Free will

The standard argument in favour of incompatibilism goes as follows. Free will requires the existence of genuine alternative possibilities. If my action was free, this means that I had several options of which I selected one. But determinism precludes the existence of alternative possibilities except the one that actually happens. More precisely, determinism states that there is only one possible evolution of the world given its initial state at a particular time and the laws. If we select the initial state of the world at a moment well before I was born, we may safely assume that my action cannot change it. So if determinism is true, I couldn’t act freely because I had no genuine alternatives: there was only one action which was possible for me to take. But compatibilists have several strategies of how to respond to this challenge. The main strategy is to deny that free will really requires the existence of alternative possibilities in the sense which is incompatible with determinism. But the question is now how to better define what free actions are. Several interpretations of the notion of freedom are available. One possible interpretation is that a free action is when there are no external constraints (physical threats, mind-controlling devices etc.). But this definition is not necessarily adequate, because there may be actions without external constraints and yet unfree (for instance, obsessive-compulsory behaviour). Moreover, some actions subjected to external constraints are nevertheless still free. If I am held at gunpoint, and the gunman demands my money, my decision to hand in my money is still free in the fundamental, metaphysical sense (I could refuse and risk death).

A marginally better analysis is that X acts freely iff X does what he/she wants (or, more carefully, that if X does what he wants, X acts freely). But it may be the case that X is forced by a mind-controlling device to do something that happens to agree with X’s wishes. We can correct this problem by adding the requirement that if X hadn’t wanted to do A, he wouldn’t have done A. Note that even in a deterministic world this statement may clearly come out true, and we don’t even have to consider a possible world which has a different past from our world – as Lewis pointed out, we can assume that a miracle made X not want to do A. But this definition of a free act raises some concerns. It may be true about a compulsive kleptomaniac that if he didn’t want to steal, he would refrain from stealing. The problem is though that this person is not free to choose what he wants, so the freedom is illusory in this case. And the incompatibilist would insist that we all are precisely in the same situation: we are not capable of choosing what we want, because our desires and beliefs are predetermined by the initial state of the world, and therefore we are not acting free exactly as the kleptomaniac from our example.

Harry Frankfurt chooses a different strategy of dealing with the argument for incompatibilism. He stresses that the existence of genuine alternatives is not necessary for the freedom of the will. He gives the following example to illustrate his point: suppose that John pushed Peter of his own will, but unbeknownst to John there was Larry nearby with a mind-controlling device who would have made John push Peter if he hadn’t wanted to do that. So actually there was no genuine open alternative to John’s action, and yet we would call him morally responsible of his action, and therefore we would have to assume that he acted freely. But the incompatibilist can formulate some objections to this argument. Firstly, it is not clear that in the described situation there were no genuine alternatives. We may argue that actually John had two options: he could push Peter of his own will, or he could be forced to do so. Secondly, even if we agree that there were no alternative possibilities for John’s action, this does not show that determinism was true in the described situation. If we assume determinism, the intervention by Larry was predetermined by the past, and the entire set-up that underlies the story is destroyed.

Finally, compatibilists attack their opponents by pointing out that actually indeterminism turns out to be incompatible with freedom of the will. This by itself does not prove that determinism is compatible with free will, for it may happen that both determinism and indeterminism are incompatible with our freedom. But this would imply that the existence of free will is impossible, and this does not look plausible. The argument is that if the world is indeterministic, then there are certain events which can be called ‘random’. These events, in turn, can cause individual actions to occur. But this means that a given action was taken not of free will, but as a result of random, indeterministic processes. It is as if our actions were governed by a throw of a die. But it may be replied that the ‘random’ event in question is not something external from the person who is about to act. Indeed, it is our decision to do this rather than that which, from the perspective of the deterministic stance is a random occurrence, in the sense that it is not uniquely fixed by the past events.

Reading:
B, Garrett, "Free will", pp. 112-117, What is this thing called metaphysics?

