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Synthese

, Volume 195, Issue 4, pp 1681–1703 | Cite as

Prioritised ceteris paribus logic for counterfactual reasoning

  • Patrick Girard
  • Marcus A. Triplett
Article

Abstract

The semantics for counterfactuals due to David Lewis has been challenged by appealing to miracles. Miracles may skew a given similarity order in favour of those possible worlds which exhibit them. Lewis responded with a system of priorities that mitigates the significance of miracles when constructing similarity relations. We propose a prioritised ceteris paribus analysis of counterfactuals inspired by Lewis’ system of priorities. By analysing the couterfactuals with a ceteris paribus clause one forces out, in a natural manner, those possible worlds which do not satisfy the requirements of the clause, thus excluding miracles. If no world can satisfy the ceteris paribus clause in its entirety, then prioritisation is triggered to select worlds that maximise agreement on those things which are favoured most.

Keywords

Ceteris paribus Counterfactuals Conditional logic 

Ceteris paribus clauses implicitly qualify many conditional statements that formulate laws of science and economics. A ceteris paribus clause adds to a statement a proviso requiring that other variables or states of affairs not explicitly mentioned in the statement are kept constant, thus ruling out benign defeaters. For instance, Avogadro’s law says that, all other things being equal, the volume and number of moles of gas are proportional. Varying the temperature or pressure could provide situations that violate the plain statement of the law, but the ceteris paribus clause accounts for those. It specifically isolates the interaction between volume and number of moles by keeping everything else equal. In the same spirit, the Nash equilibrium in game theory is a solution concept that picks strategy profiles in which none of the agents could unilaterally (i.e., keeping the actions of others equal) deviate to their own advantage. This bears similarity to the epistemological forcing of Hendricks (2006), which seeks to rule out ‘irrelevant alternatives’ in a way which allows knowledge in spite of the possibility of error. Holliday (2014) develops several interpretations of the epistemic operator K based on the relevant alternatives epistemology; namely, that in order for an agent to have knowledge of a proposition, that agent must eliminate each relevant alternative. One could see relevant worlds as those which keep things equal. When reasoning using Avogadro’s law, the relevant possible worlds are those where the temperature and pressure have not changed. Thus, in order for an agent to have knowledge, that agent must eliminate the alternatives among the worlds which ‘keep things equal.’

We will motivate our discussion by thinking through Kit Fine’s well-known ‘minor-miracles’ argument Fine (1975), a putative counterexample to David Lewis’ theory of counterfactuals that involves Nixon, a nuclear device deployable at the push of a button, and the counterfactual

If Nixon had pushed the button, there would have been a nuclear holocaust.

Fine argues that worlds in which a ‘small miracle’ prevents Nixon from successfully launching a nuclear bomb on the Soviets are more similar to the actual world than worlds in which he succeeds and initiates an atomic war and a nuclear holocaust. Lewis responded with a four-point system of priorities that guides the construction of similarity relations Lewis (1979). The system favours worlds that minimise occurrences of miracles over worlds that maximise perfect match of facts. Worlds in which Nixon successfully launches a nuclear device, Lewis argues, are excluded for failing to prioritise the right kind of facts. Refined counter-examples that involved a more judicious choice of miracles were advanced (Tooley 2003; Wasserman 2006; Elga 2001). The counter-examples are constructed to obey Lewis’ system of priorities while failing to match intuitive evaluations of counterfactuals. Schaffer (2004) defused those counter-examples by stipulating, on top of Lewis’ system of priorities, that facts causally dependent on the antecedent of conditionals be “kept equal” when evaluating similarity of worlds. This strategy of keeping other things equal is the focus of our logical analysis.

We will argue that ceteris paribus logic, suitably adapted to conditionals, provides the logical framework that underlies much of Lewis’ system of priorities. Ceteris paribus logic makes explicit the logical requirement that certain information must remain fixed during the evaluation of counterfactuals. This is implicitly thought to hold, to some degree, when one works with models which have similarity orders or systems of spheres. The conditional logic of Priest (2008) makes just that assumption, but with no syntactic assurance. We show how to equip conditional models with a prioritised ceteris paribus relation that combines a prioritisation order over sets of formulas (following Lewis), and an equivalence relation that keeps “other things” equal (following Schaffer). We introduce ceteris paribus counterfactual modalities that maximise agreement according to that relation. Worlds that keep other things equal are favoured, but failing to find such worlds, we use prioritisation to select those worlds which keep equal as much as possible.

1 Counterfactuals

Here we shall formalise counterfactuals in the style of Lewis (1973). Let \(\mathsf {Prop}\) be a set of propositional variables. We are concerned with models of the form \(\mathcal {M}= (W, \preceq , V)\) such that the following obtain.
  1. 1.

    W is a non-empty set of possible worlds.

     
  2. 2.
    \(\preceq \) is a family \(\{\preceq _w\}_{w\in W}\) of similarity orders, i.e., relations on \(W_w \times W_w\) (with \(W_w \subseteq W\)) such that:
    • \(w \in W_w\),

    • \(\preceq _w\) is reflexive, transitive and total, and

    • \(w \prec _w v\) for all \(v \in W_w\setminus \{w\}\).

     
  3. 3.

    V is a valuation function assigning a subset \(V(p) \subseteq W\) to each propositional variable \(p\in \mathsf {Prop}\).

     
Intuitively, \(W_w\) is the set of worlds which are entertainable from w. Worlds which are not entertainable from w are deemed simply too dissimilar from w to be considered. One says that u is at least as similar to w as v is when \(u \preceq _w v\), and that it is strictly more similar when \(u\prec _w v\).
If \(\mathcal {M}\) satisfies each of the three requirements we call \(\mathcal {M}\) a conditional model. For ease of exposition, we will assume that our conditional models satisfy the limit assumption: for every non-empty \(S \subseteq W\) the set
$$\begin{aligned} \mathsf {Min}^\mathcal {M}_\le (S) = \{v \in S \cap W : v \le u \text { for every } u\in S\cap W\} \end{aligned}$$
(1)
is non-empty. We will suppress the superscript \(\mathcal {M}\) if it is clear from the context which model we are discussing. Of course, we may generalise the semantics for counterfactuals in the usual way so that our results also hold for models which do not satisfy the limit assumption.

Definition 1

(Language Open image in new window ) The language Open image in new window of counterfactuals is generated by the following grammar

We define \(\varphi \wedge \psi := \lnot (\lnot \varphi \vee \lnot \psi )\), \(\varphi \rightarrow \psi := \lnot \varphi \vee \psi \), \(\varphi \equiv \psi := \varphi \rightarrow \psi \, \wedge \, \psi \rightarrow \varphi \), Open image in new window .

Definition 2

(Semantics) Let \(\mathcal {M}= (W, \preceq , V)\) be a well-founded conditional model. Then

Let \(w \in W\). If \(w \in \llbracket \varphi \rrbracket ^\mathcal {M}\) we write \(\mathcal {M}, w \models \varphi \), and if \(w \not \in \llbracket \varphi \rrbracket ^\mathcal {M}\) we write \(\mathcal {M}, w \not \models \varphi \).

2 The Nixon argument

There is a problem dating back to the 1970s (Fine 1975; Bennett 1974; Lee Bowie 1979) surrounding the semantics for counterfactuals proposed by Lewis. The argument goes as follows. Assume, during the Cold War, that President Richard Nixon had access to a device which launches a nuclear missile at the Soviets. All Nixon is required to do is press a button on the device. Consider the counterfactual if Nixon had pushed the button, there would have been a nuclear holocaust. Call it the Nixon counterfactual. It is not so difficult to see that the Nixon counterfactual could be true, or could be imagined to be true. Indeed, one could argue that the Nixon counterfactual ought to be true in any successful theory of counterfactuals (Fine 1975, p. 452) and (Lewis 1979, p. 468) agree that the counterfactual is true, and so do we. Fine, however, used the Nixon counterfactual to argue that the Lewis semantics yields the wrong verdict. This is because “a world with a single miracle but no holocaust is closer to reality than one with a holocaust but no miracle” (Fine 1975, p. 452). In response, Lewis argues that, provided the Nixon situation is modelled using a similarity relation which respects a plausible system of priorities, the counterfactual will emerge true. We will provide a response complimentary to Lewis’ using ceteris paribus counterfactuals, but first let us see how Fine and Lewis model the situation.

