Abstract
In this chapter we devise a spacetime logic and argue that temporaryism must give way to spatiotemporaryism, which latter construes variation in what exists as variation across spacetime. In Sect. 9.1 we argue that much of the rationale for thinking, in a prerelativistic setting, that what exists varies across time, should survive the finding that there is no absolute and total temporal order and rationalise the corresponding thought that what exists varies across spacetime. In Sect. 9.2 we introduce a spacetime-sensitive language and a spacetime logic with operators and relations defined over the fourfold causal structure of spacetime. In Sect. 9.3 and Sect 9.4 we use these tools to articulate and compare competing spatiotemporaryist ontologies and contrast them with spatiopermanentism according to which what exists does not vary across spacetime. In Sect. 9.5 we show which relativistic views can naturally be taken to correspond to which classical theories of time.
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Notes
- 1.
- 2.
(AR1) is used for proving the first item on this list, axioms (AR3), (AR4) and (AR2) are used for proving the third item, and (AR5) for proving the fourth item. The second item on the list already follows from the definition of ‘●’.
- 3.
Suppose @xφ. Then by (RD3R), @x¬¬φ. By (A11R), it follows that Sx & ¬@x¬φ. Using (A12R), we get ⚪φ.
- 4.
Assume E!m & E!n & Sm & Sn. Assume for reductio m prec n. By (P2R), (P3R), (A34R) and (A23R), this yields @n(E!n & ▴¬E!n & ▾¬E!n & ¬E!m). By (T2R), this yields ⚪(E!n & ▴¬E!n & ▾¬E!n & ¬E!m). Given our initial assumption, we can derive ¬▵(E!n & ▴¬E!n & ▾¬E!n & ¬E!m), ¬▿(E!n & ▴¬E!n & ▾¬E!n & ¬E!m) and ¬(E!n & ▴¬E!n & ▾¬E!n & ¬E!m). This leaves as the only option ◃(E!n & ▴¬E!n & ▾¬E!n & ¬E!m) which, it would seem, can only be rejected if (BO+) is assumed.
- 5.
Suppose φ → ◂φ is a theorem. By (AR1) and (RR), then, ◃φ → ◃◂φ is a theorem. Using (AR5) and again (AR1) and (RR), ◃◂φ → φ is a theorem. So, ◃φ → φ is a theorem. Conversely, suppose the latter is a theorem. Then by (AR1) and (RR), so is ◂◃φ → ◂φ. By (AR5), then, φ → ◂φ is a theorem.
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Correia, F., Rosenkranz, S. (2018). Spatiotemporaryism. In: Nothing To Come. Synthese Library, vol 395. Springer, Cham. https://doi.org/10.1007/978-3-319-78704-6_9
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