Fatalism and free will

For Jan Łukasiewicz logical determinism as presented in the previous lecture was unacceptable. He insisted that the fault lies in the logic that underlies the argument. Łukasiewicz claimed that in order to reject logical determinism we have to abandon classical logic and two of its fundamental principles: the principle of the excluded middle and the closely related principle of bivalence, stating that each grammatically and semantically correct sentence has one of the two logical values: truth or falsity. Łukasiewicz introduced a new, three-valued logic, in which any sentence can receive one of the three values: truth, falsity, and indeterminacy (‘possibility’). If we believe that the fact that it will rain tomorrow is not determined today, we should assign to the statement “It will rain tomorrow in Warsaw” the third logical value of indeterminacy. Łukasiewicz’s logic has a rule, according to which the disjunction of two indeterminate sentences is itself indeterminate. So the sentence “It will rain tomorrow in Warsaw or it won’t rain tomorrow in Warsaw” is neither true nor false today, and the principle of the excluded middle is not valid.

But it may be pointed out that the afore-mentioned disjunction will for certain turn out to be true, no matter what the weather will be like in Warsaw on the next day. So Łukasiewicz’s conception has an important flaw: under his interpretation a sentence about the future which for certain will receive the value “true” has to be deemed indeterminate at present. (Moreover, in Łukasiewicz’s logic it is possible to construct a sentence which is true now but it will turn out to be false in the future!) One possible way of ameliorating this situation is to cut off the link between the truth of the disjunction as a whole, and the truth of its components. We can admit that the whole disjunction “It will be p or it won’t be p” is true now, but from this it doesn’t follow that p is true now or not-p is true. The conclusion only follows if we accept the principle of bivalence, but if we admit that there is a third logical value, the argument is no longer valid. So an alternative proposal is to retain all the laws of classical logic but to extend the number of logical values to three instead of two. But one consequence of this strategy is that classical connectives will no longer be truth-functional. Compare the statement “Tomorrow it will rain or tomorrow it won’t rain” with the disjunction “Tomorrow I will write a letter or visit my friend”. Under the assumption that all sentences involved are indeterminate as of now, the first disjunction comes out true, but the second is indeterminate (it can still happen that I won’t write a letter and I won’t visit my friend).

Finally, it may be claimed that logical determinism has nothing to do with genuine determinism, and that it is based on some sort of semantic confusion. When I say that it is true at t1 that x happens at t2, it looks like I am reporting a fact of the matter which happens at t1, but actually the fact I am talking about still takes place at t2. Hence, if I say that today it is true that tomorrow it will rain, semantically it is the same as if I said simply “Tomorrow it will rain”. I am talking about a future event which exists in a different temporal sphere from the moment of my utterance (this reminds us of the eternalist conception of time).

Logical determinism is closely related to another philosophical position known as fatalism. Fatalism stresses that there is only one possible scenario which the universe will follow in its future development, and therefore any action we may want to take is futile: it will be what is supposed to be regardless. One popular version of fatalism is religious (theological) fatalism. God is omniscient (all-knowing), and therefore he knows everything about the future as well as about the past. From this it seems to follow that he knows what will happen with me regardless of my actions. Logical fatalism, in turn, is based on a similar argument to the argument for logical determinism. The principle of the excluded middle ensures that every statement about the future is already either true or false, so it looks like our current actions are superfluous. Fatalism implies that we are actually not creators of our destiny – that whatever we do does not make any difference for our future.

A typical argument supporting logical fatalism can be presented as follows. It doesn’t make sense to take any precautions, for instance to take shelter during an air raid, because it is true that either you’ll be hurt in the bombing or you won’t (again, we assume the principle of the excluded middle). If you are going to be hurt, no precautions can help (they will turn out to be ineffective), whereas if you are not going to be hurt, precautions are unnecessary. Thus it is not rational to take any precautions. This argument has a similar structure to a more plausible argument showing that present actions are irrelevant to the past events of which we don’t know yet. As the argument goes, it is pointless to pray for the survival of a friend in a catastrophe which happened yesterday when the full list of victims is not yet known, because again either the friend survived or he didn’t. If he survived, the prayer is not needed, whereas if he died, it is futile.

