Zhang, Yuxuan
(2025)
A New Formal Approach to Two-Dimensional Semantics: Building on Davies and Humberstone’s Two-Dimensional Modal Logic.
[Preprint]
Abstract
In “Two Conceptions of Necessity”, Martin Davies and Lloyd Humberstone construct a two-dimensional modal logic to formalize Gareth Evans’ distinction between superficial and deep modalities, thereby addressing Saul Kripke’s notions of “contingent a priori propositions” and “necessary a posteriori propositions”. However, Davies and Humberstone’s two-dimensional modal logic fails to account for the necessity a posteriori of identity statements involving proper names, thus falling short of satisfying the explanatory demands of two-dimensional semantics. To overcome these limitations, this paper proposes a new formalization approach for two-dimensional semantics: replacing the doubly-indexed mechanism of possible worlds with variable semantic models, transforming the vertical axis in the 2D-matrix from a designated “actual world” to specific semantic models corresponding to distinct worlds—termed “world-models”. Each possible world corresponds to a world-model that describes it, with the primary difference between world-models lying in the interpretation function’s distinct valuations to individual constants. This formal framework not only more appropriately handles Kripkean identity statements involving proper names but also aligns more closely with David Chalmers’ epistemic interpretation of two-dimensional semantics.
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