counterfactual supposition (DLCS), $S\mathcal {L}_{DLCS}$ , are recursively defined as

$$ \begin{align*}\varphi:= p\, \vert\, \neg \varphi\, \vert\, \varphi \wedge \varphi\, \vert\, K\varphi \vert\, B\varphi\, \vert\, [\Lsh \varphi]\varphi\, \vert\, \varphi> \varphi.\ \end{align*} $$

Read ‘K’ as ‘one knows that’, ‘B’ as ‘one believes that’, ‘ $[\Lsh \varphi ]$ ’ as ‘after the counterfactual supposition of $\varphi $ ’, and ‘ $>$ ’ as a counterfactual conditional, ‘if it were …then it would be that’.

Definition 12. A dynamic counterfactual model (DCM) is a tuple $M=(W, \{\leqslant _w\}_{w \in W}, \{\preccurlyeq _w\}_{w \in W}, v)$ : W is a non-empty set of worlds, $\{\leqslant _w\}_{w \in W}$ is a plausibility relation on W, $\{\preccurlyeq _w\}_{w \in W}$ is a divergence relation on W, and $v: \mathfrak {P}\rightarrow \mathcal {P}(W)$ is a valuation function that maps each propositional variable to a proposition taken as a set of worlds.

For any $P \subseteq W$ , let $f_{P}(w)= min_{P} \preccurlyeq _w$ denote the set of minimal elements of P with respect to $\preccurlyeq _w$ . In words, $f_{P}(w)$ denotes the set of closest or least divergent P-worlds to w.

We now define the model-transforming operation for counterfactual revision. Let $M=(W, \{\leqslant _w\}_{w \in W}, \{\preccurlyeq _w\}_{w \in W}, v)$ be a model and $P \subseteq W$ .

Definition 13. The counterfactual revision of M with $P \subseteq W$ is a model $M^{\Lsh P}=(W, \{\leqslant ^{\prime }_w\}_{w \in W}, \{\preccurlyeq _w\}_{w \in W}, v)$ where

(3) $$ \begin{align} w' \leqslant^{\prime}_w w" \iff \begin{cases} w' \in f_P(v) \text{ for some } v \in min \leqslant_w \\ w' \not \in f_P(v) \,\, \& \,\, w" \not \in f_P(u) \text{ for all } v, u \in min \leqslant_w \,\, \& \,\, w' \leqslant_w w". \end{cases} \end{align} $$

So counterfactual revision turns M into an $M^{\Lsh P}$ that differs from M in that the closest P-worlds to the most plausible worlds in M, i.e., intuitively, the worlds reachable from each of the belief-worlds of the agent by minimal change required to make P true, become the most plausible worlds.

The truth conditions for formulas of the language go thus:

Definition 14. With model $M=(W, \{\leqslant _w\}_{w \in W}, \{\preccurlyeq _w\}_{w \in W}, v)$ , $w \in W$ :

1. (Sp) $M, w \vDash p \iff w \in v(p).$

2. (S¬) $M, w \vDash \neg \varphi \iff M, w \nvDash \varphi .$

3. (S∧) $M, w \vDash \varphi \wedge \psi \iff M, w \vDash \varphi \,\, \mathrm { and } \,\, M, w \vDash \psi .$

4. (SK) $M, w \vDash K\varphi \iff M, v\vDash \varphi \,\, \text { for all} \, v \in W$ .

5. (SB) $M, w \vDash B\varphi \iff M, v\vDash \varphi \,\, \text { for all} \, v \in \min \leqslant _w$ .

6. (S↰) $M, w \vDash [\Lsh \varphi ] \psi \iff M^{\Lsh \varphi }, w \vDash \psi .$

7. (S>) $M, w \vDash \varphi> \psi \iff M, v \vDash \psi \,\, \text { for all} \, v \in \min _{\vert \vert \varphi \vert \vert } \preceq _w$ .

(Again ‘ $\!\! \vert \vert \varphi \vert \vert $ ’ refers to the truth set of formula $\varphi $ ; also, we use ‘ $M^{\Lsh \varphi }$ ’ as a shorthand for the clunky ‘ $M^{\Lsh \vert \vert \varphi \vert \vert }$ ’.)

So (SK) has it that knowledge is truth at all worlds. (SB) has it that belief is truth at the most plausible worlds. (S $\Lsh $ ) has it that counterfactual supposition works as a model-transformer: after the counterfactual supposition of $\varphi $ , $\psi $ is the case iff $\psi $ is true in the transformed model at w.

