McTaggart’s argument for the thesis that the A-series involves a contradiction is more complicated, and we will be only able to give its rough outline here. The argument starts with the unquestionable assumption that the three spheres constituting the A-series – the past, present and the future – are mutually exclusive. And yet McTaggart claims that the existence of the A-series requires that each event be past, present and future; thus a contradiction ensues. A typical response to this claim is that events are past, present and future not simultaneously, but in succession. The battle of Waterloo is past, but was present, and had been future. My current lecture is present, but was future and will be past. But McTaggart demands that we explain precisely what we mean by the words “was”, “had been” or “will”. One possible explication is as follows: an event x was present means that x is present at some past moment. Similarly, an event x will be past if and only if x is past at some future moment. But now we can observe that we have applied the distinction among the past, present and the future to moments, and again it can be claimed that the A-series requires that every moment is past, present and future. To avoid this conclusion we can only repeat the same procedure: we can distinguish between the situation in which a moment m was present (past, future), is present (past, future) and will be present (past, future). In order to explain the use of grammatical tenses, we have to appeal yet again to the second-order past, present and future moments at which the first-order moments can be classified as present, past and future without a contradiction, so it should be clear now that an infinite regress looms large.
It is not clear whether the above regress is vicious. Some commentators claim that the regress can be avoided without falling victim to a logical contradiction. For instance J. Lowe insists that the A-series can be expressed in a language that employs temporal adverbial modifiers “presently”, “pastly”, and “futurely”. Lowe notes that each event x has to satisfy three disjunctions: (1) x is either pastly past, or presently past, or futurely past; (2) x is either pastly present, or presently present or futurely present; and (3) x is either pastly future, or presently future or futurely future (note that the disjunctions (1) – (3) are not necessarily exclusive). Lowe’s point is that we don’t need to explain the adverbial modifiers in a way that leads to a regress, and he also thinks that the A-theorist should be content with such a characteristic of the A-series. This last claim can be questioned, though. The partition of events into Lowe’s nine temporal spheres falls short of making time move. The truth of (1)-(3) is logically compatible with a completely stationary time, in which a given event x always belongs to the same spheres. McTaggart can repeat his main point: in order for the A-series to exist, every event has to be pastly present, presently present and futurely present, pastly past, presently past and futurely past, and pastly future, presently future and futurely future.
McTaggart’s distinction gives rise to two theories of time: the A-theory and the B-theory. The main difference between them lies in their approach to the idea of the passage of time. The A-theory accepts the existence of the objective passage of time, while the B-theory rejects it. The B-theorists invoke two arguments against the passage of time. Firstly, if the passage of time existed, it would make sense to ask how fast time flows. The rate of time’s flow would have to be measured in seconds per second, which is a dimensionless quantity. Secondly, the movement of time requires some stationary background against which it can happen (similarly to ordinary motion, for which the background is precisely time itself). But this implies the existence of a second-order time, which presumably requires yet another, higher-order time and so on. B-theorists insist that we can translate our ordinary way of speaking about time into the language of the B-theory, based on the fundamental relation “earlier than”. The main challenge for the B-theory is how to express grammatical tenses in the tenseless B-language. A typical suggestion goes along the following lines. The temporal expressions, such as “past”, “present”, “now”, “yesterday”, “tomorrow”, ten days ago”, belong to the category of the so-called indexicals, i.e. expressions whose meaning depends on the context of utterance. Other words in this category are “here”, “there” “I”, etc. When I utter the word “here” while standing in Trafalgar square, this word refers to a different place than when I utter it under the Eiffel tower. Similarly, when I say “It is cold now”, I mean something like “It is cold at the moment of my utterance”. The expression “Napoleon was defeated at the battle of Waterloo” can be interpreted as “Napoleon’s defeat at the battle of Waterloo is earlier than the moment of utterance”.
The proponents of the A-theory of time do not give up easily. They point out that the experience of the passage of time is too fundamental to dismiss it as some sort of illusion. They accuse the B-theorists of interpreting time as an extra spatial dimension (so-called spatialisation of time). The defenders of the passage of time claim that it is possible to meet the B-theorists objections. The passage of time does not have to be literally interpreted as a kind of motion, to which the ordinary notion of velocity would apply. Rather, it consists in the fundamental fact that events come into being successively. Responding to the argument from the rate of flow Tim Maudlin claims that it misses the point. He points out that there is nothing fundamentally wrong with dimensionless quantities, quoting the example of an exchange rate of one currency for itself (dollars for dollars). But we may note that even if Maudlin is right and the notion of the velocity at which time passes is not meaningless, still it is quite unsettling that in his approach this velocity can assume only one value (one second per second) as a matter of conceptual necessity. In other words, time cannot speed up, nor can it slow down. A different solution to this problem has been proposed by Peter Forrest. According to his approach, time passes by adding new layers of spacetime of positive thickness to the already existing universe. The thickness of the successive new layers is the measure of the rate of the flow of time. This picture of the passage of time dispenses with the stationary background against which the passage is supposed to occur.
The A and B theories of time are naturally associated with particular positions regarding the reality of temporal spheres of events. The B-theory is typically connected with the view known as eternalism (or the block universe view). According to eternalism, all events, past, present and future, enjoy the same fundamental status of reality. The battle of Waterloo did not vanish – it exists but in a different part of spacetime than the region occupied by us. Past events are analogous to events that occur in spatially remote regions of the universe: they happen elsewhere, but are not less real because of that. The unintuitiveness of this position is best exposed in Arthur Prior’s “Thanks goodness it’s over” argument. He points out that the eternalist cannot satisfactory explain why we feel relieved when something bad comes to an end. For instance, when my teeth stop aching after taking a pain killer, I feel relieved, but why should I, given that my past pain did not cease to be real? You may reply that the pain belongs to the past now. But according to the B-theory, this means that my (real) pain is earlier than the moment of utterance. Why should I be happy about this?
The A-theory of time is compatible with more than one ontological position regarding the reality of past and future. The most radical is the view known as presentism, which claims that only present events exist. Both past and future events are not real (the former are no longer real, the latter not yet real). The universe consists just of one three-dimensional layer of events which moves as time passes. Presentism is threatened by two main arguments: one from science, and the other from semantics. It is commonly accepted that presentism is incompatible with the special theory of relativity. According to special relativity, the relation of simultaneity is relative with respect to the frame of reference (we will talk about this later). Consequently, the set of events simultaneous with my current “present” depends on the selected frame of reference. But presentism requires that only one set of mutually simultaneous events be real, hence it privileges one particular frame of reference, and this fact violates the principle of relativity. The argument from semantics turns on the fact that some statements about past events (and future events too) are true. But what is the true sentence “Napoleon lost the battle of Waterloo” about, if neither Napoleon, not the battle exists? What is its truthmaker?