But there is an important difference between the two cases. The present actions can causally influence future events, but not the past ones. It is not true that if I am not going to be hurt in an air raid, my actions will turn out to be irrelevant to this outcome. The relevance of a given action can be evaluated with the help of counterfactual conditionals only: we have to assess what would happen, had I not done what I actually did. Suppose, for instance, that I met a friend on my morning walk. I can’t say that my decision to go on a walk this morning was irrelevant to this fact: if I hadn’t gone for a walk, I would not have met my friend. Regarding the theological version of fatalism, it may be objected that God’s foreknowledge about our future actions does not preclude the fact that these actions are free. It is not the case that I am choosing to do X now because God knew that I would do X, but rather God knew this because of my current decision. I could make a different decision, but this would not confute God’s knowledge; rather it would change his knowledge. (Of course one controversial issue remains: it looks like my action can influence the past.)

It may be noted that fatalism effectively denies the existence of free will. But we can ask whether physical determinism has the same consequence. So the next issue we are going to consider is whether determinism is logically compatible with free will. There are several positions regarding this issue. Compatibilism states that determinism can be logically reconciled with the existence of free will, whereas incompatibilism professes that determinism logically excludes free will. Incompatibilism is further divided into two positions: libertarianism, which asserts that free will exists and therefore determinism must be false, and hard determinism, assuming the universal validity of determinism and denying that people have free will. On the other hand, a typical variant of compatibilism is the position known as soft determinism, which accepts both determinism and free will. Of course it is logically possible to be a compatibilist and to deny determinism or free will, or even both. For instance, some philosophers subscribe to compatibilism while maintaining that in fact determinism is false in our world and free will exists. Such a position could be called “soft libertarianism”.

Reading regarding fatalism and free will

B. Garrett, Chapter 7 "Free will", pp. 103-112, What is this thing called metaphysics?

Determinism

The doctrine of determinism can be presented in many non-equivalent ways. Below we will formulate several possible interpretations of this view, and we’ll try to select the formulations that are theoretically most interesting and conceptually fruitful, and discard the ones that lead to some confusion. One quite popular version of determinism states that every event has its cause (this view is also known as the principle of causation). The main problem with this thesis is that its meaning and truth depends on the notion of cause used, and as we know there are many radically different interpretations of causation that can be used here. For instance, if we followed the counterfactual interpretation of causation, the principle of causation would reduce to the statement that each event has a necessary condition that is distinct from it. This claim seems to be rather trivially true (for instance the Big Bang can be seen as a cause of all subsequent events in this sense), but the doctrine of determinism is usually not seen as such a weak statement. Alternatively, a cause can be interpreted as a sufficient condition, and under such an interpretation determinism would state that for each event x there is a set of conditions that are jointly sufficient for the occurrence of x. This claim may well turn out to be false: for instance according to our current scientific theories there is no sufficient condition ensuring that an individual unstable nucleus will decay at a given moment t. The process of radioactive decay is considered inherently probabilistic.


The most famous formulation of determinism is associated with the name of Pierre Simone de Laplace. Laplacean version of determinism is based on the notion of predictability. It is a well-known fact that laws of nature enable us to make successful future predictions from past states of a given system. For example, the laws of classical (Newtonian) mechanics together with the exact positions and velocities of material objects imply the future trajectories of all involved objects (of course if all acting forces are known). But in practice calculating the future evolution of the system encounters numerous obstacles, some of them associated with the fact that our knowledge about initial conditions is always inaccurate due to measurement errors. For that reason Laplace invoked a superhuman intelligence (a demon) in his thought experiment. Laplace’s thesis of determinism is that a demon who could know exactly the initial conditions of the world (the momentary positions and velocities of all objects in the world) and the forces acting upon those objects, would be able to use the universal laws of motion to predict all future states of the world. One immediate problem with this formulation is that we don’t know exactly what computational abilities this demon would possess. It is possible that in order to solve all the equations of motion we would have to use a machine more powerful than a finite Turing machine. So it is quite possible that the world could be deterministic (in the sense that we will specify soon) and yet the future states could not be computed from the past states given certain restrictions on the process of computability.