(S $>$ ) has it that $\varphi $ counterfactually implies $\psi $ , at w in M, iff $\psi $ is true in all of w’s closest $\varphi $ -worlds, i.e., iff $f_{ \vert \vert \varphi \vert \vert }(w) \subseteq \vert \vert \psi \vert \vert $ . So this is a variably strict conditional à la Stalnaker [Reference Stalnaker and Rescher32] and Lewis [Reference Lewis28].

Definition 15. We say that $\varphi $ is a logical consequence of a set of formulas $\Gamma $ , and write $\Gamma \vDash \varphi $ , if for every model M, and every world w in M, if $M, w \vDash \psi $ for all $\psi \in \Gamma $ then $M, w \vDash \varphi $ . We say that $\varphi $ is a logical truth if for every model M and every world w in M, $M, w \vDash \varphi $ .

We use the following shorthand notation:

$\bot := p \wedge \neg p \hspace {10mm} \top := p \vee \neg p \hspace {10mm} \varphi \leftrightarrow \psi := (\varphi \to \psi ) \wedge (\psi \to \varphi ).$

Via the belief operator B and dynamic operator $ [\Lsh \varphi ]$ we can define in DLCS belief under counterfactual supposition:

• • Belief under the counterfactual supposition of $\varphi $

  $$ \begin{align*}B_{\varphi} \psi:= [\Lsh \varphi] B\psi.\end{align*} $$

Notice that by definition, $B_{\varphi } \psi $ if and only if $ \underset {v \in \min \leqslant _w} \bigcup f_{\vert \vert \varphi \vert \vert }(v) \subseteq \vert \vert \psi \vert \vert $ .

Definition 16. For sentences $\varphi , \psi , \theta \in S\mathcal {L}$ , $(\psi )_{\varphi }$ is defined inductively as

$$ \begin{align*}(p)_{\varphi}&= p \\(\neg \psi)_{\varphi} &= \neg (\psi)_{\varphi}\\(\psi \wedge \theta)_{\varphi}&= (\psi)_{\varphi} \wedge (\theta)_{\varphi}\\[4pt](K \psi)_{\varphi}&= K (\psi)_{\varphi}\\[4pt](B \psi)_{\varphi}&= B (\varphi> (\psi)_{\varphi})\\[4pt](\psi>\theta)_{\varphi}&=(\psi_(\varphi)>(\theta)_\varphi)\\[4pt]([\Lsh\theta]\psi)_\varphi& = ((\psi)_\theta)_\varphi) . \end{align*} $$

As the next proposition shows, $(\psi )_{\varphi }$ is the sentence that will be translated to $\psi $ under the counterfactual assumption of $\varphi $ .

Proposition 2. For any DCM, M, and world w,

$$ \begin{align*}M^{\Lsh \varphi}, w \vDash \psi \iff M, w \vDash (\psi)_{\varphi}.\end{align*} $$

Proof. By induction on the complexity of $\psi $ .

1. (i) $\psi =p$ . Follows immediately.

2. (ii) $\psi = \theta \wedge \zeta $ . Then

   $$ \begin{align*}M^{\Lsh \varphi}, w \vDash \theta \wedge \zeta \iff M^{\Lsh \varphi}, w \vDash \theta \& M^{\Lsh \varphi}, w \vDash \zeta \iff\end{align*} $$
   $$ \begin{align*}M, w \vDash (\theta)_{\varphi} \& M, w \vDash (\zeta)_{\varphi} \iff M, w \vDash (\theta)_{\varphi} \wedge (\zeta)_{\varphi} \iff M, w \vDash (\theta \wedge \zeta)_{\varphi}.\end{align*} $$

3. (iii) $\psi = \neg \theta $ . Then

   $$ \begin{align*}M^{\Lsh \varphi}, w \vDash \neg \theta \iff M^{\Lsh \varphi}, w \nvDash \theta \iff\end{align*} $$
   $$ \begin{align*}M, w \nvDash (\theta)_{\varphi} \iff M, w \vDash \neg (\theta)_{\varphi} \iff M, w \vDash (\neg \theta)_{\varphi}.\end{align*} $$

4. (iv) $\psi = K \theta $ . Then

   $$ \begin{align*}M^{\Lsh \varphi}, w \vDash K \theta \iff M^{\Lsh \varphi}, v \vDash \theta \,\,\mathrm{for \,\, all\,\,} v \in W \iff\end{align*} $$
   $$ \begin{align*}M, v \vDash (\theta)_{\varphi} \,\,\mathrm{for \,\, all\,\,} v \in W \iff M, w \vDash K (\theta)_{\varphi} \iff M, w \vDash (K \theta)_{\varphi}.\end{align*} $$

