Abstract
Franz Berto has recently proposed two distinct semantics for the logic of ceteris paribus imagination, one which combines a theory of topics with a standard possible worlds semantics, another based on impossible worlds. An important motivation for using these tools is to handle the hyperintensionality of imagination reports. I argue however that both semantics prove inadequate for different reasons: the former fails to draw some intuitive hyperintensional distinctions, while the latter draws counterintuitive hyperintensional distinctions. I propose an alternative truthmaker semantics, guided by an independently motivated philosophical analysis of the content of imagination acts, that preserves the attractive features of both approaches, while avoiding their symmetrical defects in the treatment of hyperintensionality.
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Notes
- 1.
Giordani offers in [15] a sound and complete axiomatization of the logic determined by the class of models that satisfy (BC), without assuming (PIE).
- 2.
- 3.
One might point out that \(p'\) is implicit in what Daniel says, and I am prepared to agree. But what is merely implied or suggested by what someone says is not part of what is said. So when Zeno imagines that Daniel is right, he takes as input of his act of imagination that what is said by Daniel is true, which does not contain \(p'\).
- 4.
For instance, Berto emphasizes that “different acts of imagining the same explicit content can trigger the import of different background information depending on contexts (the time and place at which the cognitive agent performs the act, the status of its background information, etc.)” [2, p. 1287].
- 5.
As in the previous section, we do not reproduce the proofs that are already given by Berto in [2], unless they are particularly informative.
- 6.
Each node represents a world identified by its name. Names of impossible worlds are followed by a star. Under each possible world, we indicate the propositional variables that are verified and falsified respectively. All the other formulas that are verified and falsified, respectively, can be deduced with the help of the recursive conditions given in the definition of an IW-model. For impossible worlds, we also indicate verified and falsified non atomic formulas when they cannot be obtained from propositional variables by condition (IW3) and leave implicit those that can be so obtained. An arrow labeled with formula A between worlds w and \(w'\) indicates that \(wR_{A}w'\). It can be checked that the represented models satisfy all the conditions in the definition of an IW-model.
- 7.
I am indebted to an anonymous reviewer for bringing this solution to my attention.
- 8.
This definition characterizes a distributive notion of exact consequence which requires that all possible situation exactly verifying all the premises individually also verifies the conclusion. One may also consider a collective understanding of exact consequence which requires that all possible situations exactly verifying the conjunction of all the premises also verifies it conclusion. Unless otherwise specified, by exact consequence we will understand hereafter exact consequence in the distributive sense. For a recent study of exact consequence in the distributive sense see [14].
- 9.
In the two diagrams below, labeled nodes represent situations. Downward straight lines between situations represent the relation of parthood. Atomic formulas exactly verified by situations are indicated below or above the nodes representing them. Unless they are particularly relevant to the example, we do not indicate the complex formulas exactly verified or falsified by a situation when they can be computed from the atomic formulas it exactly verifies or falsifies, via the recursive clauses given above.
- 10.
Defending this particular definition of topic goes way beyond the purpose of this paper. See however [13] for a detailed defense of this account.
- 11.
A similar problem is addressed by Fine in [9] as part of the project of providing an exact semantics for intuitionnistic logic.
- 12.
A similar strategy is followed by Fine in the exposition of his truthmaker semantics for counterfactuals [8], where he only gives classical verification conditions for counterfactuals, themselves defined in terms of exact and inexact verification.
- 13.
Labeled nodes represent situations. Labels of the form \(w_{i}\) indicate situations that are possible worlds. Downward straight lines between situations represent the relation of parthood. All and only possible situations are part of a possible world. Labeled bent lines represent accessibility relations, so that \(w_{i}\) is connected to \(s_{j}\) by a bent line labeled by A just in case \(s_{j} \in f(A,w_{i})\). For each situation, the propositional variables that it exactly verifies or falsifies according to the model are indicated above or below the node representing that situation. The formulas verified exactly, inexactly and classically by each situation are not represented, but can be easily computed with the help of the recursive semantical clauses given earlier. It can be checked that each diagram satisfies the definition of a TM-model.
- 14.
Preliminary versions of this paper have been presented in Bochum, Louvain-la-Neuve, Milan and Prague. I would like to thank all audiences at these venues for their valuable feedback, as well as an anonymous reviewer for considerably helpful criticism and suggestions. This research was supported by the project “Intuitions in Science and Philosophy” funded by the Danish Council for Independent Research DFF 4180-00071.
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Saint-Germier, P. (2021). Hyperintensionality in Imagination. In: Giordani, A., Malinowski, J. (eds) Logic in High Definition. Trends in Logic, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-030-53487-5_6
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