Counterfactuals as variably strict conditionals

In document Context-indexed counterfactuals and non-vacuous counterpossibles (Page 69-72)

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 Iโ€™ll 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‘Š, for each world and ๐‘– โˆˆ โ„ โ€“ as in the case of sphere frames (๐‘Š, ๐‘†) or variable sphere frames (๐‘Š, ๐’ฎ) โ€“ each world is assigned a set of accessibility spheres $๐‘ค โŠ† โ„˜(๐‘Š) that satisfy certain restrictions. Each $๐‘ค 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‘† โˆˆ $๐‘ค, 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 ๐‘†โ€ฒ โˆˆ $๐‘ค a basis for comparative similarity of worlds to w is established. The

intuition here is that worlds in some sphere ๐‘† โˆˆ $๐‘ค, 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 Iโ€™ll 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 Iโ€™ll 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) ๐‘1>๐‘ ๐‘ž

(2) If I walked on the lawn and everyone did that, the lawn would be ruined. (ii) (๐‘1โˆง ๐‘2) >๐‘ ~๐‘ž and (iii) ~((๐‘1โˆง ๐‘2) >๐‘ ๐‘ž)

If the above counterfactuals were analysed as any strict conditional, then the stipulated scenario would be impossible, since if ๐‘1>๐‘ ๐‘ž is true, then so is (๐‘1โˆง ๐‘2) >๐‘ ๐‘ž, by

Proposition 2.1, which contradicts ~((๐‘1โˆง ๐‘2) >๐‘ ๐‘ž). In other words, there is no variable

sphere model (๐‘Š, ๐’ฎ, ๐‘‰) such that all (i)-(iii) are true at some world ๐‘ค โˆˆ ๐‘Š. However, we could assume (1) as true, based on the strict conditional >๐‘  that corresponds to an accessibility assignment ๐‘†๐‘ค๐‘  determined by an overall similarity such that worlds where

โ€˜everyone else walks on the lawnโ€™, denoted [๐‘2], would be disregarded as irrelevant to its

evaluation at world ๐‘ค. That is, the choice of ๐‘  โˆˆ โ„ would be such that ๐‘†๐‘ค๐‘  โˆฉ [๐‘2] = โˆ… for the

same reasons as worlds where kangaroos walk upright with the aid of crutches would be disregarded as irrelevant to analyzing the counterfactual โ€˜If 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‘ โ€ฒ, that corresponds to an accessibility assignment ๐‘†

๐‘ค๐‘ โ€ฒ determined by overall

similarity of worlds such that ๐‘†๐‘ค๐‘  โŠ† ๐‘†๐‘ค๐‘ โ€ฒ and ๐‘†๐‘ค๐‘ โ€ฒโˆฉ [๐‘2] โ‰  โˆ…, and all the ๐‘2-worlds in ๐‘†๐‘ค๐‘ โ€ฒ are ~๐‘ž-

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 ๐‘ค โˆˆ ๐‘Š, and the evaluation of the

counterfactual (2) at ๐‘ค is done in terms of >๐‘ โ€ฒ. In other words, a system of spheres model (๐‘Š, $, ๐‘‰) for the variably strict conditional can be fashioned such that the variable

accessibility assignment $๐‘ค = {๐‘†๐‘ค๐‘ , ๐‘†๐‘ค๐‘ โ€ฒ} satisfies the intended comparative similarity

relationship ๐‘†๐‘ค๐‘  โŠ† ๐‘†๐‘ค๐‘ โ€ฒ, thereby allowing all ๐‘1 > ๐‘ž, (๐‘1โˆง ๐‘2) > ~๐‘ž and ~((๐‘1โˆง ๐‘2) > ๐‘ž) to be

true at ๐‘ค โˆˆ ๐‘Š. The intuition that worlds in ๐‘†๐‘ค๐‘  are more similar to ๐‘ค than those in ๐‘†๐‘ค๐‘ โ€ฒ\๐‘†๐‘ค๐‘ 

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€“ thereโ€™s 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 Stalnakerโ€™s 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 wonโ€™t 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€™ โ€˜similarity sphereโ€™ semantics, which I define and discuss in detail in the next section.

In document Context-indexed counterfactuals and non-vacuous counterpossibles (Page 69-72)