2.1 The Fine model

Consider two sets of possible worlds. One set, \(\mathbf u \), consists of those worlds in which Nixon pushes the button, and the button successfully launches the missile. The second, \(\mathbf v \), consists of those worlds in which Nixon pushes the button, but some small occurrence – such as a minor miracle – prevents the button’s correct operation. Certainly those worlds where the button does not launch the missile bear more similarity to the present world than those where it does. This is Fine’s interpretation of Lewis’ semantics. Any world in \(\mathbf u \) has been devastated by nuclear warfare, countless lives have been lost, there is nuclear winter, etc., whereas worlds in \(\mathbf v \) continue on as they would have done.

To illustrate Fine’s interpretation, let psmh be the propositions:
and consider the following model, the Fine model:

An arrow from x to y indicates relative similarity to w, so for example \(u_1 \preceq _w u_2\). Arrows are transitive, and the ‘snake’ arrow between \(\mathbf u \) and \(\mathbf v \) indicates that \(v_i \preceq _w u_j\) for every ij. For each \(u_i \in \mathbf u \), one has \(\mathcal {F}, u_i \models p \wedge s \wedge h\); and for each \(v_i \in \mathbf v \), \(\mathcal {F}, v_i \models p \wedge m\). World w is intended to represent the real world: Nixon did not push any catastrophic anti-Soviet buttons, no nuclear missile was successfully launched at the Soviets, no miracle prevented any such missile, and no nuclear holocaust occurred.

World \(v_1\) is more similar to w than any world in \(\mathbf u \) is, since in any \(\mathbf u \)-world Nixon pushes the button and begins a nuclear holocaust. By (1), \(v_1\) is therefore the minimal p-world. At \(v_1\) the proposition h is false, and so Open image in new window . Therefore, Fine concludes, the Nixon counterfactual is false in Lewis’ semantics.

2.2 The Lewis model

In response, Lewis argues that the proper similarity relation to model the Nixon counterfactual should respect the following system of priorities (Lewis 1979, p. 472):
  1. 1.

    It is of the first importance to avoid big, widespread, diverse violations of law.

     
  2. 2.

    It is of the second importance to maximize the spatio-temporal region throughout which perfect match of particular fact prevails.

     
  3. 3.

    It is of the third importance to avoid even small, localized, simple violations of law.

     
  4. 4.

    It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.

     
Based on this system of priorities, world \(u_1\) emerges more similar to w than \(v_1\) because worlds in \(\mathbf v \) exhibit miracles – a small miracle that prevents Nixon from successfully launching the missile, and in some of them an additional big miracle that erases all traces of Nixon’s ploy. Indeed, Lewis argues that “perfect match of particular fact counts for much more than imperfect match, even if the imperfect match is good enough to give us similarity in respects that matter very much to us” (Lewis 1979, p. 470). The Lewis model, then, looks like this:

In the Lewis model, \(u_1\) is the world most similar to w, and in \(u_1\) the missile successfully launches, there is a nuclear holocaust, and so the Nixon counterfactual is true. Lewis thus responds to Fine by defending a similarity order that favours \(u_1\) over \(v_1\).

We will achieve a resolution similar to Lewis’ without having to defend a model different from Fine’s. After all, as Lewis says: “I do not claim that this pre-eminence of perfect match is intuitively obvious. I do not claim that it is a feature of the similarity relations most likely to guide our explicit judgments. It is not; else the objection we are considering never would have been put forward” (Lewis 1979, p. 470). Instead, we will treat the Nixon counterfactual with an explicit ceteris paribus clause, dispatching with the unintuitive pre-eminence of perfect match in constructing the similarity relation.

Our interpretation of the Nixon counterfactual is much like in preference logic, where formal ceteris paribus reasoning was first applied (van Benthem et al. 2009; Doyle and Wellman 1994; von Wright 1963). Consider the following diagram, which shows a preference of a raincoat to an umbrella as in von Wright (1963), provided wearing boots is kept constant:
Arrows point to more preferred alternatives, and are transitive. Evidently, having an umbrella and boots is preferred to having a raincoat and no boots. The variation of having boots skews the preference. If a ceteris paribus clause is enforced, guaranteeing that in either case boots will be worn or boots will not be worn, then the correct preference is recovered. A similar situation occurs in the logic of counterfactuals. The variation of certain propositions can skew the supplied similarity order. In Fine’s argument, this is done by the variation of physical laws1; i.e., a miracle. If we were to restrict the worlds considered during the evaluation of the counterfactual to those that agree with w on the proposition m, then in \(\mathcal {F}\) the world \(v_1\) would no longer assume the role of minimal p-world. Rather, \(u_1\) would. In world \(u_1\) a nuclear holocaust does occur, as a consequence of which the counterfactual becomes true. This is our approach to the Nixon argument, which we next formalise.

3 Ceteris paribus semantics

We introduce in our language a new conditional operator which generalises the usual one. In particular, it accommodates explicit ceteris paribus clauses. The authors in van Benthem et al. (2009) were the first to define object languages in this way, where they developed a modal logic of ceteris paribus preferences in the sense of von Wright (1963). For now we will take the ordinary conditional operator and embed within it a finite set of formulas \(\varGamma \) understood as containing the other things to be kept equal.2

Definition 3

(Language \(\mathcal {L}_\mathsf {CP}\)) Let \(\varGamma \) be a finite set of formulas. Then the language \(\mathcal {L}_\mathsf {CP}\) is generated by the grammar3
$$\begin{aligned} \varphi \ {:}{:}{=} \ p \ | \ \lnot \varphi \ | \ \varphi \vee \psi \ | \ [\varphi , \varGamma ]\psi . \end{aligned}$$

We understand the modality \([\varphi , \varGamma ]\psi \) as the counterfactual Open image in new window subject to the requirement that the truth of the formulas in \(\varGamma \) does not change. We define \(\varphi \wedge \psi := \lnot (\lnot \varphi \vee \lnot \psi )\), \(\varphi \rightarrow \psi := \lnot \varphi \vee \psi \), and \(\langle \varphi , \varGamma \rangle \psi := \lnot [\varphi , \varGamma ] \lnot \psi \). We call \(\varGamma \) the set of \(\mathsf {CP}\)-conditions or \(\mathsf {CP}\)-set for short. We call the conditional \([\varphi , \varGamma ]\psi \) a \(\mathsf {CP}\)-conditional, or, if the antecedent is false, a \(\mathsf {CP}\)-counterfactual.

For the semantics, we need some additional notation. Given a model \(\mathcal {M}\), a world \(w \in W\) and a set of formulas \(\varGamma \), the \(\varGamma \)-colour of w is the set \(\text {Col}^{\mathcal {M}}_{\varGamma }(w)\) of formulas in \(\varGamma \) true at w and the negation of formulas in \(\varGamma \) false at w:
For a finite set of formulas \(\varGamma \, = \, \{\gamma _1,\dots ,\gamma _n\}\), let \(\text {Pal}(\varGamma )\, = \, \{\{\pm \gamma _1,\dots ,\pm \gamma _n\} : \pm \gamma _i = \gamma _i \text { or } \pm \gamma _i = \lnot \gamma _i \}\) be the set of all possible subsets of formulas and negated formulas from \(\varGamma \). We call \(\text {Pal}(\varGamma )\) the \(\varGamma \) -palette. For example, if \(\varGamma = \{p, \lnot q\}\) then the corresponding \(\varGamma \)-palette is
$$\begin{aligned} \text {Pal}(\varGamma )= \{\{p, \lnot q\}, \{\lnot p, \lnot q\}, \{p, \lnot \lnot q\}, \{\lnot p, \lnot \lnot q\}\}. \end{aligned}$$

Fact 1

For any set of formulas \(\varGamma \) and world w, there is a unique \(\gamma \in \text {Pal}(\varGamma )\) such that \(\gamma = \text {Col}^{\mathcal {M}}_{\varGamma }(w)\).

Figuratively, given a \(\mathsf {CP}\)-set \(\varGamma \), any world w picks a unique \(\varGamma \)-colour \(\text {Col}^{\mathcal {M}}_{\varGamma }(w)\) from the available \(\varGamma \)-palette \(\text {Pal}(\varGamma )\). \(\text {Col}^{\mathcal {M}}_{\varGamma }(w)\) contains the information that needs to be “kept equal” when working out most similar worlds.

Let \(\mathcal {M}= (W, \preceq , V)\) be a conditional model and let \(w, u, v \in \mathcal {M}\). Let \(\varGamma \subseteq \mathcal {L}_\mathsf {CP}\) be finite.
  • Set \([w]_\varGamma = \{v \in W_w : \text {Col}^{\mathcal {M}}_{\varGamma }(w) = \text {Col}^{\mathcal {M}}_{\varGamma }(v)\}\), the collection of w-entertainable worlds that agree with w on \(\varGamma \).

