Abstract
When an event occurs which involves a change from a state \(\phi \) to a state not-\(\phi \), when does the change occur? This is known in the philosophical literature as the problem of the moment of change. I discuss a puzzle based on this problem raised by Sextus Empiricus in Against the Physicists. I compare two lines of solution, one provided by Hans Kamp’s work on the logic of change and the other inspired by Aristotle’s claim that change cannot happen at an instant. I end with a discussion of these solutions vis-à-vis the account of cessation implicatures proposed by Altshuler and Schwarzschild.
For feedback I received on an earlier version of this paper, I am grateful to Daniel Altshuler, Fabio Del Prete, Ferruccio Franco Repellini, Giuseppe Spolaore, and an anonymous referee.
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Notes
- 1.
Sextus Empiricus (1936).
- 2.
The first known instance of this problem is in Plato’s Parmenides 156c–157b and concerns the change from rest to motion.
- 3.
Here’s Aristotle’s formulation of the argument (Barnes 1984):
...everything that has changed from something to something has changed in a period of time. For suppose that a thing has changed from A to B in a now. Now the now in which it has changed cannot be the same as that in which it is at A (since in that case, it would be in A and B at once); for we have shown above that that which has changed, when it has changed, is not in that from which it has changed. If, on the other hand, it is a different now, there will be a period of time intermediate between the two (Physics, Book VI, 237\(^{a}\), 18–25).
- 4.
The formulation of rule (2) is from van Benthem (1983). Van Benthem attributes it to Dowty (1979). However, although at various points in the text Dowty seems to assume (2), his formulation does not require that intervals \(i'\) and \(i''\) overlap with i, it requires that \(i'\) precede i and \(i''\) follow i.
- 5.
Perhaps I am being overconfident in making this claim. An anonymous reviewer points out that we do say things like “From the time we are born, we are dying” for dramatic effect. So, in this sense, one might say that “Socrates dies” is true at an interval that spans from the time Socrates was born to the time he was dead. Moreover, suppose that the traffic light is green for a while, then, when it should turn red, it flickers between green and red a few times due to a malfunctioning, and eventually it settles on red. One might appropriately say in this case that the light turned red. I find the latter example more convincing than the former (one often says patently false things for dramatic effect). But I agree that determining for which interval it can be appropriately asserted that someone died at that interval or that a light turned red at that interval is to some extent a context-dependent matter. I think that this has to do with how we conceptualize events in different contexts. So, one way to state the semantic problems with representing change via the BECOME operator in (2) is that a purely interval-based approach fails to make this dependence on event conceptualization explicit (I come back to this in Sect. 7). See, however, Dowty (1979) and Landman (1991) for discussion of some ways in which one may cope with these problems in an interval semantics framework.
- 6.
Here I am assuming, with Dowty (1979), an instant-based interval semantics in which intervals are convex (gapless) sets of instants. If intervals are primitive, density would require that [\(t_{j}\) \(t_{k}\)] contain an infinite number of smaller intervals, and similar questions would arise.
- 7.
Indeed, for many philosophers (Medlin 1963; Hamblin 1969; Sorabji and Kretzmann 1976; Chisholm 1980; Kamp 1980; Mortensen 1985; Le Poidevin 2003; Priest 2006), the problem of the moment of change is stated thus: if state s described by \(\varphi \) holds before an instant \(t_{0}\) and state \(s'\) described by not-\(\varphi \) holds after \(t_{0}\), does s hold at \(t_{0}\) or does \(s'\) hold at \(t_{0}\)? For a discussion of the metaphysical debate on the moment of change both in ancient and in contemporary philosophy, see Strobach (1998).
- 8.
In Kamp’s paper, event models also contain a function A\(_1\) which maps events onto pairs consisting of a sentence of L and one of the values P, F, B, C. Since A\(_1\) plays no role in my discussion, I ignore it here.
- 9.
- 10.
A\(_{2}'\) is the function that corresponds to A\(_{2}\) in the completion M\('\) of the event model M.
- 11.
This is not the way Kamp states it, although it seems natural enough in his framework.
- 12.
Formally, we may prove this consequence as follows:
Show: if
, then
.Proof: suppose
. Then, \(a \in \) F\('\)(BQ, t). Thus, \(\exists e (e \in t\ \wedge \) A\(_{2}'\)(e, Q(a)) = B). Thus, given that any two events in t overlap, by the principle of consistency of overlapping events every e in t is such that, if A\(_{2}'\)(e, Q(a)) is defined, then A\(_{2}'\)(e, Q(a)) = B. Thus, given that A\(_{2}'\) is a function, there is no event e in t such that A\(_{2}'\)(e, Q(a)) = P. Thus,
.Show: if
, then
Proof: suppose
. Then, \(a \in \) F\('\)(BQ, t). Thus, \(\exists e (e \in t\ \wedge \) A\(_{2}'\)(e, Q(a)) = B). Thus, given that any two events in t overlap, by the principle of consistency of overlapping events every e in t is such that, if A\(_{2}'\)(e, Q(a)) is defined, then A\(_{2}'\)(e, Q(a)) = B. Thus, given that A\(_{2}'\) is a function, there is no event e in t such that A\(_{2}'\)(e, Q(a)) = F. Thus,
.By an analogous reasoning, it is possible to show that, if
, then
and
. - 13.
