We will start our analysis of metaphysical aspects of time with temporal relations. Temporal relations can be defined on both things and events, but because things vary greatly with respect to their duration, it is more convenient to choose events instead. Another simplifying assumption we have to make is that we will consider events as being momentary (point-like): they are assumed to occupy a single moment in time, not an interval (no duration). The main temporal relation is the earlier-than relation E. Julius Caesar’s death is earlier than the fall of Constantinople, and the fall of Constantinople is earlier than the storming of the Bastille. The main formal features of the earlier-then relation are as follows: it is irreflexive (no event is earlier than itself), asymmetric (if x is earlier than y, then y is not earlier than x), and transitive. These three characteristics ensure that the relation of being earlier is a strong ordering of the set of all events. To this we will add the requirement of linearity, which can be explicated as follows. First let us define the following relation R: R(x, y) iff neither E(x, y), nor E(y, x) (neither x is earlier than y, nor y is earlier than x). The requirement of linearity of the relation E amounts to the condition that R be a relation of equivalence, i.e. reflexive, symmetric and transitive. If that is the case, we can call R the relation of simultaneity. An example of a situation when E is not linear is a so-called branching time. If events are ordered on a tree with different branches pointing towards the future, the conditions of irreflexivity, asymmetricity and transitivity of E are satisfied, but the linearity is violated, for events located on different branches are not comparable (one is neither earlier than, nor later than, nor simultaneous with the other one). More precisely, the condition of the transitivity of the relation R is violated in this case, since we can choose two events x and z on one branch and a third one y on another branch, from which it clearly follows that R(x, y) and R(y, z), but not R(x, z). Some philosophers insist that the set of all events should be given the structure of a tree due to indeterminism, but we will continue to assume the linearity condition.
The relation of simultaneity can help us define an important category of temporal objects: moments (temporal instants). A moment at which an event x occur is just the set of all events simultaneous with x. Moments are equivalence classes of events with respect to simultaneity. Notice that the ordering relation “earlier than” between events can be naturally extended for moments. Moment m is earlier than moment n iff all events participating in m are earlier than the events constituting n. This definition is formally correct, because the relation E is invariant with respect to substitution of simultaneous events. One consequence of the proposed definition of moments is that there can be no empty moments (instants with no events). This has serious consequences for the controversy between two ontological positions regarding the nature of time: absolutism (substantivalism) and relationism. We will discuss this problem later.
Another issue is the problem of time measurement. How to measure the duration of an interval between two events x and y? One possibility is to use a cyclic (repeatable) process, for instance a pendulum, a water clock, or a sundial, and to count how many cycles happen during the interval from x to y. But the main problem of this approach is how to make sure that the subsequent cycles have the same durations. If we use another cyclic process to prove that our initial measuring device was uniform, we will end up in a regress. A solution is to adopt a conventionalist answer to the question of how to compare lengths of different time interval. But accepting conventions does not imply total arbitrariness. We should adopt conventions which make sure that the fundamental laws of physics involving time (such as the laws of Newtonian mechanics) have the simplest possible mathematical form.
Some philosophers insist that there is one important aspect of time which is missing from our analysis so far. It is the dynamical aspect of time expressed in the division of events into three spheres: past, present and future. The passage of time is not included in the earlier-than relation. It can be expressed in the observation that events move from being future into being present, and finally they turn into past events. John McTaggart introduced a fundamental distinction, to this day referred to in virtually all publications on the subject of time. It is the distinction between the A series and the B series. The B series is the set of all events together with the relation “earlier than”, while the A series is the set of all events divided into past events, present events and future events. An important difference between these two interpretations of time is that the B interpretation can be given exclusively in a tenseless language, while the A theory requires the use of tensed forms of verbs. For instance it is correct to say that the Battle of Hastings is earlier that the Battle of Waterloo, where the verb “be” has an atemporal sense, not relativised to the present moment. In contrast, the Battle of Hastings is now past, but it was present and had been future, while my current lecture is present, was future and will be past. For McTaggart this fact shows that the B series is static, “frozen in time”, eternal, while the A series is dynamic, moving, changing. We should also note that the A series descriptions cannot be definitionally reduced to the B type expressions. Using the relation of being earlier than we can define the notions of past present and future only relatively to a given event (moment). The past of an event x is the set of all events earlier than x; the present of x is the set of all events simultaneous with x, and the future of x is the set of all events such that x happens earlier from them. On the other hand, the task of reducing the B-series to the A-series has greater chances of success. For instance, we could try to give the following reductive definition of the relation E: x is earlier than y iff there is a moment of time at which x is present and y is future.