  • Define \(\preceq ^\varGamma _w \, {:}{=} \, \preceq _w \cap \ ([w]_\varGamma \times [w]_\varGamma )\), the restriction of \(\preceq _w\) to the above worlds.

Thus if \(u, v \in [w]_\varGamma \) then either \(u \preceq ^\varGamma _w v\) or \(v \preceq ^\varGamma _w u\). For the semantics of ceteris paribus conditionals, we adapt the definition of minimal worlds by restricting it to \([w]_\varGamma \):
$$\begin{aligned} \mathsf {Min}^\mathcal {M}_{\preceq ^\varGamma _w}(S) = \{v \in S \cap W \cap [w]_\varGamma : \text { there is no } u \in S \cap W \cap [w]_\varGamma \text { with } u \prec ^\varGamma _w v\}. \end{aligned}$$
(2)

Definition 4

(\(\mathsf {CP}\) Semantics) Let \(\mathcal {M}= (W, \preceq , V)\) be a conditional model. Then

The semantics for the regular connectives are the same as those in Definition 2.

Remark 1

Since \([w]_\emptyset = W_w\), the Lewis counterfactual Open image in new window is recovered with \([\varphi , \emptyset ]\psi \).

3.1 The Fine model II

Consider again the Fine model \(\mathcal {F}\).
As before we have Open image in new window (or equivalently \(\mathcal {F}, w \not \models [p,\emptyset ]h\) as expressed in \(\mathcal {L}_\mathsf {CP}\), following Remark 1). If we take the \(\mathsf {CP}\)-set \(\varGamma = \{m\}\), then we eliminate \(\mathbf v \)-worlds from the zone of w-entertainable worlds \([w]_\varGamma \), and \(u_1\) becomes the world most similar to w. Hence
$$\begin{aligned} \mathcal {F}, w \models [p, \{m\}]h. \end{aligned}$$
(3)
We thus think about the Nixon counterfactual by way of ceteris paribus reasoning. Allowing the truth of arbitrary formulas to vary during the evaluation of a counterfactual can distort the given similarity order, thereby attributing falsity to a sentence which is true. By forcing certain formulas to keep their truth status fixed, one can rule out these cases, as we have just done with the expression (3). But which formulas should be kept equal?

4 Where are \(\mathsf {CP}\)-sets from?

Instead of negotiating a choice between the Fine and the Lewis models, ceteris paribus conditionals make explicit how the truth of a conditional varies with different ceteris paribus commitments. Having created a formalism which accommodates explicit ceteris paribus clauses, one would desire a method for uniformly selecting ceteris paribus sets.

4.1 Equal past

Various aspects of similarity factor in the choice of most similar worlds. More fine-grained analyses require that events prior to the occurence of the antecedent be equal to those in the actual world. “Keeping the past equal” is a strategy that McCall (1984) exploited when proposing a theory of similarity based on physical possibility in a branching universe of possible worlds. To evaluate the truth of the counterfactual:

(1)    If Napoleon had won the battle of Waterloo, he would not have died on St. Helena.

McCall’s instructions are to inspect those worlds:

(i) in which Napoleon wins, and (ii) which branch off from the actual world as close as possible to the moment of defeat on June 18, 1815. If the closest of these worlds, or an asymptotically approaching sequence of closer and closer worlds, each fails to contain Napoleon’s death on St. Helena, then conditional (1) is true. If for every such world there is another, branching off equally close or closer to the moment of defeat, in which Napoleon dies on St. Helena after all, then (1) is false (McCall 1984, p. 467).

To what extent can a ceteris paribus logic formalise this idea of keeping the past equal? Let \(P\varphi \) stand for the temporal modality “\(\varphi \) was true at some time in the past.” We could attempt to construct \(\mathsf {CP}\)-sets that keep past events equal by allowing formulas to vary only if they described present or future states of affairs. For example, the formula \([\varphi , \{Pq : q \in \mathsf {Prop}\}]\psi \) attempts to evaluate the counterfactual Open image in new window subject to the requirement that all statements asserting the truth of a proposition at an earlier point in time cannot change in truth-value. Clearly this is not yet fine-grained enough to keep the past fully equal. Consider the formula \(Pn =\) “Nirvana released the album Nevermind.” The presence of the formula in the proposed \(\mathsf {CP}\)-set \(\varGamma \) preserves the fact that Nevermind was released, but doesn’t preserve all aspects of its release. In reality, Nevermind was released in the year 1991, but a world in which Nirvana released the album in 1992 would not be eliminated by a ceteris paribus clause \(\varGamma \) that now included Pn. We could introduce more refined temporal modalities in \(\mathcal {L}_\mathsf {CP}\) to make sure we keep the past fully equal. For instance, we could follow Carlo Proietti and Gabriel Sandu who proposed a ceteris paribus temporal language with indexed modalities to treat the Fitch (knowability) paradox (Proietti and Sandu 2010) and the problem of future contingents (Proietti 2009). We leave this for another project. As always in ceteris paribus reasoning, there is a trade-off between how much of the other things can be kept equal, and what precisely counts as being equal.

4.2 Universe of discourse and causal independence

For von Wright (1963), ceteris paribus means fixing every proposition which does not occur in the universe of discourse of an expression. More precisely, let \(\mathsf {UD}(\varphi )\) be the set of all propositions occurring in the formula \(\varphi \), defined inductively as follows:
Then the ceteris paribus counterfactual if \(\varphi \) were the case then, ceteris paribus, \(\psi \) would be the case amounts to the expression
$$\begin{aligned}{}[\varphi , \mathsf {Prop}\setminus (\mathsf {UD}(\varphi ) \cup \mathsf {UD}(\psi ))]\psi . \end{aligned}$$
(4)
Now all propositions not occurring in the universe of discourse of the counterfactual antecedent or consequent are fixed.4
In the Fine model, \(\mathsf {Prop}\setminus (\mathsf {UD}(p) \cup \mathsf {UD}(h)) = \{m,s\}\). The \(\mathsf {CP}\)-set \(\{m,s\}\), however, makes \([w]_{\{m,s\}} = \emptyset \): none of the worlds in \(\mathbf u \) or \(\mathbf v \) agree with w on the propositions m and s. This makes every conditional with \(\mathsf {CP}\)-set \(\{m,s\}\) vacuously true. In particular
$$\begin{aligned} \mathcal {F}, w \models [p, \{m, s\}]\bot . \end{aligned}$$
(5)
For von Wright’s strategy to function in a meaningful fashion, every formula occurring in \(\varGamma \) must be independent of the counterfactual antecedent. Indeed, von Wright assumes that the truth-value of propositional variables can be altered without affecting the others. But in the Fine model, m and s are dependent on p, so we can’t change the truth-value of p without affecting the truth-values of m and s, which is why the counterfactual \([p, \{m,s\}]h\) is trivially true. This is a more general problem with the ceteris paribus interpretation of counterfactuals. Lewis observed this as well when considering the counterfactual ‘if Kangaroos had no tails, they would topple over’:

We might think it best to confine our attention to worlds where kangaroos have no tails and everything else is as it actually is; but there are no such worlds. Are we to suppose that kangaroos have no tails but that their tracks in the sand are as they actually are? Then we shall have to suppose that these tracks are produced in a way quite different from the actual way. (Lewis 1973, p. 9)

We are obstructed in a similar way in our analysis of the Nixon counterfactual. Fixing the truth of even a small number of propositions while varying the truth of a formula \(\varphi \) has a cascading effect on other propositions; this may ultimately result in inconsistency. Lewis continues:

Are we to suppose that kangaroos have no tails but that their genetic makeup is as it actually is? Then we shall have to suppose that genes control growth in a way quite different from the actual way (or else that there is something, unlike anything there actually is, that removes the tails). And so it goes; respects of similarity and difference trade off. If we try too hard for exact similarity to the actual world in one respect, we will get excessive differences in some other respect. (Lewis 1973, p. 9)

These are indeed the kinds of difficulties that our ceteris paribus interpretation faces. A modeller is left without knowing which propositions can vary while keeping one particular proposition constant. One approach is to imbue a conditional model with a causal relation \(\leadsto \) over formulas, so that \(\varphi \leadsto \psi \) means “\(\psi \) is causally dependent on \(\varphi \).” This is done by Schaffer (2004) to diffuse indeterministic variants of the Nixon scenario (Tooley 2003; Wasserman 2006; Elga 2001):

Only match among those facts causally independent of the antecedent should count towards similarity. Not all matching is equal. After all, if outcome o causally depends on p or \(\lnot p\), then o should be expected to vary with p or \(\lnot p\) —its varying should hardly count for dissimilarity. (Schaffer 2004, p. 305)

We could attempt to construct \(\mathsf {CP}\)-sets by allowing propositions to vary only if they describe states that are causally independent from the states described by the antecedent of the conditional. Such counterfactuals take the form
$$\begin{aligned}{}[\varphi , \mathsf {Prop}\setminus \{p \in \mathsf {Prop}: \varphi \leadsto p\}]\psi . \end{aligned}$$
This can be generalised to allow the truth of whole formulas to vary so long as they satisfy this condition of causal independence. (After all, one could desire that the truth-value of another counterfactual is kept constant in the truth-evaluation of the counterfactual one is firstly concerned with.) Schaffer’s proposal is in fact very much in line with the prioritised ceteris paribus logic that we will propose.

