Abstract
The history of reasoning about time is filled with paradoxes and conundra; contemporary physics is no exception. Logics in general, and more specifically temporal logics, represent a rigorous formal tool in order to solve or clarify problems of this kind. In what follows we will first explain, from the point of view of philosophical logic, what is a paradox and what should count as a solution to it. After that we will illustrate A.N. Prior’s formalization of the traditional paradox of future contingency and determinism. Then we will focus on two modern paradoxes – the twin paradox and the time travel paradox – and show how an adequate temporal logic can help their framing and understanding.
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Notes
- 1.
Knowing that A implies that A is true and knowing that not-A implies that A is false. This is an intuitive property of knowledge that the medievals expressed with the motto nihil scitum nisi verum.
- 2.
Of course, (a) must hold for the argument to be conclusive and that is where Aristotle and Diodorus diverge in their analysis. Most commentators read the whole chapter IX of Aristotele (1941) as an attempt to consistently reject (a) and therefore discard the Diodorean argument.
- 3.
Our object of analysis here is the context-independent (or absolute) notion of time, as opposed to the context-dependent (or phenomenological) one (see Ismael , Chap. 2 of this volume). The latter brings in additional elements of complexity, although, as argued by Ismael , it can be framed in rigorous terms.
- 4.
The twin paradox dates back to the very early days of relativity theory. Einstein himself stressed as a “peculiar consequence” of the theory the fact that “if at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B” (in Einstein 1905 § 4, see also Einstein 1911). This scenario was popularized by Langevin (1911) with the more vivid description of a travel back and forth from Earth in a projectile. Finally, it was Weyl (1918) who upgraded the example with two twins aging differently, one staying on Earth and one travelling in the projectile. However, neither Einstein nor Langevin nor Weyl called it a paradox, since the situation is perfectly consistent with the theory. The alleged paradoxality originated mostly in philosophical discussion, due to the objection that if motion is relative then the situation should be symmetric – from the local perspective of each twin, the other is the one moving – and therefore there seems to be no reason why the twins should age at a different pace. The reader may consult During (2014) for a detailed historical account of how the example developed and crystallized into a paradox.
- 5.
The original source of the paradox, and of its name, is the science fiction novel by Barjavel (1944) whose main character , an imprudent time traveller, kills his grandfather before the latter meets the time traveller’s grandmother.
- 6.
This was not because Prior was unaware of special and general relativity, the point is that a Newtonian framework is detailed enough for dealing with traditional paradoxes and conundrums. Indeed, when dealing with a paradox, a good logical analysis consists first and foremost in simplifying the picture in a way that is fine-grained enough for the problem to arise and hopefully to articulate a solution. In the case of determinism and free-will the further complexity induced by relativistic space-time is to a large extent superfluous and therefore negligible.
- 7.
- 8.
See De Interpretatione chap. IX.
- 9.
- 10.
See Baudry (1950).
- 11.
Our approach diverges from Prior’s insofar as his construction was mostly syntactic and employed semantics just as an auxiliary tools. Prior had a specific philosophical motivation for this. However, a semantic approach fits better with our explanatory purpose.
- 12.
Allowing multiple histories is not merely a logician’s trick: in quantum mechanics a many-world interpretation is actually provided which allows parallel universes as in our scenario.
- 13.
In what follows we will not fully adhere with Belnap’s presentation. For example, Belnap does not provide truth clauses for temporal operators and does not introduce operators for possible reference frame s . However, to understand some puzzles of space-time relativity, we need to give at least a partial account of how these operators should work. For this purpose we will (freely) borrow from Strobach (2007).
- 14.
This is considered, for various reasons, to be the most coherent option by Belnap (1992).
- 15.
The truth clause of the operator P * n is defined analogously to that of F * n (see previous subsection), i.e. P * n φ is true at (e,h,f) if and only if φ is true at (e’,h,f) for some e’ such that e’ < < e and d(e,e’) = n w.r.t. the frame f.
- 16.
- 17.
This solution is pretty much the same as the one provided by Lewis (1976). In the same paper, Lewis deems as logically impossible to change one’s own past. However, such logical impossibility refers to changing one and the same history by making, e.g., c and ¬c true there. As we have shown, a contradiction may indeed be avoided only by means of two alternative histories. Of course, such an option generates additional problems of trans-world identity and causation which are the object of ongoing philosophical disputes.
References
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Acknowledgements
The author is grateful to Ilaria Jemos, Iacopo Carusotto and Jeroen Smid for useful theoretical insights and contributions to clarification, and to Andrew McFarland for a thorough language check of the manuscript. This research was supported by the Swedish Research Council.
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Proietti, C. (2017). Time of Logics and Time of Physics. In: Bouton, C., Huneman, P. (eds) Time of Nature and the Nature of Time. Boston Studies in the Philosophy and History of Science, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-319-53725-2_3
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