Two alternative views compatible with the A-theory are: the growing block theory and the shrinking block theory. The first assumes that past and present events exist, but not future events. The second accepts the opposite: present and future events exist, but past events do not. Both theories are susceptible to similar objections as presentism, although the argument from semantics is now limited to the case of future statements for the growing block theory, and the case of past statements for the shrinking block theory.
Readings:
E.J. Lowe, Chapter 17 "Tense and the reality of time", pp. 307-324, A Survey of Metaphysics.
M.J. Loux, Chapter 7 "The nature of time", pp. 205-228, Metaphysics: A Contemporary Introduction.
Tuesday, March 9, 2010
Wednesday, February 24, 2010
Temporal relations
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?
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?
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.
Wednesday, January 20, 2010
Identity and possible worlds
Is the relation of numerical identity contingent or necessary? It seems that in many cases identity statements are contingent. Gottlob Frege explained how it is possible that true identity statements can be informative. We may understand the meanings of names “a” and “b” without knowing that they refer to one and the same object, hence the statement “a = b” tells us more than the trivial truth that an object is identical with itself. It is tempting to interpret Frege’s result as implying that the identity “a = b” could be false, i.e. that there is a possible world in which “a” and “b” refer to different objects while retaining their original meanings. Saul Kripke famously questioned that suggestion. Kripke claims that all identity statements are in fact necessarily true, and he presents a formal argument in support of his claim. The argument is based on two premises. (1) Every object is necessarily identical with itself (For all x, it is necessary that x = x). (2) If x has property P and y is identical with y, then y has P. Premise (2) is a variant of Leibniz’s law. We can now use the formula “It is necessary that x = x” and, given that x = y, we can substitute y for x, obtaining “It is necessary that x = y”. In conclusion, (1) and (2) lead to the statement: (3) For all x, y, if x = y, then it is necessary that x = y. From (3) it follows that if we take any proper names “a” and “b”, then if only it is true that a = b, it is necessarily so.
But how can we reconcile this formal result with the intuition expressed at the beginning of the previous paragraph? In the actual world the names “Hesperus” and “Phosphorus” refer to the same object: the planet Venus. But couldn’t it be the case that in some other possible world Hesperus and Phosphorus were different objects? Kripke explains away this intuition by pointing out that the possible situation in which we would be tempted to say that Hesperus is not identical with Phosphorus can be reinterpreted in such a way that the identity will be preserved. According to Kripke, terms “Hesperus” and “Phosphorus” are so-called rigid designators, i.e. terms that refer to the same object in all possible worlds in which they refer to anything at all. “Hesperus” is not synonymous with the description “The brightest star on the morning sky”, nor is “Phosphorus” synonymous with “The brightest star on the evening sky”. These descriptions are used contingently in the actual world to fix the reference of both names. In another world the descriptions may not pick the object which is the referent of both terms (i.e. the planet Venus), but if Venus exists in this world, both terms “Hesperus” and “Phosphorus” will continue referring to it.
It may be pointed out that when we restrict the thesis of the necessity of identity statements to rigid designators, its truth becomes quite trivial. However, Kripke claims that his thesis has non-trivial consequences regarding for instance the mind-body controversy. Without going into too much detail, let us consider the Identity Theory, according to which mental events are numerically identical with some physical events. The identity theorists maintain that their claim is true but only contingently, i.e. in some possible worlds there are beings that possess particular neurological states but lack mental states (so-called zombies), and in other possible worlds there may be disembodied minds. But according to Kripke’s analysis, if the statement “This pain is identical with this stimulation of the nervous system” is true, it is necessary, and hence neither zombies nor disembodied minds are possible. But couldn’t we explain away these possible scenarios in a similar way we have redescribed the Hesperus-Phosphorus case? Namely, couldn’t we just say that the rigid designators “this pain” and “this stimulation of the nervous system” are contingently associated with some descriptions, which fail to pick the same object in alternative possible worlds? Unfortunately, as Kripke points out, the terms referring to mental states have no associated descriptions, because their reference is fixed directly by the person that is in a given mental state. So the case of the mind-body identification is different from the case of the Hesperus-Phosphorus identity.
What is the ontological status of possible worlds? One radical answer to this question is known under the name of modal realism (possibilism), which has been proposed by David Lewis. Modal realism consists of several claims. One of them is that possible worlds are made up of concrete, spatiotemporal objects. Possible worlds are not fictions or abstract constructions, but real places with flesh-and-blood inhabitants. Things contained in other possible worlds exist in the same fundamental sense as things in the actual world. Thus it is legitimate to assume that the variables of the existential quantifier range over all possible worlds. For the sake of convenience we may want to relativise the notion of existence to a particular world (speaking about “existing-in-a-world”), but this relativisation does not imply any significant ontological difference. The second element of the doctrine of modal realism is that there is nothing fundamental and absolute about the notion of actual world. The term “actual” is indexical (analogously to terms such as “here”, “now”, “I”, whose meaning depends on the context of utterance), which means that each possible world is actual from the perspective of its inhabitants. Finally, modal realism assumes that possible worlds are spatiotemporally and causally separated from each other. One important consequence of these assumptions is that one object cannot exist in more than one world. Transworld identity is an empty notion in modal realism. But how can we interpret the modal statement “This two-metre high tree could be five metres high” without assuming that this tree can exist in other possible worlds? Lewis solves this problem by introducing the notion of a counterpart. This tree has many counterparts in other possible worlds – trees that are sufficiently similar to it, but not numerically identical. An object could have a given property, if one of its counterparts possesses this property in some possible world.
The main motivation for modal realism comes from its radically reductive character. Lewis subscribes to nominalism and he uses the concept of concrete possible worlds to give a reductive analysis of various abstract entities, such as properties, propositions or meanings. For instance a proposition is simply defined as a class of possible worlds (the proposition “Snow is white” is the class of possible worlds in which snow is white). According to this definition, proposition p is true in a world w, if w is an element of the class of worlds p. Properties, in turn, are defined as functions which assign a set of objects to each possible world. These sets are intuitively understood as consisting of objects that possess a given property in a particular world. This definition avoids the well-known difficulty resulting from the fact that two numerically distinct properties can nevertheless have the same extension in the actual world. The property of being an elephant and the property of being the largest land animal living on Earth now may have the same extensions in the actual world, but there are possible worlds in which elephants are not the largest living land animals. However, reductions offered by modal realists are often criticised as not entirely adequate. For instance, it is pointed out that all necessary true propositions become identical. But we believe that there is a difference between the statements “2+2 = 4” and “It is snowing or it is not snowing”. Similarly, it can be maintained that the property of being triangular and being trilateral are different, and yet in all possible worlds their extensions are identical.