These preceding approaches to constructing ceteris paribus sets can be understood as an attempt to maximise the amount of information that really can be forced to stay constant. In Von Wright’s approach the construction of \(\varGamma \) is syntactic, with an underlying assumption that atomic propositions can vary unilaterally, whereas in Schaffer’s approach, it’s up to the modeller to enumerate the causal relationships between propositions. We will equip our models with a relational structure over formulas that bears similarity to Lewis’ system of priorities, but that is enforced syntactically with the framework of ceteris paribus logic.

5 Ceteris paribus prioritisation

What ought to be kept equal when we can’t keep everything else equal? Our answer to this question is based on a simple intuition: keep equal what is most valuable, as much as possible. By ‘what is most valuable,’ we mean something akin to Lewis’ system of priorities. Of all the things that we may be able to keep equal when considering possible worlds, some are more important than others. We prefer to avoid violating laws over avoiding the violation of approximate match of particular fact. And by ‘as much as possible,’ we mean something akin to Schaffer’s restrictions to causally independent facts. It might be impossible to keep everything equal, in which case the modality should maximise what can be kept equal. This is what we will do in the rest of the paper. Our strategy is to introduce an order over sets of formulas which serves as a guide in selecting possible worlds that keep equal those sets of formulas that are valued most. If \(\varGamma \) identifies all the other things we wish to keep equal, then prioritisation will help us select the things to keep equal if we can’t keep them all equal.

5.1 Prioritised ceteris paribus semantics

Let \(\unlhd \) be a preorder (i.e., a reflexive and transitive relation) over subsets of formulas. We write \(\varDelta \bowtie \varGamma \) if \(\varDelta \, \unlhd \, \varGamma \) and \(\varGamma \, \unlhd \, \varDelta \), and we write \(\varDelta \, \lhd \, \varGamma \) if \(\varDelta \, \unlhd \, \varGamma \) and \(\varGamma \, \ntrianglelefteq \, \varDelta \). We say that \(\unlhd \) is a ceteris paribus prioritisation if it satisfies the increment and equivalence conditions:
  • (Increment) For any \(\varGamma , \varDelta \subseteq \mathcal {L}_{\mathsf {CP}}\), if \(\varGamma \subseteq \varDelta \) then \(\varDelta \, \unlhd \, \varGamma \).

  • (Equivalence) If \(\varphi \equiv \psi \), then \(\varGamma \bowtie \varGamma '\), where \(\varGamma '\) is obtained from \(\varGamma \) by replacing any occurence of \(\varphi \) with \(\psi \).

Informally, the increment condition says that “the more you can keep equal, the better,” and the equivalence condition that “\(\mathsf {CP}\)-sets that differ only by logically equivalent formulas are interchangeable.”
We introduce a prioritised ceteris paribus conditional \([\varphi , \varGamma , \unlhd ]\psi \) into the language. The intended interpretation of this modality is that \(\psi \) holds in the minimal \(\varphi \)-worlds, ceteris paribus, with respect to the order \(\unlhd \). The agreement set between w and v over \(\varGamma \) is defined as the intersection of their \(\varGamma \)-colours:
Let \(\unlhd \) be a ceteris paribus prioritisation, define the relation \(\unlhd _w^\varGamma \) on \(W_w\) by setting

Thus, \(u\unlhd _w^\varGamma v\) if either the agreement set between u and w is strictly prioritised over the agreement set between v and w (i.e., the formulas in \(\varGamma \) that u keeps equal are more important than those that v keeps equal), or the agreement set between u and w is only weakly prioritised over that between v and w, and u is at least as similar to w as v is. In other words, if \(A_\varGamma ^\mathcal {M}(u,w)\) is not strictly prioritised over \(A_\varGamma ^\mathcal {M}(v,w)\) but is comparable,5 then we use the similarity order \(\preceq _w\).

Remark 2

Since we assume that \(\unlhd \) is a preorder,6 we cannot guarantee that \(\unlhd _w^\varGamma \) is a total order, even though \(\preceq _w\) is. But counterfactuals are just as meaningfully defined over non-total orders.

Fact 2

\(A_\varGamma (w,u) \unlhd A_\varGamma (w,v)\) if and only if \(A_{\varGamma '}(w,u) \unlhd A_{\varGamma '}(w,v)\), where \(\varGamma '\) is obtained from \(\varGamma \) by replacing any occurence of \(\varphi \) with \(\lnot \varphi \).

Proof

Since \(\varphi \in \varGamma \), by the definition of agreement sets,
Hence by the equivalence condition,
Thus if \(A_\varGamma (w, u) \unlhd A_\varGamma (w, v)\), then \(A_{\varGamma '}(w, u) \unlhd A_{\varGamma '}(w, v)\). The converse is similar. \(\square \)

Next we define a prioritised conditional model, \(\mathcal {M} = (W, \preceq , \unlhd , V)\), to simply be a conditional model augmented with a ceteris paribus prioritisation \(\unlhd \).

Definition 5

(\(\mathsf {PCP}\) Semantics) Let \(\mathcal {M}= (W, \preceq , \unlhd , V)\) be a prioritised conditional model. Then

We leave it to the reader to check that prioritised ceteris paribus semantics is a generalisation of the Lewis semantics, as expressed in the following fact:

Fact 3

Let \(\unlhd _\mathsf {UN}\) be the universal relation over the set of all formulas. The Lewis counterfactual is recovered with \([\varphi , \emptyset , \unlhd _\mathsf {UN}]\psi \).

5.2 Two instances of ceteris paribus prioritisation

In Ref. Girard and Triplett (2015) two special cases of ceteris prioritisation are investigated, called respectively ‘naïve counting’ and ‘maximal supersets.’ We show how to recover each as a special case of ceteris paribus prioritisation and look at their treatment of the Nixon counterfactual over the \(\mathsf {CP}\)-set \(\{m,s\}\).

5.2.1 Naïve counting

The idea behind naïve counting is simple; a world is prioritised over another one if the number of formulas in its agreement set is greater:
$$\begin{aligned} \varDelta \, \unlhd \, \varGamma \text { iff }|\varGamma | \le |\varDelta |. \end{aligned}$$
Thus we may write \([\varphi , \varGamma , \le ]\psi \) for the prioritised ceteris paribus modality that tracks cardinal agreement, based on naïve counting.7

Fact 4

Let \(\mathcal {M}= (W, \preceq , \le , V)\) be a prioritised conditional model with priority order \(\le \), and let \(w \in W\). Then the following are true, where \(\pm \alpha \) is a shorthand which uniformly stands for either \(\alpha \) or \(\lnot \alpha \):
  1. 1.

    \(\mathcal {M}, w \models (\pm \alpha \wedge \langle \varphi , \varGamma , \le \rangle (\pm \alpha \wedge \psi )) \rightarrow \langle \varphi , \varGamma \cup \{\alpha \}, \le \rangle \psi \)

     
  2. 2.

    \(\mathcal {M}, w \models \langle \varphi , \varGamma \rangle \psi \Rightarrow \mathcal {M}, w \models \langle \varphi , \varGamma , \le \rangle \psi \)

     

Remark 3

Fact 4.1 does not hold for arbitrary orders \(\unlhd \). Consider the following model \(\mathcal {M}\) with a ceteris paribus prioritisation \(\unlhd \) such that \(\{q,r\} \lhd \{p\} \lhd \{q\} \lhd \emptyset \):

Let \(\varGamma = \{p,q\}\). Since no s-world agrees with w on \(\varGamma \), but \(\{p\} \lhd \{q\}\), one has \(\mathsf {Min}_{\unlhd ^\varGamma _w}(\llbracket s \rrbracket ^\mathcal {M}) \, = \, \{v\}\). Hence \(\mathcal {M}, w\models \langle s, \varGamma , \unlhd \rangle (t \wedge r)\). But \(\mathcal {M}, w\not \models \langle s, \varGamma \cup \{r\}, \unlhd \rangle t\), as \(\mathsf {Min}_{\unlhd ^{\varGamma \cup \{r\}}_w}(\llbracket s \rrbracket ^\mathcal {M}) \, = \, \{u\}\).

