Abstract
The standard inferentialist approaches to modal logic tend to suffer from not being able to uniquely characterize the modal operators, require that introduction and elimination rules be interdefined, or rely on the introduction of possible-world like indexes into the object language itself. In this paper I introduce a hypersequent calculus that is flexible enough to capture many of the standard modal logics and does not suffer from the above problems. It is therefore an ideal candidate to underwrite an inferentialist theory of meaning for modal operators. Here I treat specifically the modal logics K, D, T, S4, B, and S5. I show that the calculi are adequate for each set of models, and show that they meet a large set of criteria that are generally thought necessary for a calculus to underwrite a theory of meaning.
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Notes
It is worth noting that a set of rules meeting these constraints may still fail to confer meaning. These formal constraints are necessary but though perhaps not sufficient for a set of rules to be meaning determining.
Here by ‘status’ I mean to speak generally enough to capture whatever someone might take to be their central meaning theoretic notion.
This is different from Cut eliminability wherein an algorithm for generating a cut free proof from any proof has been provided.
Poggiolesi (2009) has developed tree-hypersequent calculi that do all of the above and are known to be cut-admissible. The novelty of this framework is that instead of relying on the datatype of trees of sequents to directly characterize a set of models this framework shows that the datatype of lists of sequents is sufficient to characterize a range of prominent modal logics.
They also uniquely characterize the extensional connectives.
For the purposes of this argument, I forego consideration of how the structural rules of a sequent calculus contribute to the meaning of the logical constants.
There is an objection to this approach along these lines: Suppose that a priori knowability and alethic necessity obey S5. The result of Sect. 4.2 establishes that they are indistinguishable in a single hypersequent system. This speaks against the necessity of uniqueness for modal operators. The response to this is suggested by Łukasiewicz (1953) who says that these expressions are like “...identical twins who cannot be distinguished when met separately, but are instantly recognized as two when seen together”. The previous quote is metaphorical, but it suggests a solution to the problem. As in Restall (2012) these two operators can be embedded in the right structure so that both obey S5 but they do not collapse into a single operator.
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Parisi, A. A Hypersequent Solution to the Inferentialist Problem of Modality. Erkenn 87, 1605–1633 (2022). https://doi.org/10.1007/s10670-020-00264-x
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DOI: https://doi.org/10.1007/s10670-020-00264-x