Modal realism is often rejected on grounds of its extravagant ontology. An alternative position is offered in the form of actualism (moderate realism). Alvin Plantinga suggests that possible worlds are mere theoretical constructions which enable us to formulate non-reductive explanations of modal notions. The only genuine world is the actual world, and the quantification in our language should be restricted to objects in the actual world. Possible worlds different from the actual world can be defined as complete and consistent sets of propositions, or better as complete and consistent states of affairs (situations). Each proposition corresponds to a given state of affairs. All states of affairs are abstract objects which exist in the actual world, but only some of them obtain (those that correspond to true propositions). False propositions describe existing states of affairs which nevertheless don’t obtain. According to actualism, the expression “actual” has an absolute meaning: it refers to one and only world that truly exists.
One important difference between actualism and possibilism regards the notion of transworld identity. Actualism admits that one object can exist in many possible worlds. But how are we to understand this statement, if possible worlds do not exist literally, but are mere constructions out of abstract objects? Plantinga proposes the following solution. That an object a exists in a possible world w means that if w were actual, a would exist in it. Note that the explicans is a counterfactual conditional, but it cannot be interpreted in terms of possible worlds, since this would require an introduction of second-order possible worlds (a possible world in which another possible world would be actual). This fact shows that Plantinga’s conception does not offer a fully reductive analysis of modal notions, and that some modalities have to be taken as primitive.
Readings:
M.J. Loux, "The necesary and the possible", pp. 153-186, Metaphysics: A Contemporary Introduction.
E.J. Lowe, "Necessity and identity", pp. 84-95, "Possible worlds", pp. 120-133, A Survey of Metaphysics.
But how can we reconcile this formal result with the intuition expressed at the beginning of the previous paragraph? In the actual world the names “Hesperus” and “Phosphorus” refer to the same object: the planet Venus. But couldn’t it be the case that in some other possible world Hesperus and Phosphorus were different objects? Kripke explains away this intuition by pointing out that the possible situation in which we would be tempted to say that Hesperus is not identical with Phosphorus can be reinterpreted in such a way that the identity will be preserved. According to Kripke, terms “Hesperus” and “Phosphorus” are so-called rigid designators, i.e. terms that refer to the same object in all possible worlds in which they refer to anything at all. “Hesperus” is not synonymous with the description “The brightest star on the morning sky”, nor is “Phosphorus” synonymous with “The brightest star on the evening sky”. These descriptions are used contingently in the actual world to fix the reference of both names. In another world the descriptions may not pick the object which is the referent of both terms (i.e. the planet Venus), but if Venus exists in this world, both terms “Hesperus” and “Phosphorus” will continue referring to it.
It may be pointed out that when we restrict the thesis of the necessity of identity statements to rigid designators, its truth becomes quite trivial. However, Kripke claims that his thesis has non-trivial consequences regarding for instance the mind-body controversy. Without going into too much detail, let us consider the Identity Theory, according to which mental events are numerically identical with some physical events. The identity theorists maintain that their claim is true but only contingently, i.e. in some possible worlds there are beings that possess particular neurological states but lack mental states (so-called zombies), and in other possible worlds there may be disembodied minds. But according to Kripke’s analysis, if the statement “This pain is identical with this stimulation of the nervous system” is true, it is necessary, and hence neither zombies nor disembodied minds are possible. But couldn’t we explain away these possible scenarios in a similar way we have redescribed the Hesperus-Phosphorus case? Namely, couldn’t we just say that the rigid designators “this pain” and “this stimulation of the nervous system” are contingently associated with some descriptions, which fail to pick the same object in alternative possible worlds? Unfortunately, as Kripke points out, the terms referring to mental states have no associated descriptions, because their reference is fixed directly by the person that is in a given mental state. So the case of the mind-body identification is different from the case of the Hesperus-Phosphorus identity.
What is the ontological status of possible worlds? One radical answer to this question is known under the name of modal realism (possibilism), which has been proposed by David Lewis. Modal realism consists of several claims. One of them is that possible worlds are made up of concrete, spatiotemporal objects. Possible worlds are not fictions or abstract constructions, but real places with flesh-and-blood inhabitants. Things contained in other possible worlds exist in the same fundamental sense as things in the actual world. Thus it is legitimate to assume that the variables of the existential quantifier range over all possible worlds. For the sake of convenience we may want to relativise the notion of existence to a particular world (speaking about “existing-in-a-world”), but this relativisation does not imply any significant ontological difference. The second element of the doctrine of modal realism is that there is nothing fundamental and absolute about the notion of actual world. The term “actual” is indexical (analogously to terms such as “here”, “now”, “I”, whose meaning depends on the context of utterance), which means that each possible world is actual from the perspective of its inhabitants. Finally, modal realism assumes that possible worlds are spatiotemporally and causally separated from each other. One important consequence of these assumptions is that one object cannot exist in more than one world. Transworld identity is an empty notion in modal realism. But how can we interpret the modal statement “This two-metre high tree could be five metres high” without assuming that this tree can exist in other possible worlds? Lewis solves this problem by introducing the notion of a counterpart. This tree has many counterparts in other possible worlds – trees that are sufficiently similar to it, but not numerically identical. An object could have a given property, if one of its counterparts possesses this property in some possible world.
The main motivation for modal realism comes from its radically reductive character. Lewis subscribes to nominalism and he uses the concept of concrete possible worlds to give a reductive analysis of various abstract entities, such as properties, propositions or meanings. For instance a proposition is simply defined as a class of possible worlds (the proposition “Snow is white” is the class of possible worlds in which snow is white). According to this definition, proposition p is true in a world w, if w is an element of the class of worlds p. Properties, in turn, are defined as functions which assign a set of objects to each possible world. These sets are intuitively understood as consisting of objects that possess a given property in a particular world. This definition avoids the well-known difficulty resulting from the fact that two numerically distinct properties can nevertheless have the same extension in the actual world. The property of being an elephant and the property of being the largest land animal living on Earth now may have the same extensions in the actual world, but there are possible worlds in which elephants are not the largest living land animals. However, reductions offered by modal realists are often criticised as not entirely adequate. For instance, it is pointed out that all necessary true propositions become identical. But we believe that there is a difference between the statements “2+2 = 4” and “It is snowing or it is not snowing”. Similarly, it can be maintained that the property of being triangular and being trilateral are different, and yet in all possible worlds their extensions are identical.
Modal realism is often rejected on grounds of its extravagant ontology. An alternative position is offered in the form of actualism (moderate realism). Alvin Plantinga suggests that possible worlds are mere theoretical constructions which enable us to formulate non-reductive explanations of modal notions. The only genuine world is the actual world, and the quantification in our language should be restricted to objects in the actual world. Possible worlds different from the actual world can be defined as complete and consistent sets of propositions, or better as complete and consistent states of affairs (situations). Each proposition corresponds to a given state of affairs. All states of affairs are abstract objects which exist in the actual world, but only some of them obtain (those that correspond to true propositions). False propositions describe existing states of affairs which nevertheless don’t obtain. According to actualism, the expression “actual” has an absolute meaning: it refers to one and only world that truly exists.