To see how naïve counting operates in the Fine model, consider the following set-up:

Firstly, \(A^\mathcal {F}_\varGamma (w, u_1) \bowtie A^\mathcal {F}_\varGamma (u, v_1)\) because \(|A^\mathcal {F}_\varGamma (w, u_1)| = |A^\mathcal {F}_\varGamma (u, v_1)| = 1\). Moreover \(v_1 \prec _w u_1\), and so one has both \(\mathcal {F}, w\models [p, \varGamma , \le ]\lnot h\) and \(\mathcal {F}, w \not \models [p, \varGamma , \le ] h\).

Thus, naïve counting substantiates a reading of the Nixon counterfactual that interprets it as being false.

5.2.2 Maximal supersets

An approach to counterfactuals familiar to the AI community (Katsuno and Mendelzon 1991; Dalal 1988; del Cerro and Herzig 1994, 1996) makes use of a selection function which chooses the ‘closest’ world according to maximal sets of propositional variables. More specifically, each world w satisfies some set \(\mathbf P _w \subseteq \mathsf {Prop}\) of propositional variables, and a world u is a world closest to w if there is no v with \(\mathbf P _u \subset \mathbf P _v \subseteq \mathbf P _w\). Taking this as a kind of ceteris paribus formalism we obtain an instance of our prioritised ceteris paribus counterfactuals.

In this case, \(\varGamma \) is maximally preserved in the sense that worlds which preserve the same propositions as another, and furthermore preserve additional propositions from \(\varGamma \), are deemed to approximate \(\varGamma \) more closely; while worlds uv with neither \(A^\mathcal {M}_\varGamma (u, w) \subseteq A^\mathcal {M}_\varGamma (v, w)\) nor \(A^\mathcal {M}_\varGamma (v, w) \subseteq A^\mathcal {M}_\varGamma (u, w)\) are considered incomparable. We can thus use a subset relation to define a prioritisation rule:
$$\begin{aligned} \varGamma \, \unlhd \, \varDelta \quad \text { iff }\varDelta \subseteq \varGamma . \end{aligned}$$
Write \([\varphi , \varGamma , \subseteq ]\psi \) for the prioritised ceteris paribus modality with prioritisation given by set containment.

Fact 5

Let \(\mathcal {M}= (W, \preceq , \subseteq , V)\) be a prioritised conditional model with priority order \(\subseteq \). Let \(w \in W\). Then the following are true.
  1. 1.

    \(\mathcal {M}, w \models (\pm \alpha \wedge \langle \varphi , \varGamma , \subseteq \rangle (\pm \alpha \wedge \psi )) \rightarrow \langle \varphi , \varGamma \cup \{\alpha \}, \subseteq \rangle \psi \)

     
  2. 2.

    If \(\mathcal {M}, w \models \langle \varphi , \varGamma \rangle \psi ,\) then \(\mathcal {M}, w \models \langle \varphi , \varGamma , \subseteq \rangle \psi \).

     
  3. 3.

    If \(\mathcal {M}, w \models \langle \varphi , \varGamma , \subseteq \rangle \psi ,\) then \(\mathcal {M}, w \models \langle \varphi , \varGamma , \le \rangle \psi \).

     
Returning to the Fine model with \(\mathsf {CP}\)-set \(\{m,s\}\), consider the following with priority given by \(\subseteq \):
Neither \(A^\mathcal {F}_\varGamma (w, u_1) \subseteq A^\mathcal {F}_\varGamma (w, v_1)\), nor \(A^\mathcal {F}_\varGamma (w, v_1) \subseteq A^\mathcal {F}_\varGamma (w, u_1)\). So neither \(A^\mathcal {F}_\varGamma (w, u_1) \unlhd A^\mathcal {F}_\varGamma (w, v_1)\), nor \(A^\mathcal {F}_\varGamma (w, v_1) \unlhd A^\mathcal {F}_\varGamma (w, u_1)\). Therefore both \(u_1\) and \(v_1\) are elements of \(\mathsf {Min}_{\unlhd ^\varGamma _w}(\llbracket \varphi \rrbracket ^\mathcal {F})\), and we obtain an agnostic analysis of the Nixon counterfactual:
$$\begin{aligned} \mathcal {F}, w \not \models [p, \{m,s\}, \subseteq ]h \text { and } \mathcal {F}, w\not \models [p, \{m,s\}, \subseteq ]\lnot h. \end{aligned}$$
The maximal superset priority rule thus provides a more conservative ceteris paribus counterfactual that stays undecided on the truth of the Nixon counterfactual.

5.3 The Fine model III

A question remains regarding the Nixon counterfactual and ceteris paribus prioritisation with the von Wright \(\mathsf {CP}\)-set \(\{m, s\}\): what kind of prioritisation would substantiate Lewis’ reading of the Nixon counterfactual? To answer this question, we need to look for a prioritisation along the lines of Lewis’ system of priorities, and one that makes the counterfactual true. Here’s a proposal, with Open image in new window specifying the required prioritisation:
By Fact 2, Open image in new window implies that Open image in new window . HenceThus we have that Open image in new window . Hence Open image in new window , and so we recover the truth of the Nixon counterfactual.
Let us recapitulate. Working with the \(\mathsf {CP}\)-set \(\varGamma = \{m,s\}\) and following von Wright’s strategy of keeping the complement of the universe of discourse of the Nixon counterfactual equal, a ceteris paribus semantics that requires complete agreement on the ceteris paribus set trivialises the counterfactual; a prioritised ceteris paribus semantics with a priority rule determined by naïve counting makes the Nixon counterfactual false; prioritised ceteris paribus semantics with a priority rule determined by maximal supersets stays agnostic on the counterfactual; and the order Open image in new window recovers Lewis’ interpretation with a prioritisation that disfavours occurrences of small miracles. This information is summarised in the following table, where the abbreviations \(\mathsf {CP}, \mathsf {NC},\) and \(\mathsf {MS}\) respectively stand for \(\mathsf {CP}\) semantics, \(\mathsf {PCP}\) semantics with priority determined by naïve counting, and \(\mathsf {PCP}\) semantics with priority determined by a maximal superset rule.

Counterfactual

Clause

\(\mathsf {CP}\)

\(\mathsf {NC}\)

\(\mathsf {MS}\)

Open image in new window

Open image in new window

\(\{m,s\}\)

\(\mathsf {true}\)

\(\mathsf {false}\)

\(\mathsf {false}\)

\(\mathsf {true}\)

Open image in new window

\(\{m,s\}\)

\(\mathsf {true}\)

\(\mathsf {true}\)

\(\mathsf {false}\)

\(\mathsf {false}\)

The choice of prioritisation orders impacts on the truth-value of counterfactuals, just like the choice of similarity orders and ceteris paribus sets do. The preceding table shows that by including ceteris paribus prioritisation in models, we allow for a more fine-grained analysis of counterfactuals from within the object language of conditional logic.

6 Completeness

In this final section we establish the completeness of ceteris paribus logic with and without prioritisation. In what follows, we provide reductions from languages that have ceteris paribus modalities to languages that only have the usual counterfactual modality Open image in new window , or the comparative possibility operator \(\preceq \). In each case, one may take a common axiomatisation for the underlying logic (e.g., Lewis’ VC (Lewis 1973)) and obtain an axiomatisation for the ceteris paribus logic via the reduction scheme.

6.1 Ceteris paribus completeness

We first recast the original definition of \(\mathcal {L}_\mathsf {CP}\) in a way that avoids non-well-founded ceteris paribus sets (Seligman and Girard 2011).

Definition 6

For each ordinal \(\alpha \) let \(\mathcal {L}_\alpha \) be given by
$$\begin{aligned} \varphi \ {:}{:} {=} \ p \ | \ \bot \ | \ \lnot \varphi \ | \ \varphi \vee \psi \ | \ [\varphi ,\varGamma ] \psi \end{aligned}$$
where \(\varGamma \subseteq \mathcal {L}_\beta \) is finite and \(\beta < \alpha \). \(\mathcal {L}_{\mathsf {CP}}\) is then defined to be \(\bigcup _\alpha \mathcal {L}_\alpha \).

Lemma 1

The modal operator \([\varphi , \varGamma ]\psi \) is definable in Open image in new window .

