Strict conditionals and comparative similarity of worlds

The counterfactual proposed by Lewis, is closely related to ceteris paribus conditionality, and as such is not fundamentally different from the logics C and C+ discussed earlier. The

difference lies in how the ceteris paribus clause is explained. It is evident from the passage below, that the kind of strict conditional Lewis has in mind is effectively based on

accessibility that satisfies certain ceteris paribus constrains, which he then suggests are best expressed in terms of comparative similarity of worlds.

Counterfactuals are related to a kind of strict conditional based on comparative similarity of possible worlds. A counterfactual 𝜑 > 𝜓 is true at a world w if and only if 𝜓 holds at certain 𝜑-worlds; but certainly not all 𝜑-worlds matter. ‘If kangaroos had no tails, they would topple over’ is true (or false, as the case may be) at our world, quite without regard to those possible worlds where kangaroos walk around on crutches, and stay upright that way. Those worlds are too far away from ours. What is meant by the counterfactual is that, things being pretty much as they are –the scarcity of crutches for kangaroos being pretty much as it actually is, and so on – if kangaroos had no tails they would topple over. Lewis (1973, pp.8-9)

Lewis also observes that in our consideration regarding what kind of restricted necessity should underlie the strict conditional that best captures counterfactual reasoning, aside from ruling out worlds that are grossly dissimilar from the actual world (e.g. where tailless kangaroos use crutches to stay upright), we also must avoid deeming as accessible worlds that are too similar (or at least be careful when doing so). For if we include into the

accessibility sphere worlds where kangaroos have no tails, but otherwise everything else is exactly the same as the actual world, then kangaroos despite being tailless would

nevertheless leave tail tracks in the sand, and their genetic make-up, being as it actually is, would nevertheless somehow still code for different phenotypical traits (absence of tail). Lewis concludes that counterfactuals are apparently based on a strict conditional

corresponding to an accessibility assignment determined by an overall similarity of worlds, where respects of difference and respects of similarity are “somehow” balanced off against each other.

In the light of this, let’s consider again our earlier example of tailless-kangaroos necessity. It is clear from the above concerns regarding the intended notion of comparative similarity of worlds that the strict conditional □𝐾_{(𝐾 ⊃ 𝑇) would fall short of being the adequate model for}

the evaluation of the counterfactual ‘If Kangaroos had no tails, they would Topple over’. This inadequacy stems from the fact that the restriction corresponding to □𝐾_{, namely [𝐾]}ℳ0_, includes all possible 𝐾-worlds, and as such it may include worlds that have just been argued to be “too far” to be regarded as relevant to the evaluation of the counterfactual in question.

What we require, is a restriction that meets the similarity criteria that Lewis has in mind. That is, we are interested in 𝑠 ∈ ℐ corresponding to an accessibility restriction 𝑆𝑤𝑠 ∈ 𝒮ℳ0

determined by an overall similarity of worlds such that 𝑆𝑤𝑠 ∩ [𝐾]ℳ0 doesn’t contain worlds

where kangaroos manage to stay upright with the aid of crutches, or worlds where kangaroos leave a tail track behind in the sand as they actually do (despite their taillessness). That is, all 𝐾-worlds in 𝑆𝑤𝑠 are sufficiently similar to 𝑤 to count as relevant in the evaluation of the

counterfactual with antecedent 𝐾 at 𝑤. Then the strict conditional □𝑠(𝐾 ⊃ 𝑇) would meet the accessibility assignment requirement as determined by an overall similarity of worlds in the manner intended by Lewis. It’s apparent that Lewis intends there to be a fitting restriction 𝑠 ∈ ℐ of this kind in general, for any antecedent 𝐴 ∈ 𝐹𝑜𝑟 of any strict conditional based on

This would yield the following account of the counterfactual when considered as a single strict conditional based on the comparative similarity of worlds, i.e. the intended model truth conditions for the counterfactual in terms of a single strict conditional based on the

comparative similarity of worlds are as follows:

Definition 2.12: The counterfactual ‘If 𝐴, then 𝐵’ is true at a world 𝑤 iff ℳ0, 𝑤 ⊩ 𝐴 >𝑠𝐵.

Recalling that ℳ0, 𝑤 ⊩ 𝐴 >𝑠𝐵 iff ∀𝑢 ∈ 𝑆𝑤𝑠: ℳ0, 𝑢 ⊩ 𝐴 ⊃ 𝐵 for all 𝐴, 𝐵 ∈ 𝐹𝑜𝑟, 𝑤 ∈ 𝑊, and

where all worlds in 𝑆𝑤𝑠 are regarded as sufficiently similar to 𝑤 to count as relevant in the

evaluation of the counterfactual with antecedent 𝐴 at 𝑤.

To be precise, the intended accessibility restriction 𝑆𝑤𝑠 corresponding to 𝑠 ∈ ℐ isn’t constant,

but rather a function of the antecedent. Lewis’ argument that the counterfactual cannot be any strict conditional amounts to showing that this can’t be done on the intended model, where we are only equipped with a class of strict conditionals of fixed strictness. The accessibility restriction index denoted with 𝑠, used throughout the discussion in the next section is just to remind us that we’re not talking about arbitrary elements of ℐ, but ones that correspond to accessibility restrictions determined by an overall similarity of worlds. I devote the next section to Lewis’ argument, and given that it is expressed with direct reference to the intended model ℳ0, I shall omit any model-denoting superscripts.