McTaggart makes two significant claims regarding the two approaches to the concept of time. One is that the existence of the A series is necessary for time to exist, and therefore for the existence of the B series as well. The other claim is more radical: McTaggart insists that the A series is contradictory. From these two claims it follows that time does not exist. McTaggart fully embraces this consequence. McTaggart argues in support of the first claim as follows. His main point is that the A series is necessary in order to express the notion of change. For McTaggart the notion of change applies to events only: each event changes from being in the far future to being in the closer future, then to being present, and then to being past. But an objection can be raised that there is a legitimate notion of change which is applicable to things, not events, and which can be expressed in the B-theory. B. Russell used the following example: a poker put in a fireplace changes from being cold at t1 to being hot at t2. More generally, a change is the fact that a given sentences about an object is true at t1 and false at a later time t2. This interpretation of change does not require the A-series. But McTaggart retorts that the change described in Russell’s example is spurious. In fact there is no real change here at all, since it is always true that the poker is cold ad t1 and hot at t2. To bolster his claim, McTaggart uses an argument from analogy. Consider the zero meridian and two points on it: one m1 in England and one m2 in France. The meridian can be ordered in the same way events (moments) are ordered in the B-series. But now the sentence “This point on the meridian lies in England” is true at t1 but false at t2. So, according to Russell’s definition, there is a change happening here. But clearly we see that nothing really changes, hence Russell’s definition is incorrect.
In response to McTaggart argument it can be pointed out that the analogy he uses is incomplete and therefore it does not warrant the conclusion. First, the meridian case lacks a counterpart of the poker in Russell’s example: a thing that retains its identity in spite of the change in properties. The statement which is supposed to be true at m1 but false at m2 is not about any thing which exists at both points. One may try to correct McTaggart’s argument in the following way: suppose that the required counterpart of the thing in the temporal case is the entire Great Britain, and the sentence considered is “Great Britain is sunny”, which happens to be true at some point on the meridian but false at a different point. But even here the analogy is not sufficiently strong: we couldn’t claim by any stretch of the imagination that the whole Great Britain is wholly present at any single point on the meridian, whereas it is typically assumed that things are wholly present at temporal instants. The second objection to McTaggart’s argument is that it arbitrarily selects one method of ordering points on the meridian (either from the South Pole to the North Pole or vice versa), whereas the temporal case has an objective temporal direction independent of our decision.
Readings:
B. Garrett, "Time: The fundamental issue", pp. 69-82, What is this thing called metaphysics?
Wednesday, February 24, 2010
Thursday, February 18, 2010
Events
Events constitute a separate category of spatiotemporal objects which is different from the category of things. The main difference between events and things lies in their different ways of existing in time. Things, according to the common intuition, persist in time, while events happen, occur, or take place. Things are continuants, while events are occurents. This difference can be explained as follows. Compare the battle of Waterloo, which is an event, with Napoleon, a thing. Both Napoleon and the battle of Waterloo coexisted during a certain period of time, but at each moment of the battle Napoleon was fully present, while only a small part of the battle takes place at a given moment. Events are not repeatable – they occur as a whole only once – but things exist at different times without losing their identity. (It has to be added though that there are non-standard conceptions of how things persist in time, according to which at a given moment of time only a part of the thing is present, exactly as in the case of events. We will talk more about this later.)
