# General limitations: “The Granularity Problem”

In document Context-indexed counterfactuals and non-vacuous counterpossibles (Page 31-35)

## 1.3.2 Wider applications of impossible worlds 1 Content as intension, via possible worlds

### 1.3.2.2 General limitations: “The Granularity Problem”

However, this approach falls short of giving an adequate analysis of contexts containing hyperintensional phenomena, i.e. contexts where intensional equivalence is insufficient for identity, or contexts that do not respect logical equivalence (Cresswell, 1975, p.25), or more broadly, contexts that do not respect necessary equivalence (Nolan, 2013, p.366).39 Before demonstrating how intensions fall short of delivering hyperintensional distinctions, I will briefly discuss the analogous phenomenon, of the inadequacy that extensions display in drawing intensional distinctions. This will place the following discussion in a broader context. Consider the following example. ‘Canberra is the capital of Australia’ – the referent of ‘Canberra’ and ‘The capital of Australia’ is the city of Canberra.40 Now, despite its

apparent innocuity, the counterfactual ‘If Brisbane were the capital of Australia, then Brisbane would be the capital of Australia’ gives rise to an intensional context41, where

substitutivity of co-extensive expressions (and co-referential expressions in particular) isn’t guaranteed to be truth preserving. Consider an instance of substituting the co-referring singular terms, in this case ‘Canberra’ for ‘the capital of Australia’ in the consequent of (2).

1. If Brisbane were the capital of Australia, then Brisbane would be the capital of Australia.

2. If Brisbane were the capital of Australia, then Brisbane would be Canberra. Although the first counterfactual is an instance of counterfactual identity, the truth of the

39 Nolan’s generalizes the definition by lifting the restriction to logical equivalence – necessity need not be just

logical, e.g. it may also be metaphysical, which need not be the same as logical necessity: ‘A position in a

sentence is said to be sensitive to hyperintensional differences, if the truth value of the sentence can be altered by replacing the expressions in that position with one that necessarily applies to the same things’ (Nolan, 2013, p.366).

40 In this example I am treating co-referential terms as a special case of co-extensive predicates, where being the

capital of some country is thought of as having that property.

second one is context dependent. In particular there are contexts where it is false, e.g. when we intend (in the hypothetical scenario) both cities to remain where they actually are.

Brisbane being the capital of Australia, and both cities remaining where they actually are is a perfectly possible scenario, which the reading of (2) doesn’t rule out. The general point I wish to stress is that just as co-reference (extensional equivalence) is inadequate for drawing intensional distinctions (‘Canberra’ doesn’t mean ‘the capital of Australia’), so intensional equivalence falls short of drawing hyperintensional distinctions. Consequently, just as extensional equivalence is inadequate for identity conditions that guarantee substitutivity salva veritate on intensional contexts, intensional equivalence is inadequate for identity conditions that guarantee substitutivity salva veritate on hyperintensional contexts. That is, the approach fails to distinguish sentences that are either necessarily true, or necessarily false – the former are identified with the set of all possible worlds, since necessity is modelled as truth at all possible worlds, and the latter with the empty set since necessary falsehoods are not true at any possible world.42 Let us denote the proposition expressed by sentence 𝐴 with [𝐴]. That is let us adopt the notation [𝐴] ≔ {𝑤 ∈ 𝑊 ∶ ℐ𝐴(𝑤) = 𝑡𝑟𝑢𝑒} for the reminder of our discussion, to distinguish [𝐴] from the sentence that expresses it. To illustrate this more formally, in terms of the Kripke models this would mean that for any two sentences 𝐴 and 𝐵 that express a necessary truth, the following identity holds for all models (𝑊, 𝑅, 𝑉), [𝐴] = [𝐵] = 𝑊, and likewise, for any two sentences 𝐴 and 𝐵 that express a necessary falsehood, the following identity holds for all models [𝐴] = [𝐵] = ∅ for all models.

Consequently, propositional content modelled as intension leads to cointensive expressions being analyzed as meaning the same thing, which is strongly counterintuitive. Consider the following pairs of sentences, which on the just described possible world, meaning-as- intension, analysis are analyzed as expressing the same proposition:

1. There are no married bachelors. 2. There are infinitely many primes. 3. Some bachelors are married. 4. There are finitely many primes.

42 There is no general consensus regarding what kind of necessity represents absolute necessity, but there is a tendency of taking logical, mathematical, and metaphysical necessity as close approximations. The issue which of these necessities is more fundamental is also controversial (Berto & Jago, 2019, §1.2). In this thesis I will not make any assumptions in this regard.