One important difference between actualism and possibilism regards the notion of transworld identity. Actualism admits that one object can exist in many possible worlds. But how are we to understand this statement, if possible worlds do not exist literally, but are mere constructions out of abstract objects? Plantinga proposes the following solution. That an object a exists in a possible world w means that if w were actual, a would exist in it. Note that the explicans is a counterfactual conditional, but it cannot be interpreted in terms of possible worlds, since this would require an introduction of second-order possible worlds (a possible world in which another possible world would be actual). This fact shows that Plantinga’s conception does not offer a fully reductive analysis of modal notions, and that some modalities have to be taken as primitive.
Readings:
M.J. Loux, "The necesary and the possible", pp. 153-186, Metaphysics: A Contemporary Introduction.
E.J. Lowe, "Necessity and identity", pp. 84-95, "Possible worlds", pp. 120-133, A Survey of Metaphysics.
Wednesday, January 13, 2010
Modality
Modal notions, such as possibility and necessity, play an important role in metaphysical considerations. Intuitively, we can distinguish two ways of speaking about possibilities. We can say that it is now possible that an object might change in the future. For instance this chair may be painted in a colour different from the one it has right now. This type of possibility can be called temporal. But in a different sense it is possible that the chair might have a different colour right now – if it had been painted this colour before. This kind of possibility, which applies to the present time as well as to the past (I can say that I might have been born in a different town), will be referred to as counterfactual possibility. It is interesting to notice that counterfactual and temporal notions of possibility are logically independent, i.e. one does not imply the other. From the fact that some state of affairs is counterfactually possible it does not follow that this state of affairs is possible temporarily. A given sculpture could have a different shape now, but once it receives its actual shape it cannot be turned into a different statue in the future (Rodin’s sculpture “The kiss” could have been “The thinker” in the counterfactual sense, but not in the temporal sense). Conversely, although a seed can grow into a tree in the future, it could not be a tree right now. It should be added that counterfactual possibility, in spite of what the term suggests, does not exclude actuality. Actual states of affair are considered possible in the counterfactual sense.
Counterfactual possibility is often presented in the language that uses the concept of possible worlds. A proposition is possible if it is true in some possible worlds. Possible worlds themselves are usually interpreted as complexes (sums) of situations (states of affairs). An example of a possible situation may be that Poland has a king now. Possible worlds have to satisfy two conditions: the condition of consistency and the condition of completeness. A situation s is consistent if there is no proposition p such that p and not-p are true in s. A situation s is complete if for all propositions p, either p is true in s or not-p is true in s. From these two conditions it follows that two numerically distinct possible worlds are mutually exclusive (incompatible), i.e. there is a proposition p such that p is true in one world, and p is false in the other one. Situations that are not complete don’t have to be exclusive. An example: that this ball is red and that this ball is round. The world that we live in is called the actual world, and it is interpreted in the same way as other possible worlds. It is natural to assume that the actual world is complete, i.e. every proposition is either true or its negation is true in the actual world.
Let us see how we can use the notion of possible worlds in order to explicate some modal terms, such as possibility, necessity and contingency. These notions can be applied to propositions as well as to objects. Proposition p is possible iff p is true in some possible world. Proposition p is necessary iff p is true in all possible worlds. And p is contingent iff p is true in some possible worlds and it’s false in some possible worlds. Similarly we can define possible, necessary and contingent objects. A possible object is an object that exists in some possible worlds. A necessary object exists in all possible worlds, and a contingent object exists in some worlds, but in some it does not. The usual examples of necessary truths are the laws of logic and of mathematics. Necessary beings, in turn, typically include mathematical objects and other abstract objects. Some also cite God as an example of a necessary being. It is open to a debate whether there are any spatiotemporal necessary objects (perhaps the universe as a whole can satisfy this requirement).
Let us make an important distinction between modality de re and de dicto. Modality de dicto applies to the entire sentence, whereas modality de re is attributed to a given object. The sentence “It is possible that some man is the present king of Poland” belongs to the de dicto type, whereas “Some man is possibly the king of Poland” is de re. The first sentence can be presented in a semi-formal way as “It is possible that for some x, x is a man and x is the king of Poland”, and this sentence in turn is true if and only if there is a possible world in which Poland has a king. The second sentence translates into “For some x, x is a man and it is possible that x is the king of Poland”, and in order for this proposition to be true, there has to exist someone in the actual world who, in another possible world is the king of Poland. (These explications presuppose of course that one and the same object can exist in different possible worlds.) The second proposition logically implies the first, but the implication in the opposite direction is a matter of some controversy (the validity of the so-called Barcan law). Another example illustrating the de re/de dicto distinction is as follows: “The number of planets in the solar system is necessarily divisible by 2” (de re) and “It is necessary that the number of planets in the solar system is divisible by 2” (de dicto). The first translates into “There is an x such that x is the number of planets in the solar system and it is necessary that x is divisible by 2”, and the second reads “It is necessary that there is an x such that x is the number of planets in the solar system and x is divisible by 2”. The truth of the first sentence follows from the simple arithmetical fact that 8 is (necessarily) divisible by 2, but for the second sentence to be true, the number of planets in all possible worlds would have to be even.
We can now define an important notion of an essential property. P is an essential property of object a iff for every possible world w, if a exists in w, a has P in w. Loosely speaking, if an object loses its essential property, it ceases to be itself. Napoleon’s essential property is being a human, but being the victor from Austerlitz belongs to his accidental properties (in some possible worlds Napoleon lost the battle of Austerlitz). It is interesting to ask whether things have individual essences, i.e. essential properties such that only one object can possess them all. More specifically, the individual essence of object a is a set S of essential properties of a such that in any possible world w, if x possesses all properties from S, x is identical with a. Some philosophers claim that the individual essence of an object a is the property of being identical with a. However, this interpretation prevents us from using the notion of essence in order to explicate transworld identity. According to a different view, an object’s individual essence is its origin, i.e. the cause of its existence. In the case of human beings, their essence would be determined by the zygote (the fertilized egg) that developed into a particular person. Yet another version of essentialism insists that an object’s essence is its constitution, i.e. all parts the object consists of.
Further reading:
E.J. Lowe, Chapter 5, “Necessity and identity”, pp. 79-84; Chapter 6 “Essentialism”, pp. 98-114, in: A Survey of Metaphysics.