A slightly different version of epistemic determinism has been formulated by Karl Popper. Popper uses the notion of an ideal scientist rather than a demon, and he assumes that the scientist is capable of knowing the initial state of the world only with a finite degree of precision. The question is: is it possible to make predictions regarding the future on the basis of an inexact knowledge about the past conditions? It turns out that in so-called chaotic systems even a minute change in the initial conditions can lead to dramatic differences in the future behaviour of the system. Hence, predictability based on an imprecise description of the initial state is impossible, although technically determinism may be still valid.


Most philosophers agree that the thesis of determinism should be about the world, and not about our cognitive abilities to predict the future course of events. One possible way to explicate this ontological concept of determinism is as follows. We can say that the world is deterministic iff the complete state of the world at any time t fixes, or determines, the states of the world at all times later than t. The notion of ‘determination’ or ‘fixing’ can be explained in turn in the following fashion: the state S1 at t1 determines the later state of the world at t2 iff there is exactly one state S2 that the world will be in at t2, given that it is in S1 at t1. But this explication is still unsatisfactory. It is a trivial fact that the universe can be in only one state at a given time or, equivalently, that there is only one evolution that the world is actually following. But determinism cannot be trivially true. The idea is that when we fix the state at any moment t, it is impossible for the later states to be different from what they actually are. The notion of possibility that is involved in this statement is obviously the physical (nomological) one, so the thesis of determinism can be formulated with reference to the laws of nature. We will say that the world is deterministic iff the state of the world at any moment t and the laws of nature jointly imply that there is only one allowed state of the world at any later moment.


Two remarks can be made about this version of determinism. First, an analogous claim can be made with respect to past states (that the state of the world determines all the past states). The historical version of determinism follows from the futuristic version if the laws of nature are symmetric in time. Second, instead of the global thesis of determinism we can formulate a local version in which we limit ourselves to a given system and its states, not the entire universe. In such a local claim of determinism it is usually assumes that the system in question is isolated from the rest of the world (if we didn’t make this assumption, the thesis of determinism would be trivially false). We can also limit the question of determinism not to the world or its parts, but rather to particular laws or theories. A theory can be called deterministic iff the complete description of the state of a system at a given moment that is available in this theory, and all the laws of the theory, uniquely determine the future states of the system.


Another but equivalent way of expressing the doctrine of global determinism is in terms of possible worlds. Let W be the set of all possible worlds (including the actual world) which obey the same laws as the actual world. We can say that the world is (futuristically) deterministic iff if any two worlds w1 and w2 from W agree at any moment t (i.e. the state of both world at t is the same), they agree completely at any later time. It has to be observed that the notion of a state of the world at a given time has to be understood in a way which prevents a trivialisation of determinism. More specifically, the specification of the state at a given time t should not contain properties which refer implicitly to other temporal moments. For instance, if we accepted as part of the state at time t the property of the world that it will be at some state S five minutes later, than it would follow trivially that the state of the world at t would fix the state of the world at t + 5. But the problem is that one parameter which is usually included in the description of the momentary state of the world, i.e. instantaneous velocity of individual objects, actually makes reference to moments of time other than t (instantaneous velocity is defined as the derivative of position, and the derivative of a function f at a point x contains the information about the function’s behaviour in the infinitesimal interval around x).


Another perspective on the multifaceted problem of determinism is offered in so-called logical determinism. Logical determinism does not appeal to any laws of nature, and therefore it’s totally independent of the issue of the nomological structure of the world. Suppose that we consider some future event, for instance that tomorrow it will rain in Warsaw. If it is already true that it will rain, we can say that the event itself is determined as of today, whereas if it is not true yet, we may claim that it is not determined. But consider the law of the excluded middle: it is true today that it will rain tomorrow or it will not rain tomorrow. From this it follows that either the sentence “It will rain tomorrow in Warsaw” or the sentence “It won’t rain tomorrow in Warsaw” is true now. Whichever is the case, it seems that determinism comes out true. The thesis of logical determinism can thus be presented as follows: if an event x occurs at t, it was true at any time previous to t that x would occur. This statement seems to follow from logic itself. Given that “x occurs at t” is true, and given the principle of the excluded middle “Either it is true at an earlier moment t’ that x would occur at t, or it is true at t’ that x won’t occur at t”, it logically follows that at t’it is true that x will occur at t.