Each sentence in the first pair expresses some necessary truth, so each is true in all possible worlds. So, their intensions are identical, and consequently both sentences are analyzed as having the same content, i.e. expressing the same proposition. This doesn’t seem right, since they appear to be saying different things – (2) says nothing about marriage or bachelorhood. The latter pair of sentences suffers from the same inadequacy of distinguishing their meaning due to the pair being cointensive. This fundamental shortcoming is inherited in the analysis of propositional attitudes that give rise to hyperintensional contexts, where substitutions of necessary equivalent terms in a sentence need not be truth preserving. Take the following pairs of necessarily true sentences:

1. The axioms of Peano Arithmetic are true.

2. [Any sentence that expresses a theorem of PA] is true. 3. Water is water.

4. Water is H2O.

And consider the following substitutions of those sentences in sentences expressing doxastic and epistemic propositional attitudes:

5. Giuseppe believes that the axioms of Peano Arithmetic are true.

6. Giuseppe believes that [any sentence that expresses a theorem of PA] is true. 7. It is known a priori that water is water.

8. It is known a priori that water is H2O.

It becomes clear that in such contexts, necessary sentences are not expected to be

substitutable salva veritate. Surely, Giuseppe Peano believed the truth of his own axioms (that’s what it means to be an axiom – a truth that is immediately evident), but it doesn’t seem true that he believed all sentences, of arbitrarily complexity, that happen to express a consequence of those axioms, i.e. all theorems of PA. So, although (5) is (very likely) true, sentence (6) should be false. Also, whereas sentence (7) is true (it is probably among the least contested a priori truths out there), sentence (8) is false, since knowing that water is H2O

requires empirical knowledge of the molecular structure of water.

Counterpossible reasoning is another context where hyperintensional phenomena arise. A counterpossible conditional is a subjunctive conditional whose antecedent expresses a necessary falsehood. By direct analogy with the earlier example involving counterfactuals,

which illustrated the inadequacy of appeals to actually co-referring terms (material

equivalence) in drawing intensional distinctions, it can be shown that appeals to necessarily co-referring terms (logical equivalence) are inadequate in drawing hyperintensional

distinctions. This inadequacy in distinguishing between cointensive impossible expressions carries over to accounts of the counterpossible that restrict the analysis to intensions only.43 All analyses of counterfactuals whose truth is cashed out in terms of the corresponding material conditional’s truth at possible worlds will result in all counterpossibles being evaluated as vacuously true. This is because antecedents of counterpossibles are not true at any possible world, by definition. A notable example of such an approach – one given by Lewis (1973, 1986) – meets the same predicament, by treating all counterpossibles as logically equivalent (in the case of Lewis, as true). As mentioned earlier, I will discuss Lewis’ analysis of counterfactuals and counterpossibles in depth in chapter 2, but for the purposes of the present introduction his analysis can be can be given informally: the counterfactual ‘if it were the case that …, then it would be the case that …’ is true at a possible world w just in case the consequent is true at all the most similar possible worlds to w where the antecedent is true. Since sentences expressing a necessary falsehood are true at no possible world, and in particular a possible world satisfying some additional similarity conditions, each counterpossible is analysed as vacuously true. Consider the following pair of counterpossibles. Whereas (1) is clearly true in all contexts, (2) could be false.

1. If Sally were to square the circle, then Sally would have squared the circle. 2. If Sally squared the circle and I doubled the cube, then I would be Sally.44 So, on Lewis’ analysis (1) and (2) are logically equivalent (both are true at all possible worlds), which seems wrong. For a more emphatic demonstration that counterpossibles do

43 Observation: (2) appears more readily read as false than its counterfactual analogue of Australia’s capital(s), since the property of being the capital of Australia counterfactually ascribed to Brisbane – which actually belongs to Canberra – is unique, whereas in the counterpossible (2) there is explicit talk of two properties, which only are identified as meaning the same thing by the underlying (content-as-intension) analysis. Also,

independently of such reasons for the apparent disparity in readiness with which we would be inclined to read (2) and the earlier counterfactual example as false, it seems that we tend to “hold on” to numerical identity more than any other properties of objects, i.e. shifts in numerical identity seem to be contextually the most far- fetched. Of course, there are contexts where (2) and its counterfactual analogue would be true, but they don’t seem to be among the first ones that we’re willing to consider.

44 Squaring the circle refers to constructing a square of the same area as some circle in a finite number of steps. This construction is mathematically impossible. Doubling the cube is a related, impossible construction, whereby given the edge of a cube one is required to construct the edge of another cube that has twice the volume of the first one.

give rise to hyperintensional contexts, let us consider (3), where we substitute the consequent of (1) with a logically equivalent sentence, and observe that unlike (1), (3) could be false.

3. If Sally were to square the circle, then Mariusz would have doubled the cube. As a matter of fact, Lewis’ analysis of the counterpossibles can be viewed as emblematic of the inadequacy of intensions in drawing hyperintensional distinctions. This example

highlights how the matter of non-vacuous counterpossibles and the matter of adequate

hyperintensional distinctions are closely related – counterpossibles do create hyperintensional contexts.

In document Context-indexed counterfactuals and non-vacuous counterpossibles (Page 31-35)