Counterfactual possibility is often presented in the language that uses the concept of possible worlds. A proposition is possible if it is true in some possible worlds. Possible worlds themselves are usually interpreted as complexes (sums) of situations (states of affairs). An example of a possible situation may be that Poland has a king now. Possible worlds have to satisfy two conditions: the condition of consistency and the condition of completeness. A situation s is consistent if there is no proposition p such that p and not-p are true in s. A situation s is complete if for all propositions p, either p is true in s or not-p is true in s. From these two conditions it follows that two numerically distinct possible worlds are mutually exclusive (incompatible), i.e. there is a proposition p such that p is true in one world, and p is false in the other one. Situations that are not complete don’t have to be exclusive. An example: that this ball is red and that this ball is round. The world that we live in is called the actual world, and it is interpreted in the same way as other possible worlds. It is natural to assume that the actual world is complete, i.e. every proposition is either true or its negation is true in the actual world.
Let us see how we can use the notion of possible worlds in order to explicate some modal terms, such as possibility, necessity and contingency. These notions can be applied to propositions as well as to objects. Proposition p is possible iff p is true in some possible world. Proposition p is necessary iff p is true in all possible worlds. And p is contingent iff p is true in some possible worlds and it’s false in some possible worlds. Similarly we can define possible, necessary and contingent objects. A possible object is an object that exists in some possible worlds. A necessary object exists in all possible worlds, and a contingent object exists in some worlds, but in some it does not. The usual examples of necessary truths are the laws of logic and of mathematics. Necessary beings, in turn, typically include mathematical objects and other abstract objects. Some also cite God as an example of a necessary being. It is open to a debate whether there are any spatiotemporal necessary objects (perhaps the universe as a whole can satisfy this requirement).
Let us make an important distinction between modality de re and de dicto. Modality de dicto applies to the entire sentence, whereas modality de re is attributed to a given object. The sentence “It is possible that some man is the present king of Poland” belongs to the de dicto type, whereas “Some man is possibly the king of Poland” is de re. The first sentence can be presented in a semi-formal way as “It is possible that for some x, x is a man and x is the king of Poland”, and this sentence in turn is true if and only if there is a possible world in which Poland has a king. The second sentence translates into “For some x, x is a man and it is possible that x is the king of Poland”, and in order for this proposition to be true, there has to exist someone in the actual world who, in another possible world is the king of Poland. (These explications presuppose of course that one and the same object can exist in different possible worlds.) The second proposition logically implies the first, but the implication in the opposite direction is a matter of some controversy (the validity of the so-called Barcan law). Another example illustrating the de re/de dicto distinction is as follows: “The number of planets in the solar system is necessarily divisible by 2” (de re) and “It is necessary that the number of planets in the solar system is divisible by 2” (de dicto). The first translates into “There is an x such that x is the number of planets in the solar system and it is necessary that x is divisible by 2”, and the second reads “It is necessary that there is an x such that x is the number of planets in the solar system and x is divisible by 2”. The truth of the first sentence follows from the simple arithmetical fact that 8 is (necessarily) divisible by 2, but for the second sentence to be true, the number of planets in all possible worlds would have to be even.
We can now define an important notion of an essential property. P is an essential property of object a iff for every possible world w, if a exists in w, a has P in w. Loosely speaking, if an object loses its essential property, it ceases to be itself. Napoleon’s essential property is being a human, but being the victor from Austerlitz belongs to his accidental properties (in some possible worlds Napoleon lost the battle of Austerlitz). It is interesting to ask whether things have individual essences, i.e. essential properties such that only one object can possess them all. More specifically, the individual essence of object a is a set S of essential properties of a such that in any possible world w, if x possesses all properties from S, x is identical with a. Some philosophers claim that the individual essence of an object a is the property of being identical with a. However, this interpretation prevents us from using the notion of essence in order to explicate transworld identity. According to a different view, an object’s individual essence is its origin, i.e. the cause of its existence. In the case of human beings, their essence would be determined by the zygote (the fertilized egg) that developed into a particular person. Yet another version of essentialism insists that an object’s essence is its constitution, i.e. all parts the object consists of.
Further reading:
E.J. Lowe, Chapter 5, “Necessity and identity”, pp. 79-84; Chapter 6 “Essentialism”, pp. 98-114, in: A Survey of Metaphysics.
Wednesday, January 6, 2010
Reductionist theories of particulars
Realists admit a two-type ontology including universals and particulars, while nominalists insist that there is only one category of objects, namely particulars. Realists have sufficient resources to attempt to reduce the category of particulars to that of universals. One way of reducing particulars to universals is known as the bundle theory, according to which particulars are constituted by all of their properties. This approach is reminiscent of Berkeley’s conception of things as clusters of ideas (‘sensations’), except that properties are assumed to be independent of the perceiving subject. Another possibility is to reduce particulars to properties plus an extra object, called the bare substratum, whose role is to be the literal bearer of the properties without actually possessing them. This conception is not surprisingly called the substratum theory.
Let us consider the bundle theory in some detail. The first problem we have to face is that not every set of properties constitutes a particular object. For instance, the set consisting of the property of being a horse and the property of being winged does not identify any particular, because no object is a winged horse. In order to deal with this problem, the relation of co-instantiation (compresence, collocation) is introduced. The above-mentioned properties are not co-instantiated, hence they cannot constitute an individual. A particular is constituted by properties which are mutually co-instantiated. However, there is a small technical problem here. If we treat co-instantiation as a two-place relation, then the condition that every property in a set is co-instantiated with every other property from this set is not sufficient to ensure that all the properties are co-instantiated together. It is possible to find an example of three properties P, Q, and R such that P is co-instantiated with Q, Q is co-instantiated with R, and P is co-instantiated with R, and yet P, Q and R fail to occur together (for example we can choose R as the property of not being P or not being Q, and assume that P co-occurs with Q, P co-occurs with not-Q, and Q co-occurs with not-P). One solution can be to assume the hierarchy of higher-level relations of compresence binding the relations of compresence between properties, but this leads to the proliferation of numerically distinct relations of compresence. Another way out is to accept that the relation of co-instantiation can admit a varying number of arguments, but it is doubtful if this solution is formally correct. The substratum theory avoids this difficulty, because the substratum acts as the “glue” clumping together all the properties.
Another issue is completeness. Not every set of co-instantiated properties can be identified with one individual object. The set {redness, smoothness} is exemplified by many objects (for example some red apples). We can reduce particulars to complete sets of properties only. A complete set of properties can be broadly characterised as a set such that if we added a new property to it, we would get an inconsistent set. But it is unclear what type of inconsistency is involved in this definition: logical, nomological, or perhaps metaphysical. Also, it may be questioned whether sets of all properties possessed by particular objects are in this sense complete. Why does it create an inconsistency to add to all the properties possessed by this horse the property of having wings? The existence of inconsistency can be perhaps defended if we assumed that the set of all properties of a given object contains also negative properties (in the case of the horse the property in question would be the property of not being winged).