Events are ubiquitous in natural language, as well as in the language of philosophy and of science. We talk without hesitation about battles, treaties, births, deaths, weddings, earthquakes etc. In philosophy events are typically considered as arguments of the causal relation. It is also common to talk about mental events. In physics events of coincidence play an important role in relativity theory, while measurements constitute the foundation of quantum mechanics. It is difficult to imagine a language which would not make reference to events. And yet some philosophers deny that events exist as a separate category of entities. To counter this claim, Donald Davidson has suggested a linguistic argument in support of the admission of events into one’s ontology. Consider the following sentence: (1) Jones slowly buttered a piece of toast with a knife in the kitchen at midnight. It is quite obvious that from this sentence we can logically derive several consequences, for instance that Jones buttered a piece of toast, that Jones buttered a piece of toast at midnight, or that Jones did something with a knife in the kitchen at midnight. And yet it is extremely difficult to formalise these valid inferences within standard first-order logic when we assume that the variables of our language range over things only. For example, the statement “Jones walked slowly” is formalised as P(a), where P represents the complex predicate “walks slowly” and a stands for the name “Jones”. But this method of interpretation treats the sentence “Jones walked” as containing a new predicate “walks” (Q) different from the adverbially modified expression “walks slowly”, and therefore cannot account for the unquestionable entailment between the two sentences (formula Q(a) cannot be logically derived from P(a)).
Davidson suggests that we should rephrase the above sentences in a language containing reference to events. The initial sentence (1) can be interpreted as follows: “There is an x such that x is a buttering of a piece of toast, x is done by Jones, x is done slowly, x is done with a knife, x is done in the kitchen, x is done at midnight”. By eliminating some elements of the multiple conjunction we can easily obtain required logical consequences, such as “There is an x such that x is a buttering of a piece of toast, and x is done by Jones” (“Jones buttered a piece of toast”), or “There is an x such that x is done by Jones, x is done with a knife, x is done in the kitchen and x is done at midnight” (“Jones did something with a knife in the kitchen at midnight”).
Accepting events as part of our ontology requires that we be able to give some criteria of identity and difference for them. When are two events numerically identical? One possible answer may be that the sufficient and necessary condition for the identity of events is their spatiotemporal coincidence. But there are convincing examples of numerically distinct events which nevertheless coincide in space and time. A typical example is that of a metal sphere which simultaneously rotates around its axis and heats up. The events of rotating and of heating up are clearly numerically distinct, and yet they occupy the same region of spacetime. One way of saving this intuition is to adopt Davidson’s causal criterion of identity: events x and y are numerically identical iff x and y have the same causes and the same effects. Clearly the rotation of the sphere and its heating up have different causes, and different effects (for instance the former causes the sphere to flatten a bit due to the centrifugal forces, while the latter causes it to expand uniformly). But there is one big problem with Davidsonian criterion – it is namely circular. Let us suppose that we have events x and y of which we don’t know yet whether they are identical or distinct, and let us suppose that x is caused by another event u, while y is caused by w. For simplicity’s sake we assume that x and y don’t stand in causal relation to any other events. Now, in order to decide whether x = y, we have to verify whether their causes u and w are one or two events. But to do that we have to apply Davidson’s criterion again, and this requires that we know whether x and y are identical (as they are effects of u and w). Here the circle closes, and apparently we have no way of solving our initial problem.
However, it turns out that under certain assumptions it is actually possible to decide in each case the issue of identity for a group of events using Davidson’s criterion. Here I follow the suggestion made by Leon Horsten. Suppose that we have a graph containing points representing descriptions of events (not events themselves!) and arrows representing causal relation. Moreover, let us assume that our graph satisfies the condition of completeness, i.e. for each pair of events e and e’, if e is a cause of e’, then each description of e is connected by an arrow with each description of e’. Under this assumption it turns out that each graph satisfying Davidson’s criterion is solvable, i.e. for each two descriptions it is decidable whether they refer to one or two distinct events. However, it may be pointed out that the assumption of completeness is too strong (if a graph is complete, this fact by itself already fixes some identity relations). A more reasonable assumption is that of semi-completeness: for all events e and e’, if e is a cause of e’, then each description of e is connected by an arrow with some description of e’. But it can be showed that semi-complete graphs are not always solvable, and therefore the problem of circularity remains.
