Berto’s context-indexation suggestion

It is difficult to say whether the approach I chose – expanding the language by introducing additional syntactic parameters – to address those concerns is optimal, but it does appear natural. I have found some supporting evidence for this in a recurring suggestion made by Berto (2014, 2017). It was in Berto’s work on the analysis of conceivability and imagination that I have found a parallel of what I have been considering in counterfactuals. It was the manner in which Berto (2014, 2017) chose to analyze ‘representational acts’ underlying our conceivability and imagination, and ‘imagination acts’ that initially captured my attention, and in particular, his suggestion regarding how one may go about contextualizing those acts. Let me outline those features of Berto’s (2017) semantics of imagination that are relevant to his suggestion how one may go about contextualizing the object language.62

In Berto’s (2017) analysis of imagination, a basic propositional modal language is expanded by the inclusion of a family of sententially indexed modal operators [𝐴], where 𝐴 ranges over formulae that express possible acts. Expressions central to the analysis [𝐴]𝐵, are read as ‘It is imagined in act 𝐴 that 𝐵 or, more accurately ‘It is imagined in the act whose explicit content is 𝐴, that 𝐵’, where 𝐵 is any well-formed formula.63 _{On Berto’s analysis [𝐴] acts like a}

relative necessity operator, ranging over possible and impossible worlds, and [𝐴]𝐵 receives an analysis akin to expressions of sententially indexed modality described by Chellas (1975, p.138). Fundamentally Berto’s analysis of [𝐴]𝐵 rests on the same idea as the one employed in the analysis of ceteris paribus conditionals such as 𝐴 > 𝐵.

62 _{Berto (2014, p.113, f.9) makes the exactly the same suggestion in the context of conceivability.} 63 _{Berto (2017, §4).}

This brings conceiving in [sic] the vicinity of ceteris paribus conditionals. The explicit content of a representation may play a role similar to a conditional antecedent. (Berto 2014, p.8)

The explicit fictional content corresponds to the explicit content of our

imagined scenarios, and works, in Lewis’ approach, too, like the antecedent of a ceteris paribus conditional. (Berto 2017, p.7)

Berto (2014, p.113, f.9; 2017, §5) identifies the same worry in the analysis of intensional states like imagination that had been identified by Quine (1960), concerning the contextual ambiguity of counterfactuals with the same antecedents. “Is it so that, when one imagines in one act [𝐴] that 𝐵 and that 𝐶, one automatically imagines that 𝐵 ∧ 𝐶?” – he asks. On a given act of imagination [𝐴] with the same explicit content of Caesar being in command of the US troops in Korea, one can imagine Caesar using the atom bomb 𝐵, and one can imagine that he uses catapults 𝐶, however it doesn’t follow that one would thereby imagine Caesar using both the bomb and catapults. Berto observes that naturally, one could also imagine that, but the inference pattern [𝐴]𝐵, [𝐴]𝐶 ⊨ [𝐴](𝐵 ∧ 𝐶), should not be an automatic logical entailment.

The heart of the problem rests in the fact that different acts of imagining the same explicit content can give rise to imagining a different scenario in different contexts – in general, the imported information that makes [𝐴]𝐵 true is not the same as the imported information that makes [𝐴]𝐶 true. That is, it seems clear that different contexts underlie the truth of [𝐴]𝐵 and [𝐴]𝐶, and so it is not obvious that [𝐴](𝐵 ∧ 𝐶) should follow, unless we restrict what contexts should be at play throughout the inference. This can be done via a modification of the object language, by indexing representational acts [𝐴] with contexts, e.g. [𝐴]_𝑥, [𝐴]_𝑦, which would allow for an explicit syntactic restriction of inferences to a single context, for example [𝐴]_𝑥 𝐵, [𝐴]𝑥 𝐶 ⊨ [𝐴]𝑥 (𝐵 ∧ 𝐶). Below is how Berto expresses this idea.

I think that Adjunction can be maintained by fixing some contextual parameter. The formalism may represent this, if wanted, by adding a set of contexts to the interpretations and variables ranging on them in the language, and by directly indexing representational acts with contexts: [𝐴]_𝑥, [𝐴]_𝑦, for instance, will stand for two distinct acts with the same explicit content, 𝐴, performed in contexts x

and y. Once the adjunctive inference is parameterized to same-indexed contents, it should work fine. Berto (2017, p.11) 64

Naturally, this solution has its counterpart in the analysis of counterfactuals. Given the already noted similarity between the analysis of Berto’s intensional expressions [𝐴]𝐵 and ceteris paribus (or Lewisean) conditionals expressed as 𝐴 > 𝐵, the move to introduce a family {>_𝑥: 𝑥 ∈ 𝒞} of context-indexed conditional connectives that range over a set of context indices 𝒞 seems natural, and it is precisely the method I adopt. One could apply an analogous restriction to the counterpart inference patterns, i.e. 𝐴 >𝑥𝐵, 𝐴 >𝑥𝐶 ⊨ 𝐴 >𝑥(𝐵 ∧ 𝐶) in order

to ensure truth preservation. In fact I show, in agreement with Berto, that this and other generally accepted inferences65 hold for the contextualized language, whenever all instances of the counterfactual appearing in the inference are restricted to a single context index. However, I chose to not apply such a restriction in general, which I feel departs from an opportunity to make the logic relevance-sensitive in an interesting way (not only single context index premise sets).

My proposal allows for premises to range over more than one context index, so in this sense the restrictions that are in place are weaker than the one suggested by Berto. However, on top of the usual truth (preservation) condition for validity, a contextual information preserving condition is introduced. That is, I demand the existence of relevant content connection

(properly defined and developed in the model theory) between the context indices over which the premises range and the conclusion context index. For example, in the case of

𝐴 >𝑥𝐵, 𝐴 >𝑦𝐶 ⊨ 𝐴 >𝑧 (𝐵 ∧ 𝐶) for the inference to be valid, aside from truth preservation at

all worlds in all models, it is additionally required that the conclusion context index z preserves the mutual contextual information of context indices x and y, which make the premises true. Clearly, this condition is met trivially with Berto’s restriction in place.

Naturally, the model theory ensures that all the relevant terms such as context, context index, contextual information preservation, and mutual contextual information are properly and carefully motivated and defined. So although standard inferences fail to hold in general, when we lift Berto’s restriction, all of their instances that are said to preserve contextual

information (have a suitably contextualized form) do hold.

64 _{My emphasis. By Adjunction Berto means the semantic condition that guarantees the validity of the inference} pattern [𝐴]𝐵, [𝐴]𝐶 ⊨ [𝐴](𝐵 ∧ 𝐶). I shall refer to it as Adjunction of Consequents.