Several objections can be raised to the bundle theory, according to which complete sets of co-instantiated properties constitute particular objects. One such objection is that subject-predicate sentences about particulars become necessarily true. In the true sentence “This table is wooden” the noun phrase “this table” refers to a particular set of properties T, and the sentence can be translated as stating that the property of being wooden belongs to this set T. But each set possesses its elements necessarily, so the sentence cannot be false. If this table, understood as a set, didn’t have the property of being wooden, it would be a numerically different object. Another way of expressing this objection is that all properties of particulars become essential. We could try to circumvent the problem by modifying the interpretation of the above sentence so that its subject is identified with the set T minus the property of being wooden, and the thought expressed in the sentence is that the property of being wooden is co-instantiated with the rest of the properties in T. But a downside of this solution is that an attribution of a different property to the same table, for instance “This table is white”, will have to be interpreted as a sentence about a different object, i.e. the set T minus the property of being white. It is debatable whether the substratum theory is affected by a similar problem. If the subject of the sentence “This table is wooden” is identified as the set of all properties plus the bare substratum of the table, then the same argument in favour of the necessity of the property ascription goes through. However, the sentence can be interpreted as having the bare substratum as its subject, and in this case it will be contingent (the bare substratum exemplifies its properties contingently).
Another difficulty for the bundle theory is related to the problem of change. It is commonly accepted that objects can change in time by acquiring new properties or losing old ones, without losing their numerical identity. But sets of different properties are numerically distinct. The same problem seems to affect the substratum theory, unless we decide to explicate the relation of identity between various temporal stages of an object in terms of the preservation of its bare substratum (but in that case it seems that the identification of a particular object with the set of all its properties plus the substratum turns out to be vacuous – instead, we simply take the substratum as identical with the object). One possible solution may be to treat an object as the bundle of all its past, present and future properties indexed by moments of time. This suggestion corresponds well to the conception of the existence in time known as perdurantism, of which we will talk more later in the course.
Finally, the bundle theory is criticised for implying the necessary truth of the Principle of the Identity of Indiscernibles (PII). This follows immediately from the extensionality of sets: two sets containing the same elements are numerically identical. Hence there can be no numerically different bundles of the same properties. This consequence is unwelcome because of the strong arguments in favour of the possibility of the PII being false (and even in favour of its actual falsity) which we discussed earlier in the course. Again, the substratum theory has the upper hand because it is possible to have two individuals with exactly the same properties, if only their substrata are numerically distinct (the possibility of making the PII false seems to be the main advantage of the substratum theory). The bundle theory may be rescued if we agreed to replace properties with tropes. It is possible to have two individuals made up of perfectly similar tropes, hence the spirit of the PII is preserved.
The substratum theory assumes that when we “subtract” all properties from a given object what is left is a property-less pure object, called a bare substratum. The substratum cannot posses any properties, because metaphysically it has to be ready to accept any properties as their bearer. If the substratum possessed any properties, this would lead to a regress, since it would need its own bare substratum presumably with its properties and so on. The only attribution that can be made directly about the substratum regards its numerical distinctness from other substrata. Hence it can be maintained that bare substrata ground the numerical identity and distinctness of objects, known as their “thisness” or haecceity.
The main criticism of the substratum theory comes from the empiricists, who point out that because bare substrata lack any properties they cannot be perceived or in any way identified or known. It is also claimed that the notion of a bare substratum is inconsistent. On the one hand it is assumed that bare substrata don’t posses any properties, but on the other hand we characterize them in some ways: we say that they are ‘bare’, that they act as literal possessors of properties, that they ground numerical identity and difference. But aren’t these things properties of bare substrata?
A solution which tries to combine the strengths of the bundle theory and the substratum theory without their weaknesses is called the nuclear theory (proposed by Peter Simons). According to it, properties of every object can be divided into two parts: the inner nucleus and the outer fringe. The inner nucleus contains properties that are essential, and therefore such that an object cannot lose them without losing its identity (we will characterize precisely the notion of an essential property in the next lecture). Thus the nucleus plays the role of a bare substratum. The outer fringe, on the other hand, contains properties that can be lost and gained without a change in the numerical identity. But this solution has to be worked out in details in order to make sure that it copes with the problems we talked about earlier. In particular, if we elect to identify particular objects with their inner nuclei (this seems to be necessary in order to deal with the problem of necessary attributions and the problem of change), we have to make sure that no two distinct objects possess all the properties from the nucleus.
Reading:
M.J. Loux, Chapter 3 "Concrete particulars I", pp. 84-107, Metaphysics: A Contemporary Introduction
Also recommended:
J. Van Cleve, "Three Versions of the Bundle Theory".
P. Simons, "Particulars in Particular Clothing: Three Trope Theories of Substance".
Let us consider the bundle theory in some detail. The first problem we have to face is that not every set of properties constitutes a particular object. For instance, the set consisting of the property of being a horse and the property of being winged does not identify any particular, because no object is a winged horse. In order to deal with this problem, the relation of co-instantiation (compresence, collocation) is introduced. The above-mentioned properties are not co-instantiated, hence they cannot constitute an individual. A particular is constituted by properties which are mutually co-instantiated. However, there is a small technical problem here. If we treat co-instantiation as a two-place relation, then the condition that every property in a set is co-instantiated with every other property from this set is not sufficient to ensure that all the properties are co-instantiated together. It is possible to find an example of three properties P, Q, and R such that P is co-instantiated with Q, Q is co-instantiated with R, and P is co-instantiated with R, and yet P, Q and R fail to occur together (for example we can choose R as the property of not being P or not being Q, and assume that P co-occurs with Q, P co-occurs with not-Q, and Q co-occurs with not-P). One solution can be to assume the hierarchy of higher-level relations of compresence binding the relations of compresence between properties, but this leads to the proliferation of numerically distinct relations of compresence. Another way out is to accept that the relation of co-instantiation can admit a varying number of arguments, but it is doubtful if this solution is formally correct. The substratum theory avoids this difficulty, because the substratum acts as the “glue” clumping together all the properties.
Another issue is completeness. Not every set of co-instantiated properties can be identified with one individual object. The set {redness, smoothness} is exemplified by many objects (for example some red apples). We can reduce particulars to complete sets of properties only. A complete set of properties can be broadly characterised as a set such that if we added a new property to it, we would get an inconsistent set. But it is unclear what type of inconsistency is involved in this definition: logical, nomological, or perhaps metaphysical. Also, it may be questioned whether sets of all properties possessed by particular objects are in this sense complete. Why does it create an inconsistency to add to all the properties possessed by this horse the property of having wings? The existence of inconsistency can be perhaps defended if we assumed that the set of all properties of a given object contains also negative properties (in the case of the horse the property in question would be the property of not being winged).