Jaegwon Kim proposed a different interpretation of events as property exemplifications. More specifically, an event for Kim is a triple <a, P, t>; where a is an object, P is a property, and t is a time at which a possesses P. From this definition it follows that two events are identical iff they happen on the same object, at the same time, and they involve the same property. The last requirement ensures that the rotation and the heating of the sphere are numerically distinct. But Kim’s conception has several controversial consequences. First of all, it multiplies events beyond what is ordinarily acceptable. If Jones is walking slowly, his walking and his walking slowly constitute two distinct events (actually, there are as many different events of walking involved as there are ways to describe the individual style of Jones’ walking). This fact can actually threaten the analysis of logical inferences proposed by Davidson and sketched above, as in each sentence we are talking about a different event. Moreover, according to Kim’s approach it is an essential feature of an event that it occurs on a given object, at a given time, and that it involves a given property. From this it follows that my lecture on ontology given on Wednesday, February 17, at 11:30 could not have been given by someone else, could not have been a rock concert, and could not have started five minutes later. Especially the last consequence seems to be rather controversial. Other criticism of Kim’s conception is based on the observation that events can involve more than one object (relational events) or no object at all (spontaneous excitations of vacuum predicted in quantum field theory).
Event-ontologies are based on the assumption that events are the fundamental kind of objects and that other categories of objects can be reduced to events. According to one type of event-ontology, things are just sequences of events. A person, for instance, is a collection of all events from his/her birth to the death. It is worth noticing that such a reductive definition cannot be accepted by Kim, for in his conceptions events are defined in terms of things, so there would be obvious circularity. Alternatively, we could interpret events as consisting of properties and moment of time only, or we could rely on Davidson’s criterion, provided that its own circularity problem could be overcome.
Readings:
E.J. Lowe, Chapter 12 "Actions and events", pp. 214-231, A Survey of Metaphysics.
M.J. Loux, "Facts, states of affairs, and events", pp. 142-150, Metaphysics. A Contemporary Introduction.
Events are ubiquitous in natural language, as well as in the language of philosophy and of science. We talk without hesitation about battles, treaties, births, deaths, weddings, earthquakes etc. In philosophy events are typically considered as arguments of the causal relation. It is also common to talk about mental events. In physics events of coincidence play an important role in relativity theory, while measurements constitute the foundation of quantum mechanics. It is difficult to imagine a language which would not make reference to events. And yet some philosophers deny that events exist as a separate category of entities. To counter this claim, Donald Davidson has suggested a linguistic argument in support of the admission of events into one’s ontology. Consider the following sentence: (1) Jones slowly buttered a piece of toast with a knife in the kitchen at midnight. It is quite obvious that from this sentence we can logically derive several consequences, for instance that Jones buttered a piece of toast, that Jones buttered a piece of toast at midnight, or that Jones did something with a knife in the kitchen at midnight. And yet it is extremely difficult to formalise these valid inferences within standard first-order logic when we assume that the variables of our language range over things only. For example, the statement “Jones walked slowly” is formalised as P(a), where P represents the complex predicate “walks slowly” and a stands for the name “Jones”. But this method of interpretation treats the sentence “Jones walked” as containing a new predicate “walks” (Q) different from the adverbially modified expression “walks slowly”, and therefore cannot account for the unquestionable entailment between the two sentences (formula Q(a) cannot be logically derived from P(a)).
Davidson suggests that we should rephrase the above sentences in a language containing reference to events. The initial sentence (1) can be interpreted as follows: “There is an x such that x is a buttering of a piece of toast, x is done by Jones, x is done slowly, x is done with a knife, x is done in the kitchen, x is done at midnight”. By eliminating some elements of the multiple conjunction we can easily obtain required logical consequences, such as “There is an x such that x is a buttering of a piece of toast, and x is done by Jones” (“Jones buttered a piece of toast”), or “There is an x such that x is done by Jones, x is done with a knife, x is done in the kitchen and x is done at midnight” (“Jones did something with a knife in the kitchen at midnight”).
