Abstract
I will show how a metaphysical problem of Arthur Prior’s can be solved by a logical tool he developed himself, but did not put to any foundational use: metric logic. The broader context is given by the key question about the metaphysics of time: Is time tenseless, i.e., is time just a structure of instants; or is time tensed, because some facts are irreducibly tensed? I take sides with Prior and the tensed theory. Like him, I therefore I have to deal with a more specific metaphysical question: How can the instants of tenseless time be reduced to tensed facts? This is the point where, on the technical level, hybrid logic and metric logic come in. For present purposes, both can be seen as species of tense logic; and both are creations of Prior. In his argument for the tensed theory of time, Prior used hybrid tense logic to reduce instants. But, as he himself pointed out, this reduction runs into deep problems, because it immediately generalizes to other categories, for example and most importantly to persons. My main aim is to show that metric logic does not run into similar difficulties: It will help the tensed theory reduce instants, but it leaves persons untouched. I will also give reasons for preferring a metric to a hybrid logic of time that are independent of the metaphysical issue of reduction, but concern temporal reasoning, natural language semantics, and the epistemic side of time-keeping.
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Notes
Taken together, logical and metaphysical reduction amount to the converse of what Russell called logical construction.
Prior’s project and Prior’s problem are described in detail in Blackburn (2006), “Arthur Prior and hybrid logic”.
Since Prior, much work has been done on hybrid and metric logic from the perspective of the logician qua logician (cf., e.g. the work of Blackburn, as well as Montanari and de Rijke 1997 and Cresswell 2013). In my comparison of these two families of logical systems I will nevertheless (aside from a few exceptions) stick to Prior’s own contributions. They give the logician qua metaphysician enough to work with.
Here, ‘\(\Diamond \)’ and ‘\(\Box \)’ are the universal modalities always and sometimes.
Strictly speaking, it is a theory and not a logic, because it is hardly a logical matter that the contingent propositions c is true exactly once.
More precisely, what is needed is a speech act of someone of calendarian authority. E.g., on the day now known as October the 15th of the year 1582, Pope Gregory XIII proclaimed something like: “Today is October the 15th of the year 1582” and thereby instituted our current system of time-keeping. Prior observed this: “To say that a certain event occurred in A.D. 1066 is to say, approximately, that it occurred 1066 years after the birth of Christ (or the putative birth of Christ, i.e. after a time so many years before the Church gave the calendar its present shape).” (Prior 1967, p. 104; my emphasis).
‘\(\Diamond \)’ and ‘\(\Box \)’ are again the universal modalities always and sometimes.
He wrote about a formula like ‘\(\exists t \ P_{t} \ p\)’ that “quantifications of this sort do not imply that intervals are entities. [...] ‘It was the case at some time or other that p’ is just a generalization of remarks like ‘It was the case this time yesterday that p’, in which there are no named entities except any which may be named by expressions within p.” (Prior 1967, p. 96).—To put this point differently, we should understand an expression like ‘\(\exists t \ P_{t} \ p\)’ as a quantification not into subscript position, but into operator position.
I admit that my argument here is addressed to a restricted audience: I presuppose not only a tensed orientation, but also that metric and hybrid tense logic are the main candidates.
The asterisk ‘*’ marks that a verb is used tenselessly.
As
, these arguments are of the logical forms p, \(P_{t} \ c\)
\(\vdash \)
\(\Diamond (p \wedge P_{t} \ c)\) and \(\Diamond (p \wedge P_{t_{1}} \ c)\), \(P_{t_{2}} \ c\)
\(\vdash \)
\(F_{t_{2}-t_{1}} \ p\). As \(\Diamond (c \wedge \lnot Pc \wedge \lnot Fc)\) (Prior 1967, p. 105), they are correct according to the anchored metric tense logic used here (Prior 1957, p. 13, 1967, p. 97).Note that requiring interpretational naturalness is not the same as doing natural language semantics (Sect. 8).
This argument is similar to Kripke’s modal argument against the description theory of proper names (Kripke 1980, 48f.).
Or should it rather be ‘\(\exists i \ F(i \wedge Pp)\)’, with the troublesome quantification into sentence position made explicit?
In the case of metric tense logic, the neutral change of perspective is expressed by ‘\(F_{0}\)’ or ‘\(P_{0}\)’.
I.e., each operator is self-dual, or, to use Blackburn’s phrase, each operator is “both is a box and a diamond” (Blackburn 2006, p. 346).
For this condition to capture (Double Generality), the operators ‘\(\Box \)’ and ‘\(\Diamond \)’ must of course be universal. Without universal modalities, (Double Generality) cannot be formulated internally.
This desideratum could be strengthened, for example by requiring the group to be abelian and ordered, as in Montanari and de Rijke (1997).
E.g., ‘Here, now, and at-this-velocity, it is raining.’
As the Earth is not an inertial frame, ‘Here is Greenwich’ and ‘At-this-velocity is Earth’ are only rough examples.
Ancestry is understood as the basis for the relation of accessibility of a modal logic in Pleitz and Strobach (2014)—but for one that does not satisfy (Identificatoriness).
What this means for the openness of the future is future work.
I call this kind of anti-realism ‘Buddhist’ not only because I sympathize, but also to mark the important difference to eliminative materialism: It reduces persons not to physical entities that are given in an external way, but to internal perspectives as that expressed in ‘It Priorizes’, which can probably be dissolved further into all those perspectival statements that characterize a self-less stream of consciousness: ‘It itches’, ...
, these arguments are of the logical forms p, References
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Acknowledgments
I would like to thank the following people for discussions that have helped me develop this paper: Patrick Blackburn, Ben Caplan, Andreas Hüttemann, Johannes Korbmacher, Roman Klauser, Thomas Müller, Rosemarie Rheinwald, Peter Rohs, Raja Rosenhagen, David Sanson, Oliver Scholz, Stewart Shapiro, Peter Simons, Niko Strobach, and Neil Tennant.
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Pleitz, M. Solving Prior’s problem with a Priorean tool. Synthese 193, 3567–3577 (2016). https://doi.org/10.1007/s11229-015-0931-x
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DOI: https://doi.org/10.1007/s11229-015-0931-x