Proof

For a set of formulas \(\gamma \), let \(\overline{\gamma }= \bigwedge \gamma _i\) for \(\gamma _i \in \gamma \). We show that
is valid over the class of conditional models. For ease of notation we write Open image in new window rather than \([\varphi , \emptyset ]\psi \), so the right side of the equivalence (6) is really a formula of \(\mathcal {L}_\mathsf {CP}\). Because of this equivalence, however, equation (6) will be our reduction from \(\mathcal {L}_\mathsf {CP}\)-formulas to Open image in new window -formulas with the appropriate substitution.
Take an arbitrary model \(\mathcal {M}\) and an arbitrary world \(w \in W\). We must show thatBy Fact 1, there is a unique \(\lambda \in \text {Pal}(\varGamma )\) such that \(\lambda = \text {Col}^{\mathcal {M}}_{\varGamma }(w)\), so it is enough to show thatfor which it is sufficient to show
$$\begin{aligned} \mathsf {Min}_{\preceq ^\varGamma _w}(\llbracket \varphi \rrbracket )=\mathsf {Min}_{\preceq _w}(\llbracket \varphi \wedge \overline{\lambda }\rrbracket ). \end{aligned}$$
We omit the \(\mathcal {M}\) superscript in \(\llbracket \cdot \rrbracket ^\mathcal {M}\) for brevity.
\(\Rightarrow :\) Take \(x \in \mathsf {Min}_{\preceq ^\varGamma _w}(\llbracket \varphi \rrbracket )\). We show that x is a \(\preceq _w\)-minimal \(\varphi \wedge \overline{\lambda }\)-world. From the definition of \(\mathsf {Min}_{\preceq ^\varGamma _w}(\llbracket \varphi \rrbracket )\) (cf. equation (2)), we have
$$\begin{aligned}&\mathcal {M}, x \models \varphi , \end{aligned}$$
(7)
$$\begin{aligned}&\text {Col}^{\mathcal {M}}_{\varGamma }(w) = \text {Col}^{\mathcal {M}}_{\varGamma }(x), \text { and} \end{aligned}$$
(8)
$$\begin{aligned}&\text {there is no } y \in \llbracket \varphi \rrbracket \text { such that } y \prec _w^\varGamma x. \end{aligned}$$
(9)
From (8) and Fact 1 one has \(\mathcal {M}, x \models \overline{\lambda }.\) This establishes that x is a \(\varphi \wedge \overline{\lambda }\)-world, so it remains to show that it is \(\preceq _w\)-minimal among the members of \(\llbracket \varphi \wedge \overline{\lambda }\rrbracket \). To this end, let \(y \in \llbracket \varphi \wedge \overline{\lambda }\rrbracket \). If \(y \prec _w x\) then in particular \(y \in W_w\). But since \(\mathcal {M}, y \models \overline{\lambda }\) one has \(y \in [w]_\varGamma \), in which case \(y \prec ^\varGamma _w x\), which is impossible by (9). Thus \(x \in \mathsf {Min}_{\preceq _w}(\llbracket \varphi \wedge \overline{\lambda }\rrbracket )\) as desired.
\(\Leftarrow :\) Let \(x \in \mathsf {Min}_{\preceq _w}(\llbracket \varphi \wedge \overline{\lambda }\rrbracket )\). Then
$$\begin{aligned}&\mathcal {M}, x\models \varphi \wedge \overline{\lambda }, \text { and} \end{aligned}$$
(10)
$$\begin{aligned}&\text {there is no } y \in \llbracket \varphi \wedge \overline{\lambda }\rrbracket \text { such that } y \prec _w x. \end{aligned}$$
(11)
Thus \(x \in \llbracket \varphi \rrbracket \), and we must show that x is \(\preceq ^\varGamma _w\)-minimal. Let \(y \in \llbracket \varphi \rrbracket \). On the one hand if \(\text {Col}^{\mathcal {M}}_{\varGamma }(y) \ne \text {Col}^{\mathcal {M}}_{\varGamma }(x)\), then \(y \not \in [w]_\varGamma \) since it disagrees with w on the colour of \(\varGamma \). Thus \(y\not \prec _w^\varGamma x\). On the other hand, if \(\text {Col}^{\mathcal {M}}_{\varGamma }(y) = \text {Col}^{\mathcal {M}}_{\varGamma }(x)\) then \(\mathcal {M}, y\models \varphi \wedge \overline{\lambda }\). It then follows from (11) that \(y\not \prec _w x\), and so in particular \(y\not \prec _w^\varGamma x\). Hence there is no \(y \in \llbracket \varphi \rrbracket \) such that \(y\prec _w^\varGamma x\). Therefore \(x \in \mathsf {Min}_{\preceq ^\varGamma _w}(\llbracket \varphi \rrbracket )\), which completes the proof. \(\square \)

By a conditional frame we mean a pair \(F = (W, \preceq )\), such that (FV) is a conditional model for any valuation function V. Let \(\mathbf C \) be the class of conditional frames. Using the notation from Blackburn et al. (2001), we write \(\varLambda ^L_\mathbf C \) for the set of L-formulas valid over \(\mathbf C \). Additionally, notice that if Open image in new window , then the right hand sides of the equivalences established above are in Open image in new window . This allows us to recursively apply the translation (6) to an \(\mathcal {L}_\mathsf {CP}\) formula from the inside-out, with the resulting formula belonging to Open image in new window . Thus Lemma 1 provides the necessary reduction, and we obtain the following theorem.

Theorem 1

The logic \(\varLambda ^{\mathcal {L}_\mathsf {CP}}_\mathbf C \) is complete.

6.2 Prioritised ceteris paribus completeness

The proof works by translating formulas of prioritised ceteris paribus logic into formulas of the comparative possibility language, in the style of Lewis (1973, § 2.5), and axiomatising the equivalent logic. This permits a clearer reduction of prioritised ceteris paribus modalities to basic comparative possibility operators, albeit with a translation exponential in the size of \(\varGamma \).

Definition 7

For each ordinal \(\alpha \) let \(\mathcal {L}_\alpha \) be given by
$$\begin{aligned} \varphi \ {:}{:} {=} \ p \ | \ \bot \ | \ \lnot \varphi \ | \ \varphi \vee \psi \ | \ [\varphi ,\varGamma , \unlhd ] \psi \end{aligned}$$
where \(\varGamma \subseteq \mathcal {L}_\beta \) is finite, \(\unlhd \) is a ceteris paribus prioritisation over \(\mathcal {L}_\beta \), and \(\beta < \alpha \). \(\mathcal {L}_{\mathsf {PCP}}\) is then defined to be \(\bigcup _\alpha \mathcal {L}_\alpha \).
This ensures the sets \(\varGamma \) are well-defined. One can define a language \(\mathcal {L}\) of comparative possibility in a similar style, though we will only specify the following grammarWe further set \(\varphi \prec \psi := \lnot (\psi \preceq \varphi )\), \(\varphi \lhd ^\varGamma \psi := \lnot (\psi \unlhd ^\varGamma \varphi )\), \(\Diamond \varphi := \varphi \prec \bot \), and \(\Box \varphi := \lnot \Diamond \lnot \varphi \).

Definition 8

(Semantics) Let \(\mathcal {M}= (W, \preceq , \unlhd , V)\) be a prioritised conditional model. Then
$$\begin{aligned} \llbracket p \rrbracket ^\mathcal {M}&= V(p)\\ \llbracket \bot \rrbracket ^\mathcal {M}&= \emptyset \\ \llbracket \lnot \varphi \rrbracket ^\mathcal {M}&= W\setminus \llbracket \varphi \rrbracket ^\mathcal {M}\\ \llbracket \varphi \vee \psi \rrbracket ^\mathcal {M}&= \llbracket \varphi \rrbracket ^\mathcal {M}\cup \llbracket \psi \rrbracket ^\mathcal {M}\\ \llbracket \varphi \preceq \psi \rrbracket ^\mathcal {M}&= \{w \in W :\forall u \in W_w \ \exists v \in W_w \text { such that if }u \in \llbracket \psi \rrbracket ^\mathcal {M},\\&\quad \text { then } v \in \llbracket \varphi \rrbracket ^\mathcal {M}\text { and } v \preceq _w u\} \\ \llbracket \varphi \, \unlhd ^\varGamma \, \psi \rrbracket ^\mathcal {M}&= \{w \in W :\forall u \in W_w \ \exists v \in W_w \text { such that if } u \in \llbracket \psi \rrbracket ^\mathcal {M},\\&\quad \text { then } v \in \llbracket \varphi \rrbracket ^\mathcal {M}\text { and } v \, \unlhd _w^\varGamma \, u\} \\ \llbracket \varphi \, \lhd ^\varGamma \, \psi \rrbracket ^\mathcal {M}&= \{w \in W :\exists u \in W_w \ \forall v \in W_w \text { such that } u \in \llbracket \varphi \rrbracket ^\mathcal {M},\\&\quad \text { and if } v \, \unlhd _w^\varGamma \, u \text { then } v \not \in \llbracket \psi \rrbracket ^\mathcal {M}\}. \end{aligned}$$

Lemma 2

The modal operator \([\varphi , \varGamma , \unlhd ]\psi \) is definable in \(\mathcal {L}\).