5. (v) $\psi = B \theta $ . Then

   $$ \begin{align*}M^{\Lsh \varphi}, w \vDash B \theta \iff M^{\Lsh \varphi}, v \vDash \theta \,\,\mathrm{for \,\, all\,\,} v \in \min \leqslant_{w}^{\Lsh \varphi} \iff\end{align*} $$
   $$ \begin{align*}M, v \vDash (\theta)_{\varphi} \,\,\mathrm{for \,\, all\,\,} v \in \min \leqslant_{w}^{\Lsh \varphi} \iff\end{align*} $$
   $$ \begin{align*}M, v \vDash (\theta)_{\varphi} \,\,\mathrm{for \,\, all\,\,} v \in \bigcup_{u \in \min \leqslant_w} f_{\vert\vert\varphi\vert\vert}(u) \iff\end{align*} $$
   $$ \begin{align*}\,\,\mathrm{for \,\, all\,\,} u \in \min \leqslant_w, \left(M, v \vDash (\theta)_{\varphi} \,\,\mathrm{for \,\, all\,\,} v \in f_{\vert\vert\varphi\vert\vert}(u) \right) \iff\end{align*} $$
   $$ \begin{align*}\,\,\mathrm{for \,\, all\,\,} u \in \min \leqslant_w, \left(M, u \vDash \varphi> (\theta)_{\varphi} \right) \iff M, u \vDash B( \varphi> (\theta)_{\varphi}).\\[-31pt] \end{align*} $$

6.2 Ramsey Test

Now we get a Ramsey Test for our counterfactual conditional—the proof is simple.

Proposition 3. For all $\varphi , \psi \in S\mathcal {L}_{DLCS}$ ,

$$ \begin{align*}B_{\varphi}(\psi) \iff B(\varphi> (\psi)_{\varphi}).\end{align*} $$

Proof. Follows from Proposition 2 immediately.

Corollary 1. For any propositional sentence $\psi \in S\mathcal {L}_{DLCS}$ ,

$$ \begin{align*}B_{\varphi}(\psi) \iff B(\varphi> \psi).\end{align*} $$

Proof. Notice that for a propositional sentence $\psi $ without knowledge or belief operators, $(\psi )_{\varphi }= \psi $ .

So counterfactuals are nicely mapped to counterfactual supposition in the way envisaged by Ramsey: one believes that if it were that $\varphi $ , it would be that $\psi $ , precisely when one comes to believe that $\psi $ after revising one’s belief system under the counterfactual supposition of $\varphi $ .

7 Conclusion and further work

We have studied supposition as simulated belief revision in a qualitative setting. We have focused on counterfactual supposition, contrasting it with supposition ‘in the indicative mood’: we have come up with a counterfactual dynamics that, by recapturing KGM, complies with the four principles governing supposition in general, namely Certainty, Minimality, Consistency (in Comparisons), and Possibility. We have then marked the difference between indicative and counterfactual supposition by establishing that the former, but not the latter, preserves certainties. We have introduced a DLCS: a language featuring knowledge and belief operators, a dynamic operator for counterfactual supposition, and a counterfactual conditional. We have then given a semantics for the DLCS language, out of which we have established a Ramsey Test: one comes to believe a counterfactual ‘If it were that $\varphi $ , it would be that $\psi $ ’ when one comes to believe that $\psi $ after dynamically revising one’s beliefs on the counterfactual supposition of $\varphi $ . We have provided a set of axioms for DLCS, for which we have proved soundness.

One area of further investigation is: belief in Q on the counterfactual supposition of P from §4, $Bel_P(Q)$ , is the counterpart of belief in Q on the indicative supposition of P, $Bel^P(Q)$ , whereby one simulates updating one’s beliefs on learning P with certainty. Are there natural counterfactual counterparts of indicative updates on (simulating) learning soft information, as per, e.g., radical-lexicographic and conservative upgrades? How would such ‘soft’ counterfactual updates look like?

Another area for further work is tied to our DLCS: coming up with a completeness proof for the proposed axiomatic system. As explained above, what so far has been standing in our way in the construction of the canonical model is the difficulty of coming up with a divergence relation that is a well-total order and satisfies the required conditions. The counterfactual supposition of P is more complex to deal with than its indicative counterpart because, in addition to involving plausibility, i.e., the notion of what situations an agent finds most plausible given P, it also involves precisely divergence, i.e., the notion of minimal departure required to make P true.

Work we have already carried out has to do with investigating counterfactual supposition as simulated revision of credences or degrees of belief. This will be the topic of our sequel: Dynamics of Counterfactual Supposition II.