In the last section we gave an overview of Lewisโ reasons for claiming that strict conditionals best corresponding to counterfactuals are those whose accessibility assignment (conceived of as a sphere of accessibility) is determined by an overall similarity of worlds, and we also saw Lewisโ argument that no single strict conditional suffices to give an adequate account of the counterfactual. This means that a fortiori no single strict conditional whose accessibility assignment is determined by an overall similarity of worlds suffices to give an adequate account of the counterfactual. In this section I present Lewisโ solution to this in the form of a variably strict conditional account of the counterfactual, and then Iโll move onto giving a comprehensive overview (and discussion) of various conceptions of comparative similarity in terms of the model theory offered by Lewis for that account.
The systems of similarity spheres model theory developed by Lewis (1973) is a proposal to analyze he counterfactual as a variably strict conditional, rather than any single strict conditional of fixed strictness. On that novel proposal, instead of a single sphere of
accessibility, being a subset of ๐, for each world and ๐ โ โ โ as in the case of sphere frames (๐, ๐) or variable sphere frames (๐, ๐ฎ) โ each world is assigned a set of accessibility spheres $๐ค โ โ(๐) that satisfy certain restrictions. Each $๐ค is a variable accessibility assignment. On
this picture, the criteria deeming worlds as relevant to the evaluation of a counterfactual at some world w is expressed in terms of comparative overall similarity of worlds, whereby any given sphere of accessibility ๐ โ $๐ค, determined by such considerations, is thought to contain
worlds that are similar to w to some fixed degree. This way, in conjunction with other spheres of accessibility ๐โฒ โ $๐ค a basis for comparative similarity of worlds to w is established. The
intuition here is that worlds in some sphere ๐ โ $๐ค, around some world w, are more similar to
of the idea of systems of similarity spheres as representations of comparative similarity of worlds:
The system of spheres used in interpreting counterfactuals is meant to carry information about the comparative overall similarity of worlds. Any particular sphere around a world w is to contain just the worlds that resemble w to at least a certain degree. This degree is different for different spheres around w. The smaller the sphere the more similar to w must a world be to fall within it. [W]henever one world lies within some sphere around w and another world lies outside that sphere, the first world is more closely similar to w than the second. (Lewis 1973, p.14)
Before giving the formal definition, we can illustrate the core idea of such systems of spheres with the aid of the notion of a hierarchy of strictness, introduced earlier in Definition 2.11. This illustration will employ elements of Lewisโ explanation of the ceteris paribus clause in terms of comparative similarity of worlds, and key observations he made in the argument, which has been discussed at the end of the previous section. Consider the two counterfactuals (1) and (2) given below, which Iโll presently use to set up a scenario highlighting the link between Lewisโ view that counterfactuals are strict conditionals corresponding to an
accessibility assignment determined by an overall similarity of worlds, and the limitations of strict conditionals to meet this task (as discussed at the end of the previous section). In this example Iโll only rely on the first two stages employed in Lewisโ argument, i.e. with the stipulation that all (i), (ii), and (iii) are true (which agrees with intuition).
(1) If I walked on the lawn, no harm would come of it. (i) ๐1>๐ ๐
(2) If I walked on the lawn and everyone did that, the lawn would be ruined. (ii) (๐1โง ๐2) >๐ ~๐ and (iii) ~((๐1โง ๐2) >๐ ๐)
If the above counterfactuals were analysed as any strict conditional, then the stipulated scenario would be impossible, since if ๐1>๐ ๐ is true, then so is (๐1โง ๐2) >๐ ๐, by
Proposition 2.1, which contradicts ~((๐1โง ๐2) >๐ ๐). In other words, there is no variable
sphere model (๐, ๐ฎ, ๐) such that all (i)-(iii) are true at some world ๐ค โ ๐. However, we could assume (1) as true, based on the strict conditional >๐ that corresponds to an accessibility assignment ๐๐ค๐ determined by an overall similarity such that worlds where
โeveryone else walks on the lawnโ, denoted [๐2], would be disregarded as irrelevant to its
evaluation at world ๐ค. That is, the choice of ๐ โ โ would be such that ๐๐ค๐ โฉ [๐2] = โ for the
same reasons as worlds where kangaroos walk upright with the aid of crutches would be disregarded as irrelevant to analyzing the counterfactual โIf kangaroos had no tails, they would topple overโ.
As for (2), we could analyse it by a stricter conditional than >๐ (recall Definition 2.11),
denoted >๐ โฒ, that corresponds to an accessibility assignment ๐
๐ค๐ โฒ determined by overall
similarity of worlds such that ๐๐ค๐ โ ๐๐ค๐ โฒ and ๐๐ค๐ โฒโฉ [๐2] โ โ , and all the ๐2-worlds in ๐๐ค๐ โฒ are ~๐-
worlds. On a system of spheres, we would analyse the counterfactuals (1) and (2) in the above scenario by a variably strict conditional >, fashioned in a manner so that the evaluation of the counterfactual (1) is done in terms of >๐ at some world ๐ค โ ๐, and the evaluation of the
counterfactual (2) at ๐ค is done in terms of >๐ โฒ. In other words, a system of spheres model (๐, $, ๐) for the variably strict conditional can be fashioned such that the variable
accessibility assignment $๐ค = {๐๐ค๐ , ๐๐ค๐ โฒ} satisfies the intended comparative similarity
relationship ๐๐ค๐ โ ๐๐ค๐ โฒ, thereby allowing all ๐1 > ๐, (๐1โง ๐2) > ~๐ and ~((๐1โง ๐2) > ๐) to be
true at ๐ค โ ๐. The intuition that worlds in ๐๐ค๐ are more similar to ๐ค than those in ๐๐ค๐ โฒ\๐๐ค๐
appears to be preserved, because on the supposition that actually nobody walks on the lawn, it seems strongly intuitive that worlds where only I walk on the lawn are more similar to the actual world than those worlds where the lawn is stampeded by everyone in the neighborhood (or by everyone on Earth).
Given that much of which worlds are deemed relevant to the analysis is determined by the antecedent, there is no need to adjoin any indices to the variably strict conditional โ thereโs only one. I have already touched on this in the previous section, when discussing the relevant antecedent worlds (sufficiently similar worlds) to the evaluation of the counterfactual and noting that Lewis intends there to be a fitting restriction of that kind for any antecedent of any strict conditional based on comparative similarity of worlds. I will also say more about this in the next section, once the formal model theory has been defined, but what has been illustrated by the above example should suffice for an intuitive outline of the rationale underlying the systems of spheres models for the variably strict conditional. Such a framework turns out to
be robust enough to express, not only Stalnakerโs theory80 of the counterfactual, but also
other notable reformulations, and analogies in temporal and deontic logic, of which Lewis (1973) gives a comprehensive analysis. I wonโt discuss those general correspondences here โ it will suffice to say that most theories of the counterfactual (as variably strict conditional), relevant to the current chapter, can be expressed in terms of Lewisโ โsimilarity sphereโ semantics, which I define and discuss in detail in the next section.