Proof

The proof follows Lewis (1973, p. 53). We show that
$$\begin{aligned} \mathcal {M}, w \models [\varphi , \varGamma , \unlhd ]\psi \text { if and only if } \mathcal {M}, w \models \Diamond \varphi \rightarrow (\varphi \wedge \psi ) \lhd ^\varGamma (\varphi \wedge \lnot \psi ), \end{aligned}$$
for every prioritised conditional model \(\mathcal {M}\).

\(\Rightarrow :\) Let \(\mathcal {M}= (W, \preceq , \unlhd , V)\) be a prioritised conditional model such that \(\mathcal {M}, w \models [\varphi , \varGamma , \unlhd ]\psi \), and suppose that \(\mathcal {M}, w \models \Diamond \varphi \). Then there is some \(\varphi \)-world, and the limit assumption guarantees that there is in fact a \(\unlhd ^\varGamma _w\)-minimal such \(\varphi \)-world u. Hence \(\mathcal {M}, v \models \psi \) by the assumption. Let \(v \in W_w\) such that \(v \unlhd _w^\varGamma u\). If \(v \in \llbracket \varphi \rrbracket \), then it must be that \(v \in \mathsf {Min}_{\unlhd ^\varGamma _w}(\llbracket \varphi \rrbracket )\) since u is \(\unlhd ^\varGamma _w\)-minimal. So \(v \in \llbracket \psi \rrbracket \) and hence \(v\not \in \llbracket \varphi \wedge \lnot \psi \rrbracket \).

\(\Leftarrow :\) By contrapositive. Assume \(\mathcal {M}, w\not \models [\varphi , \varGamma , \unlhd ]\psi \). Then by the semantic definition, there is a \(v \in \mathsf {Min}_{\unlhd ^\varGamma _w}(\llbracket \varphi \rrbracket )\) such that \(v \not \in \llbracket \psi \rrbracket \). So \(\mathcal {M}, w\models \Diamond \varphi \), and for every \(x \in W_w\) there is \(y\in W_w\) (namely v) such that if \(x \in \llbracket \varphi \wedge \psi \rrbracket \), then \(y\in \llbracket \varphi \wedge \lnot \psi \rrbracket \) with \(y\unlhd _w ^\varGamma x\). Hence \(\mathcal {M}, w \models (\varphi \wedge \lnot \psi ) \unlhd ^\varGamma (\varphi \wedge \psi )\), and so \(\mathcal {M}, w\not \models (\varphi \wedge \psi ) \lhd ^\varGamma (\varphi \wedge \lnot \psi )\). \(\square \)

Using Lemma 2 we can reduce formulas from \(\mathcal {L}_{\mathsf {PCP}}\) to the language \(\mathcal {L}\), and in what follows we provide a reduction from \(\mathcal {L}\) to the language of comparative possibility without ceteris paribus qualifiers. To this end, denote by \(\mathcal {L}^-\) the \(\mathcal {L}\)-fragment generated by the grammar
$$\begin{aligned} \varphi \ {:}{:}{=} \ p \ | \ \bot \ | \ \lnot \varphi \ | \ \varphi \vee \psi \ | \ \varphi \preceq \psi . \end{aligned}$$

Remark 4

For any \(w, v \in W\) one has \(\mathcal {M}, w\models \overline{A^\mathcal {M}_{\varGamma }(w, v)}\) and \(\mathcal {M}, v\models \overline{A^\mathcal {M}_{\varGamma }(w, v)}\).

Lemma 3

The modal operator \(\unlhd ^\varGamma \) of \(\mathcal {L}\) is expressible in \(\mathcal {L}^-\).

Proof

We show that for every prioritised conditional model \(\mathcal {M}\) we have
$$\begin{aligned}&\mathcal {M}, w \models \varphi \, \unlhd ^\varGamma \, \psi \text { iff } \mathcal {M}, w \\&\quad \models \bigwedge \limits _{\gamma \,\in \, \text {Pal}(\varGamma )} ( \overline{\gamma } \rightarrow \bigwedge \limits _{\lambda \subseteq \gamma }[\bigwedge \limits _{\gamma \supseteq \lambda ' \lhd \lambda } \lnot \Diamond (\varphi \wedge \overline{\lambda '}) \rightarrow \bigvee \limits _{\gamma \supseteq \lambda ''\unlhd \lambda }(\varphi \wedge \overline{\lambda ''}) \preceq (\psi \wedge \overline{\lambda })]). \end{aligned}$$
There is only one \(\gamma \in \text {Pal}(\varGamma )\) such that \(\mathcal {M}, w \models \overline{\gamma }\), and \(\gamma = \text {Col}^{\mathcal {M}}_{\varGamma }(w)\) by Fact 1. Hence it is enough to show that \( \mathcal {M}, w \models \varphi \, \unlhd ^\varGamma \, \psi \) if and only if
$$\begin{aligned} \mathcal {M}, w \models \bigwedge \limits _{\lambda \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(w)}[\bigwedge \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda ' \lhd \lambda } \lnot \Diamond (\varphi \wedge \overline{\lambda '}) \rightarrow \bigvee \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w)\supseteq \lambda ''\unlhd \lambda }(\varphi \wedge \overline{\lambda ''}) \preceq (\psi \wedge \overline{\lambda })]). \end{aligned}$$
\(\Rightarrow :\) Suppose \(\mathcal {M}, w \models \varphi \unlhd ^\varGamma \psi \) and take \(\lambda \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(w)\) such that:
$$\begin{aligned} \mathcal {M}, w \models \bigwedge \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda ' \lhd \lambda } \lnot \Diamond (\varphi \wedge \overline{\lambda '}). \end{aligned}$$
(12)
Take \(v \in W_w\) arbitrary such that \(\mathcal {M}, v \models \psi \wedge \overline{\lambda }\). Notice that \(\lambda \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(w)\) and \(\mathcal {M}, v\models \overline{\lambda }\) imply that \(\lambda \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(v) \cap \text {Col}^{\mathcal {M}}_{\varGamma }(w) = A_\varGamma ^\mathcal {M}(v,w) \). So \(A_\varGamma ^\mathcal {M}(v,w) \unlhd \lambda \) by the increment condition. By hypothesis, since \(\mathcal {M}, v\models \psi \), there is \(u \in W_w\) such that \(u \unlhd ^\varGamma _w v\) and \(\mathcal {M}, u \models \varphi \). Now \(u \unlhd ^\varGamma _w v\) implies that
Suppose \(A_\varGamma ^\mathcal {M}(u,w) \lhd A_\varGamma ^\mathcal {M}(v,w)\). Since \(A_\varGamma ^\mathcal {M}(v,w) \unlhd \lambda \), one has \(A_\varGamma ^\mathcal {M}(u,w) \lhd \lambda \) by transitivity of \(\unlhd \). Furthermore from Remark 4,
$$\begin{aligned} \mathcal {M}, u\models \varphi \wedge \overline{A_\varGamma ^\mathcal {M}(u,w)}. \end{aligned}$$
Take \(\lambda ' {:}{=} A_\varGamma ^\mathcal {M}(u,w)\). Then \(\mathcal {M}, u\models \varphi \wedge \overline{\lambda '}\), and hence
$$\begin{aligned} \mathcal {M}, w\models \lnot \bigwedge \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda ' \lhd \lambda } \lnot \Diamond (\varphi \wedge \overline{\lambda '}). \end{aligned}$$
This contradicts (12). Therefore \(A_\varGamma ^\mathcal {M}(u,w) \unlhd A_\varGamma ^\mathcal {M}(v,w)\) and \(u\preceq _w v\) by (\(\dagger \)). Finally, since \(A_\varGamma ^\mathcal {M}(v,w) \unlhd \lambda \), we have \(A_\varGamma ^\mathcal {M}(u,w) \unlhd \lambda \). Take \(\lambda '' {:}{=} A_\varGamma ^\mathcal {M}(u,w)\). Then \(\mathcal {M}, u \models \varphi \wedge \overline{\lambda ''}\), and hence
$$\begin{aligned} \mathcal {M}, u\models \bigvee \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda '' \unlhd \lambda } (\varphi \wedge \overline{\lambda ''}) \end{aligned}$$
as required.
\(\Leftarrow :\) Assume that
$$\begin{aligned} \mathcal {M}, w \models \bigwedge \limits _{\lambda \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(w)}[\bigwedge \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda ' \lhd \lambda } \lnot \Diamond (\varphi \wedge \overline{\lambda '}) \rightarrow \bigvee \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w)\supseteq \lambda ''\unlhd \lambda }(\varphi \wedge \overline{\lambda ''}) \preceq (\psi \wedge \overline{\lambda })]. \end{aligned}$$
Take \(u\in W_w\) such that \(\mathcal {M}, u\models \psi \), and consider \(A_\varGamma ^\mathcal {M}(u,w)\).