Accepting events as part of our ontology requires that we be able to give some criteria of identity and difference for them. When are two events numerically identical? One possible answer may be that the sufficient and necessary condition for the identity of events is their spatiotemporal coincidence. But there are convincing examples of numerically distinct events which nevertheless coincide in space and time. A typical example is that of a metal sphere which simultaneously rotates around its axis and heats up. The events of rotating and of heating up are clearly numerically distinct, and yet they occupy the same region of spacetime. One way of saving this intuition is to adopt Davidson’s causal criterion of identity: events x and y are numerically identical iff x and y have the same causes and the same effects. Clearly the rotation of the sphere and its heating up have different causes, and different effects (for instance the former causes the sphere to flatten a bit due to the centrifugal forces, while the latter causes it to expand uniformly). But there is one big problem with Davidsonian criterion – it is namely circular. Let us suppose that we have events x and y of which we don’t know yet whether they are identical or distinct, and let us suppose that x is caused by another event u, while y is caused by w. For simplicity’s sake we assume that x and y don’t stand in causal relation to any other events. Now, in order to decide whether x = y, we have to verify whether their causes u and w are one or two events. But to do that we have to apply Davidson’s criterion again, and this requires that we know whether x and y are identical (as they are effects of u and w). Here the circle closes, and apparently we have no way of solving our initial problem.
However, it turns out that under certain assumptions it is actually possible to decide in each case the issue of identity for a group of events using Davidson’s criterion. Here I follow the suggestion made by Leon Horsten. Suppose that we have a graph containing points representing descriptions of events (not events themselves!) and arrows representing causal relation. Moreover, let us assume that our graph satisfies the condition of completeness, i.e. for each pair of events e and e’, if e is a cause of e’, then each description of e is connected by an arrow with each description of e’. Under this assumption it turns out that each graph satisfying Davidson’s criterion is solvable, i.e. for each two descriptions it is decidable whether they refer to one or two distinct events. However, it may be pointed out that the assumption of completeness is too strong (if a graph is complete, this fact by itself already fixes some identity relations). A more reasonable assumption is that of semi-completeness: for all events e and e’, if e is a cause of e’, then each description of e is connected by an arrow with some description of e’. But it can be showed that semi-complete graphs are not always solvable, and therefore the problem of circularity remains.
Jaegwon Kim proposed a different interpretation of events as property exemplifications. More specifically, an event for Kim is a triple <a, P, t>; where a is an object, P is a property, and t is a time at which a possesses P. From this definition it follows that two events are identical iff they happen on the same object, at the same time, and they involve the same property. The last requirement ensures that the rotation and the heating of the sphere are numerically distinct. But Kim’s conception has several controversial consequences. First of all, it multiplies events beyond what is ordinarily acceptable. If Jones is walking slowly, his walking and his walking slowly constitute two distinct events (actually, there are as many different events of walking involved as there are ways to describe the individual style of Jones’ walking). This fact can actually threaten the analysis of logical inferences proposed by Davidson and sketched above, as in each sentence we are talking about a different event. Moreover, according to Kim’s approach it is an essential feature of an event that it occurs on a given object, at a given time, and that it involves a given property. From this it follows that my lecture on ontology given on Wednesday, February 17, at 11:30 could not have been given by someone else, could not have been a rock concert, and could not have started five minutes later. Especially the last consequence seems to be rather controversial. Other criticism of Kim’s conception is based on the observation that events can involve more than one object (relational events) or no object at all (spontaneous excitations of vacuum predicted in quantum field theory).
Event-ontologies are based on the assumption that events are the fundamental kind of objects and that other categories of objects can be reduced to events. According to one type of event-ontology, things are just sequences of events. A person, for instance, is a collection of all events from his/her birth to the death. It is worth noticing that such a reductive definition cannot be accepted by Kim, for in his conceptions events are defined in terms of things, so there would be obvious circularity. Alternatively, we could interpret events as consisting of properties and moment of time only, or we could rely on Davidson’s criterion, provided that its own circularity problem could be overcome.
Readings:
E.J. Lowe, Chapter 12 "Actions and events", pp. 214-231, A Survey of Metaphysics.
M.J. Loux, "Facts, states of affairs, and events", pp. 142-150, Metaphysics. A Contemporary Introduction.
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