Several objections can be raised to the bundle theory, according to which complete sets of co-instantiated properties constitute particular objects. One such objection is that subject-predicate sentences about particulars become necessarily true. In the true sentence “This table is wooden” the noun phrase “this table” refers to a particular set of properties T, and the sentence can be translated as stating that the property of being wooden belongs to this set T. But each set possesses its elements necessarily, so the sentence cannot be false. If this table, understood as a set, didn’t have the property of being wooden, it would be a numerically different object. Another way of expressing this objection is that all properties of particulars become essential. We could try to circumvent the problem by modifying the interpretation of the above sentence so that its subject is identified with the set T minus the property of being wooden, and the thought expressed in the sentence is that the property of being wooden is co-instantiated with the rest of the properties in T. But a downside of this solution is that an attribution of a different property to the same table, for instance “This table is white”, will have to be interpreted as a sentence about a different object, i.e. the set T minus the property of being white. It is debatable whether the substratum theory is affected by a similar problem. If the subject of the sentence “This table is wooden” is identified as the set of all properties plus the bare substratum of the table, then the same argument in favour of the necessity of the property ascription goes through. However, the sentence can be interpreted as having the bare substratum as its subject, and in this case it will be contingent (the bare substratum exemplifies its properties contingently).
Another difficulty for the bundle theory is related to the problem of change. It is commonly accepted that objects can change in time by acquiring new properties or losing old ones, without losing their numerical identity. But sets of different properties are numerically distinct. The same problem seems to affect the substratum theory, unless we decide to explicate the relation of identity between various temporal stages of an object in terms of the preservation of its bare substratum (but in that case it seems that the identification of a particular object with the set of all its properties plus the substratum turns out to be vacuous – instead, we simply take the substratum as identical with the object). One possible solution may be to treat an object as the bundle of all its past, present and future properties indexed by moments of time. This suggestion corresponds well to the conception of the existence in time known as perdurantism, of which we will talk more later in the course.
Finally, the bundle theory is criticised for implying the necessary truth of the Principle of the Identity of Indiscernibles (PII). This follows immediately from the extensionality of sets: two sets containing the same elements are numerically identical. Hence there can be no numerically different bundles of the same properties. This consequence is unwelcome because of the strong arguments in favour of the possibility of the PII being false (and even in favour of its actual falsity) which we discussed earlier in the course. Again, the substratum theory has the upper hand because it is possible to have two individuals with exactly the same properties, if only their substrata are numerically distinct (the possibility of making the PII false seems to be the main advantage of the substratum theory). The bundle theory may be rescued if we agreed to replace properties with tropes. It is possible to have two individuals made up of perfectly similar tropes, hence the spirit of the PII is preserved.
The substratum theory assumes that when we “subtract” all properties from a given object what is left is a property-less pure object, called a bare substratum. The substratum cannot posses any properties, because metaphysically it has to be ready to accept any properties as their bearer. If the substratum possessed any properties, this would lead to a regress, since it would need its own bare substratum presumably with its properties and so on. The only attribution that can be made directly about the substratum regards its numerical distinctness from other substrata. Hence it can be maintained that bare substrata ground the numerical identity and distinctness of objects, known as their “thisness” or haecceity.
The main criticism of the substratum theory comes from the empiricists, who point out that because bare substrata lack any properties they cannot be perceived or in any way identified or known. It is also claimed that the notion of a bare substratum is inconsistent. On the one hand it is assumed that bare substrata don’t posses any properties, but on the other hand we characterize them in some ways: we say that they are ‘bare’, that they act as literal possessors of properties, that they ground numerical identity and difference. But aren’t these things properties of bare substrata?
A solution which tries to combine the strengths of the bundle theory and the substratum theory without their weaknesses is called the nuclear theory (proposed by Peter Simons). According to it, properties of every object can be divided into two parts: the inner nucleus and the outer fringe. The inner nucleus contains properties that are essential, and therefore such that an object cannot lose them without losing its identity (we will characterize precisely the notion of an essential property in the next lecture). Thus the nucleus plays the role of a bare substratum. The outer fringe, on the other hand, contains properties that can be lost and gained without a change in the numerical identity. But this solution has to be worked out in details in order to make sure that it copes with the problems we talked about earlier. In particular, if we elect to identify particular objects with their inner nuclei (this seems to be necessary in order to deal with the problem of necessary attributions and the problem of change), we have to make sure that no two distinct objects possess all the properties from the nucleus.
Reading:
M.J. Loux, Chapter 3 "Concrete particulars I", pp. 84-107, Metaphysics: A Contemporary Introduction
Also recommended:
J. Van Cleve, "Three Versions of the Bundle Theory".
P. Simons, "Particulars in Particular Clothing: Three Trope Theories of Substance".
Wednesday, December 16, 2009
Two notions of sets
Sets are considered fundamental mathematical objects, because in principle all other mathematical objects can be defined in terms of sets (can be reduced to sets). However, the notion of a set can be given at least two unequivalent interpretations. In one interpretation, the set of physical objects X is a physical complex whose spatiotemporal parts are all objects X. Sets of that sort are called mereological (or collective). Mereological sets have the following characteristic properties. First, the mereological set consisting of one object is identical with this object. Second, there is no mereological empty set (for instance, the collection of all centaurs does not exist). Third, two mereological sets built out of numerically different objects can nevertheless be identical. For instance, the mereological set of two hydrogen atoms is identical with the mereological set of two protons and two electrons constituting those atoms. This example also illustrates the fact that the relation of belonging to a mereological set is transitive. Clearly, this follows from the fact that the relation of being a member of a mereological set is identical with the part-whole relation, and the latter is transitive (if x is a part of y and y is a part of z, then x is a part of z).
The second interpretation of sets is called distributive (or set-theoretical). Distributive sets are analogous to linguistic concepts. The distributive set of all people has people as its only elements. No proper part of a person (such as a hand or a leg) belongs to this set, since proper body parts are not humans. As concepts differ from the objects subsumed under them, the set consisting of only one object is numerically different from this object. There is an empty set (a set with no elements), since there are empty concepts (such as the concept of a unicorn). Actually, as we will soon see, it can be proven that there is exactly one empty set. If you have two sets which have different numbers of elements, you can be sure that these sets are different. Hence the set of two hydrogen atoms is distinct from the set of their two protons and two electrons. Distributive sets satisfy the principle of extensionality: two sets are identical if and only if they have exactly the same elements. (Actually, mereological sets satisfy this principle too: two mereological sets are identical iff they have the same parts. But it is still possible to describe one and the same mereological set as consisting of numerically different objects, as in the example with two hydrogen atoms). From the condition of extensionality it follows that there is exactly one empty set. The relation of membership is not transitive in the case of distributive sets: if x is a member of y and y is a member of z, x does not have to be a member of z (although it may). Example: 1 є {1} and {1} є {{1}, {2}}, but it’s not the case that 1 є {{1}, {2}}.
