Prioritised ceteris paribus logic for counterfactual reasoning
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 logicCeteris 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.’
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 fourpoint 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 counterexamples that involved a more judicious choice of miracles were advanced (Tooley 2003; Wasserman 2006; Elga 2001). The counterexamples are constructed to obey Lewis’ system of priorities while failing to match intuitive evaluations of counterfactuals. Schaffer (2004) defused those counterexamples 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.If Nixon had pushed the button, there would have been a nuclear holocaust.
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
 1.
W is a nonempty set of possible worlds.
 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.
V is a valuation function assigning a subset \(V(p) \subseteq W\) to each propositional variable \(p\in \mathsf {Prop}\).
Definition 1
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
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.
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 i, j. 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 antiSoviet 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 pworld. 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
 1.
It is of the first importance to avoid big, widespread, diverse violations of law.
 2.
It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails.
 3.
It is of the third importance to avoid even small, localized, simple violations of law.
 4.
It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.
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 preeminence 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 preeminence of perfect match in constructing the similarity relation.
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
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.
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.

Set \([w]_\varGamma = \{v \in W_w : \text {Col}^{\mathcal {M}}_{\varGamma }(w) = \text {Col}^{\mathcal {M}}_{\varGamma }(v)\}\), the collection of wentertainable 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.
Definition 4
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
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
McCall’s instructions are to inspect those worlds:(1) If Napoleon had won the battle of Waterloo, he would not have died on St. Helena.
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 truthvalue. Clearly this is not yet finegrained 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 tradeoff between how much of the other things can be kept equal, and what precisely counts as being equal.(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).
4.2 Universe of discourse and causal independence
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: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)
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):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)
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 formOnly 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)
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

(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 \(\psi \).
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 nontotal 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
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
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
Fact 4
 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.
\(\mathcal {M}, w \models \langle \varphi , \varGamma \rangle \psi \Rightarrow \mathcal {M}, w \models \langle \varphi , \varGamma , \le \rangle \psi \)
Remark 3
Let \(\varGamma = \{p,q\}\). Since no sworld 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 \(\mathsf {Min}_{\unlhd ^{\varGamma \cup \{r\}}_w}(\llbracket s \rrbracket ^\mathcal {M}) \, = \, \{u\}\).
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.
Fact 5
 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.
If \(\mathcal {M}, w \models \langle \varphi , \varGamma \rangle \psi ,\) then \(\mathcal {M}, w \models \langle \varphi , \varGamma , \subseteq \rangle \psi \).
 3.
If \(\mathcal {M}, w \models \langle \varphi , \varGamma , \subseteq \rangle \psi ,\) then \(\mathcal {M}, w \models \langle \varphi , \varGamma , \le \rangle \psi \).
5.3 The Fine model III
Counterfactual  Clause  \(\mathsf {CP}\)  \(\mathsf {NC}\)  \(\mathsf {MS}\)  

\(\{m,s\}\)  \(\mathsf {true}\)  \(\mathsf {false}\)  \(\mathsf {false}\)  \(\mathsf {true}\)  
\(\{m,s\}\)  \(\mathsf {true}\)  \(\mathsf {true}\)  \(\mathsf {false}\)  \(\mathsf {false}\) 
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 nonwellfounded ceteris paribus sets (Seligman and Girard 2011).
Definition 6
Lemma 1
The modal operator \([\varphi , \varGamma ]\psi \) is definable in Open image in new window .
Proof
By a conditional frame we mean a pair \(F = (W, \preceq )\), such that (F, V) 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 Lformulas 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 insideout, 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
Definition 8
Lemma 2
The modal operator \([\varphi , \varGamma , \unlhd ]\psi \) is definable in \(\mathcal {L}\).
Proof
\(\Rightarrow :\) 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 \(\mathcal {M}, w\not \models (\varphi \wedge \psi ) \lhd ^\varGamma (\varphi \wedge \lnot \psi )\). \(\square \)
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
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 \(\unlhd _w^\varGamma \).
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 counterexamples 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.
See Schurz (2002) on comparative ceteris paribus laws.
 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.
We redefine the language more precisely as Definition 6 in the appendix, particularly avoiding nonwellfounded sets \(\varGamma \). For simplicity we work with the one now stated.
 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 cofinite set. Cofinite 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.
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.
We provide an example of a prioritisation order that allows for incomparability in Sect. 5.2.2.
 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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