Indeed, Kamp (1980) states at the end of the paper:
This brings me to the last major issue which must eventually be discussed but about which too I have been almost entirely silent: the means of expressing change that are available in natural languages, in particular English. This is a complex problem, and the little that I may have achieved in this first part of the paper can at best serve as a formal setting in which various aspects of that problem might be more fruitfully discussed than would be possible without it. (p. 178)
- 14.
Instantaneity in Kamp’s theory need not be absolute: an event e’s being instantaneous depends ultimately on how the set of events in an event model is specified, which reflects the conceptual scheme adopted in singling out events.
- 15.
Of course, Kamp may question the example I used to provide an intuitive understanding of the distinction between not being the case and failing to be. In this case, however, some alternative way to motivate the distinction intuitively should be provided.
- 16.
I presented some arguments in favor of this conclusion in Zucchi (1999).
- 17.
Here I am not suggesting that Kamp’s predicate operator B is appropriate to represent progressive predicates like “dying”. Notice that one may truthfully assert that Socrates is dying while Socrates is still alive. So, “Bdead(s)” cannot mean the same as “Socrates is dying”, since “Bdead(s)” in Kamp’s theory is false at instants at which Socrates is alive (fails to be dead).
- 18.
This assumption is made explicit, for example, in [Parsons 1989, p. 219], [Parsons 1990, p. 119], [Higginbotham 2000b, p. 37], [Higginbotham 2000a, p. 117], and [Rothstein 2004, p. 155]. Parsons allows both complete and incomplete events in the denotation of the predicate “die”; if the events are complete, however, the patient is dead at the end of the event.
- 19.
- 20.
Translation (25) is not sufficient to guarantee that this condition is met, since a state may be a proper part of another state of the same kind. This is why we need to add (28), if we want to exclude that the state of being dead precedes the last moment of the becoming dead event.
- 21.
See Parsons (1980) for a way of deriving (31) compositionally.
- 22.
The English sentence in (32) seems to have a paradoxical flavor, perhaps because it suggests that there is an instant at which Socrates is neither alive nor dead. The true missing alternative, according to the account I am considering, is not this, but one which may be less concisely expressed in English by saying that Socrates died at an interval i such that he was neither alive throughout i nor dead throughout i.
- 23.
- 24.
Is there a way of weakening the ontological commitment of the revised theory further to allow an instantaneous moment of change preceding \(t_{k}\) at which PI holds, namely at which Socrates is neither dead nor fails to be dead? Suppose \(t_{0}\) were such an instant. Since \(t_{0}\) is the moment of change, Socrates is alive before \(t_{0}\) and dead after \(t_{0}\). Then, if time is dense, there is an instant between \(t_{0}\) and \(t_{k}\) at which Socrates is dead. So, supposing that a moment like \(t_{0}\) exists would require to give up principle (28). However, under the revised theory, this would have the undesirable consequence that one may die while being already dead.
- 25.
See Dowty (1979) for a detailed discussion of this point.
- 26.
Here I am ignoring relevant literature on progressive achievements. While Moens and Steedman (1988), Rothstein (2004), and Martin (2011) agree on the view that progressive achievements involve coercing achievement predicates into derived accomplishment predicates, other authors, for example Piñón (1997) and Gyarmathy (2015), do not. Yet, as I understand it, both Piñòn and Gyarmathy agree (with what I am going to assume, namely) that the events described by achievement predicates do not include as parts the gradual processes leading to their occurrence.
- 27.
I am assuming that temporal at-phrases are VP modifiers.
- 28.
The account is developed further in Altshuler (2016).
- 29.
- 30.
Of course, the speaker answering (43b) to (43a) might also implicate that Scotty is no longer anxious, besides implicating that the doctor thinks so, but the latter implicature may occur without the former, for example in case it is common knowledge that the speaker believes the doctor to be incompetent.
- 31.
A third line of solution which I did not discuss here is suggested by the accounts proposed by Piñón (1997) and Gyarmathy (2015). According to these authors, the event described by the sentence “Socrates died” is an instantaneous event (or an event taken as instantaneous in context, in Gyarmathy’s view) occurring at a time when Socrates was dead (when the state of Socrates’ being dead begins). This proposal bites the bullet by rejecting Sextus’ premise C and accepting that Socrates died when he was dead.
, then
.
. Then, 
.
, then
. Then, 
.
, then
and
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Zucchi, S. (2019). Did Socrates Die? A Note on the Moment of Change. In: Altshuler, D., Rett, J. (eds) The Semantics of Plurals, Focus, Degrees, and Times. Springer, Cham. https://doi.org/10.1007/978-3-030-04438-1_13
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