From the ontological point of view it is important to ask what kind of objects sets are and whether they can be accepted by a nominalist. Mereological sets don’t create much of a problem, since they are just spatiotemporal objects, provided that their elements are spatiotemporal. The only contentious issue is whether we should admit the existence of arbitrary collections of objects. Is there an individual object that consists of my left pinkie, the planet Venus, and the left hind leg of some particular dinosaur? But the status of distributive sets is more controversial. Arguably, distributive sets cannot be identified with spatiotemporal objects. The singleton consisting of one physical object x is not identical with x, and because it cannot be identical with any other physical object y (since its existence would be contingent upon the existence of y, and the only acceptable ontological dependence of {x} is on x), hence {x} cannot be a physical object. Thus the most common interpretation of distributive sets is that they are abstract objects, and as such are not acceptable to the nominalist. However, nominalists can make use of the notion of distributive set in certain contexts, for which it is possible to give a nominalistic paraphrase. For instance, the statement “Socrates belongs to the set of all philosophers” can be interpreted nominalistically as “Socrates is a philosopher”. The sentence “The set of all philosophers is a subset of the set of all people” is interpreted as “All philosophers are humans”, and an analogous interpretation of the sentence “The set of all people is disjoint from the set of all elephants” can be given as “No humans are elephants”. Thus it can be claimed that the nominalist can accept first-order sets of physical objects (so-called classes). But higher-order sets, and especially those founded on the empty set, are not so easy to eliminate from the discourse.
At the beginning of the lecture we mentioned the fact that mathematical objects can be reduced to (distributive) sets. But, as Paul Benacerraf has famously noticed, such reductions are not unique. For instance, natural numbers can be interpreted as sets in at least two ways. One interpretation is given by following identifications: 0 = Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, etc. But an alternative interpretation can look like this: 0 = Ø, 1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc. These two interpretations, taken literally, cannot be true, for this would imply mathematical falsehoods, such as {{Ø}} = {Ø, {Ø}} (you can prove that this identity is false, given the principle of extensionality and the assumption that {x} is different from x). But mathematical practice does not tell us which identification should be preferred. It looks like some questions regarding numerical identity between mathematical objects are fundamentally undecidable, which calls into question the ontological status of mathematical objects as independent entities. One solution to this problem is encompassed in the so-called structuralist interpretation of mathematics. According to this interpretation, the fundamental objects that mathematical theories speak about are whole structures, not individual objects. There is no number 1 as an entity that exists separately from the entire structure of natural numbers, hence it does not make sense to ask what this number is identical with. Mathematical objects are just positions in a given structure. There is more than one way to interpret one structure (e.g. the structure of natural numbers) within another structure (the structure of sets). But an interpretation is just a homomorphism, i.e. a mapping which preserves the structure. There is no identity involved. The essence of natural numbers is exhausted in the structure of a linear, discrete order.
The second interpretation of sets is called distributive (or set-theoretical). Distributive sets are analogous to linguistic concepts. The distributive set of all people has people as its only elements. No proper part of a person (such as a hand or a leg) belongs to this set, since proper body parts are not humans. As concepts differ from the objects subsumed under them, the set consisting of only one object is numerically different from this object. There is an empty set (a set with no elements), since there are empty concepts (such as the concept of a unicorn). Actually, as we will soon see, it can be proven that there is exactly one empty set. If you have two sets which have different numbers of elements, you can be sure that these sets are different. Hence the set of two hydrogen atoms is distinct from the set of their two protons and two electrons. Distributive sets satisfy the principle of extensionality: two sets are identical if and only if they have exactly the same elements. (Actually, mereological sets satisfy this principle too: two mereological sets are identical iff they have the same parts. But it is still possible to describe one and the same mereological set as consisting of numerically different objects, as in the example with two hydrogen atoms). From the condition of extensionality it follows that there is exactly one empty set. The relation of membership is not transitive in the case of distributive sets: if x is a member of y and y is a member of z, x does not have to be a member of z (although it may). Example: 1 є {1} and {1} є {{1}, {2}}, but it’s not the case that 1 є {{1}, {2}}.
From the ontological point of view it is important to ask what kind of objects sets are and whether they can be accepted by a nominalist. Mereological sets don’t create much of a problem, since they are just spatiotemporal objects, provided that their elements are spatiotemporal. The only contentious issue is whether we should admit the existence of arbitrary collections of objects. Is there an individual object that consists of my left pinkie, the planet Venus, and the left hind leg of some particular dinosaur? But the status of distributive sets is more controversial. Arguably, distributive sets cannot be identified with spatiotemporal objects. The singleton consisting of one physical object x is not identical with x, and because it cannot be identical with any other physical object y (since its existence would be contingent upon the existence of y, and the only acceptable ontological dependence of {x} is on x), hence {x} cannot be a physical object. Thus the most common interpretation of distributive sets is that they are abstract objects, and as such are not acceptable to the nominalist. However, nominalists can make use of the notion of distributive set in certain contexts, for which it is possible to give a nominalistic paraphrase. For instance, the statement “Socrates belongs to the set of all philosophers” can be interpreted nominalistically as “Socrates is a philosopher”. The sentence “The set of all philosophers is a subset of the set of all people” is interpreted as “All philosophers are humans”, and an analogous interpretation of the sentence “The set of all people is disjoint from the set of all elephants” can be given as “No humans are elephants”. Thus it can be claimed that the nominalist can accept first-order sets of physical objects (so-called classes). But higher-order sets, and especially those founded on the empty set, are not so easy to eliminate from the discourse.
At the beginning of the lecture we mentioned the fact that mathematical objects can be reduced to (distributive) sets. But, as Paul Benacerraf has famously noticed, such reductions are not unique. For instance, natural numbers can be interpreted as sets in at least two ways. One interpretation is given by following identifications: 0 = Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, etc. But an alternative interpretation can look like this: 0 = Ø, 1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc. These two interpretations, taken literally, cannot be true, for this would imply mathematical falsehoods, such as {{Ø}} = {Ø, {Ø}} (you can prove that this identity is false, given the principle of extensionality and the assumption that {x} is different from x). But mathematical practice does not tell us which identification should be preferred. It looks like some questions regarding numerical identity between mathematical objects are fundamentally undecidable, which calls into question the ontological status of mathematical objects as independent entities. One solution to this problem is encompassed in the so-called structuralist interpretation of mathematics. According to this interpretation, the fundamental objects that mathematical theories speak about are whole structures, not individual objects. There is no number 1 as an entity that exists separately from the entire structure of natural numbers, hence it does not make sense to ask what this number is identical with. Mathematical objects are just positions in a given structure. There is more than one way to interpret one structure (e.g. the structure of natural numbers) within another structure (the structure of sets). But an interpretation is just a homomorphism, i.e. a mapping which preserves the structure. There is no identity involved. The essence of natural numbers is exhausted in the structure of a linear, discrete order.
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