Case 1: There is an \(x\in W_w\) such that \(A_\varGamma ^\mathcal {M}(x,w) \lhd A_\varGamma ^\mathcal {M}(u,w)\) and \(M, x\models \varphi \). Then \(x\unlhd _w^\varGamma u\), by definition of \(\unlhd _w^\varGamma \).

Case 2: There is no \(x\in W_w\) such that \(A_\varGamma ^\mathcal {M}(x,w) \lhd A_\varGamma ^\mathcal {M}(u,w)\) and \(\mathcal {M}, x\models \varphi \). Towards a contradiction, suppose there is \(y\in W_w\) and a set \(\lambda '\) with \(\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda ' \lhd A_\varGamma ^\mathcal {M}(u,w)\) such that \(\mathcal {M}, y\models \varphi \wedge \overline{\lambda '}\). But \(\lambda ' \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(w)\) and \(\mathcal {M}, y \models \overline{\lambda '}\) imply that \(\lambda ' \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(y) \cap \text {Col}^{\mathcal {M}}_{\varGamma }(w) = A_\varGamma ^\mathcal {M}(y,w) \). So \(A_\varGamma ^\mathcal {M}(y,w) \unlhd \lambda ' \lhd A_\varGamma ^\mathcal {M}(u,w)\), and \(\mathcal {M}, y\models \varphi \), contradicting the case assumption. Hence no such y can exist, from which it follows that
$$\begin{aligned} \mathcal {M}, w\models \bigwedge \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda ' \lhd A_\varGamma ^\mathcal {M}(u,w)} \lnot \Diamond (\varphi \wedge \overline{\lambda '}). \end{aligned}$$
Together with the initial assumption this implies that
$$\begin{aligned} \mathcal {M}, w\models \bigvee \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda '' \unlhd A_\varGamma ^\mathcal {M}(u,w)}(\varphi \wedge \overline{\lambda ''}) \preceq (\psi \wedge \overline{A_\varGamma ^\mathcal {M}(u,w)}). \end{aligned}$$
Since \(\mathcal {M}, u\models \psi \wedge \overline{A_\varGamma ^\mathcal {M}(u,w)}\), there is an \(x\preceq _w u\) such that
$$\begin{aligned} \mathcal {M}, x\models \bigvee \limits _{\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \lambda '' \unlhd A_\varGamma ^\mathcal {M}(u,w)}(\varphi \wedge \overline{\lambda ''}). \end{aligned}$$
So \(\mathcal {M}, x\models \varphi \wedge \overline{\varTheta }\) for some set \(\varTheta \) with \(\text {Col}^{\mathcal {M}}_{\varGamma }(w) \supseteq \varTheta \unlhd A_\varGamma ^\mathcal {M}(u,w)\). But \(\varTheta \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(w)\) and \(\mathcal {M}, x \models \overline{\varTheta }\) imply that \(\varTheta \subseteq \text {Col}^{\mathcal {M}}_{\varGamma }(x) \cap \text {Col}^{\mathcal {M}}_{\varGamma }(w) = A_\varGamma ^\mathcal {M}(x,w)\), and so \(A_\varGamma ^\mathcal {M}(x,w) \unlhd \varTheta \) by the increment condition. We thus have that \(A_\varGamma ^\mathcal {M}(x,w) \unlhd A_\varGamma ^\mathcal {M}(u,w)\) by transitivity of \(\unlhd \). Finally since \(x\preceq _w u\) one has \(x\unlhd _w^\varGamma u\).

Hence in any case there is \(x\unlhd _w^\varGamma u\) such that \(\mathcal {M}, x\models \varphi \), as desired. \(\square \)

As with Theorem 1, the translation from \(\mathcal {L}\)-formulas to \(\mathcal {L}^-\)-formulas permits a reduction to an underlying complete logic, so that we obtain the following theorem.

Theorem 2

The logic \(\varLambda ^{\mathcal {L}_\mathsf {PCP}}_\mathbf C \) is complete.

7 Concluding remarks

This paper has introduced a prioritised ceteris paribus logic for counterfactual reasoning by adapting the formalism in van Benthem et al. (2009). The prioritisation of our logic is inspired by Lewis’ system of priorities, which he offered to restrict the construction of similarity orders. Whereas his system could be rejected for being ad hoc (McCall 1984, p. 470), or “not intuitively obvious” (Lewis 1979, p. 470), our formalism shows that it can be formalised in a systematic way, without having to depart from primitive similarity orders over possible worlds. Naïve similarity orders like in the Fine model can be rectified by ceteris paribus prioritisation. Instead of rejecting the Fine model for simplistically mischaracterising the Nixon counterfactual, we add a correcting ceteris paribus clause to the analysis of the counterfactual. To diffuse counter-examples to Lewis’ theory of counterfactuals in a systematic way, other things must be kept equal, as much as possible. Deviant worlds that exploit variance of other things that might look innocuous should be minimised, and ceteris paribus logic does exactly this.

We have provided completeness theorems which demonstrate that the ceteris paribus logics so obtained ultimately reduce to the underlying counterfactual logic; in our case Lewis’ VC. In general, ceteris paribus reasoning requires keeping equal as much information as possible, and sometimes unknown information (for example, unanticipated defeaters of laws). Keeping everything else equal may indeed mean keeping equal an indefinite, and possibly infinite, set of things. Exploring ceteris paribus logic without cardinality restrictions to \(\varGamma \) is thus more than a mere technical exercise. But it is not so straightforward to extend the present framework to accommodate the presence of infinite \(\varGamma \). The translations presented in the completeness proofs only carry over to the infinite case for infinitary languages, which is not much of a solution. We instead suggest following the \(\delta \)-flexibility approach of Seligman and Girard (2011).

Footnotes

  1. 1.

    See Schurz (2002) on comparative ceteris paribus laws.

  2. 2.

    The choice of \(\varGamma \) finite is largely technical. We will mention some possibilities and difficulties regarding the case where the ceteris paribus set \(\varGamma \) may be infinite in Sect. 7.

  3. 3.

    We redefine the language more precisely as Definition 6 in the appendix, particularly avoiding non-well-founded sets \(\varGamma \). For simplicity we work with the one now stated.

  4. 4.

    Though the set \(\mathsf {Prop}\setminus (\mathsf {UD}(\varphi ) \cup \mathsf {UD}(\psi ))\) is an infinite set when \(\mathsf {Prop}\) is, it is not as ambitious as the previous infinite sets we’ve considered, because it is a co-finite set. Co-finite sets are a special cases of flexible sets as defined and axiomatised in Seligman and Girard (2011). To keep focus, we will stick with finite \(\mathsf {CP}\)-sets in this paper, and leave an exploration of the flexibility approach to infinite sets in conditional logic for future research.

  5. 5.

    It could be that neither \(A_\varGamma ^\mathcal {M}(v,w) \unlhd A_\varGamma ^\mathcal {M}(u,w)\) nor \(A_\varGamma ^\mathcal {M}(u,w) \unlhd A_\varGamma ^\mathcal {M}(v,w)\), as \(\unlhd \) is a preorder.

  6. 6.

    We provide an example of a prioritisation order that allows for incomparability in Sect. 5.2.2.

  7. 7.

    In \([\varphi ,\varGamma ,\le ]\), we use \(\le \) instead of \(\unlhd \) to indicate that the modality is based on the specific ceteris prioritisation given by naïve counting. We will use notation in a similar fashion below.

Notes

Acknowledgements

We wish to thank the participants at the Australasian Association of Logic and the Analysis, Randomness and Applications meetings held in New Zealand in 2014. A preliminary version of this paper was presented at TARK XV and we wish to thank the participants for helpful suggestions on improving the paper. We also wish to thank Sam Baron, Andrew Withy, Balder ten Cate and the anonymous referees for valuable comments.

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of AucklandAucklandNew Zealand
  2. 2.University of AucklandAucklandNew Zealand

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