Counterfactual theory of causation

Now we have to say a couple of words about the relation of closeness (or similarity) between possible worlds. Formally, it is a two-place relation relativised to the actual world: “world w1 is more similar to the actual world than world w2”, and it is assumed to possess standard properties, such as asymmetricity, transitivity and linearity, plus minimality (the actual world is closer to itself than any other world). But what properties of possible worlds should be taken into account when evaluating their relative similarity with respect to the actual world? Let us observe that two aspects of similarity can be taken into account: similarity with respect to individual facts and similarity with respect to laws. It may seem that the similarity with respect to laws should be seen as more important than the similarity with respect to individual facts, and consequently that a world with laws different than those in the actual world should be seen as more distant than any world with the same laws but different individual facts. But this assumption leads to unintuitive consequences, as Lewis points out. Suppose that we are working under the assumption of determinism, i.e. the assumption that the complete state of the world at a given moment t, together with the laws, uniquely determine all the later states. From this it follows that if we consider a world w which differs from the actual world at a moment t, and has all the actual laws, w would have to differ from the actual world at all moments preceding t. But this implies the following counterfactual: “If I sneezed now, the state of the universe would be different at any past moment t”. Counterfactuals for which the antecedent describes an event happening later than the event described by the conditional are called “backtracking”. Lewis maintains that backtracking counterfactuals are usually considered incorrect in standard discourse. In his approach backtracking counterfactuals come out false even under determinism, because possible worlds in which a small violation of laws (“a miracle”) makes it possible for the antecedent-event to occur are usually closer to the actual world than the worlds in which the differences in individual facts stretch infinitely into the past. The world in which we evaluate the counterfactual “If I sneezed at t, then ...” is exactly identical with the actual world up to moment t, when a small miracle occurs making it possible for me to sneeze.

Let us return to the analysis of causation done with the help of counterfactual conditionals. The elimination of backtracking counterfactuals advocated by Lewis solves the main problems affecting the regularity approach: the problem of mixing up causes and effects and the problem of how to distinguish causal relations from the common cause correlations. If an event x of type A causes an event y of type B, the counterfactual “If y had not happened, x would not have happened beforehand” is not (typically) true, because it is a backtracking counterfactual. To evaluate it, we take a possible world which is identical with the actual one up to the moment when y is supposed to occur, and in such a world x happens, but a small miracle prevents y from happening. Regarding the common cause case in which A causes B and then C, it is not true that if B hadn’t occur, C would not have occurred, because A would still be present, causing C to happen. The elimination of B is achieved again not by eliminating its cause A via a backtracking counterfactual (which would eliminate C as well), but by assuming a small miracle which happens just before B and makes it disappear.

But Lewis’s account of causation has its own share of troublesome cases. The identification of the causal relation with the relation of counterfactual dependence between distinct events leads to difficulties with the cases of pre-emption. Suzie’s throw is clearly a cause of the bottle’s shattering, and yet there is no counterfactual dependence between the two events, due to the presence of Billy and his stone. And the counterfactual “If Suzie hadn’t thrown her stone, Billy would have thrown his” is a normal, forward-looking counterfactual which does not require backtracking. Lewis’s response to this case is the following modification of the definition of causal relation. Event x is a cause of event y iff there are events x1, x2, ..., xn such that x1 is counterfactually dependent on x (meaning that if x hadn’t happened, x1 would not have happened), x2 is counterfactually dependent on x1, ..., and y is counterfactually dependent on xn. In the pre-emption case this modification works as follows. We pick an event X between the act of throwing the stone and the shattering such that at its moment Billy has already given up on his throw (this event may be that the stone is flying to its target at a certain speed). Now we can observe that X is counterfactually dependent on Suzie’s throw (if she hadn’t thrown, her stone would not have been flying towards the target), while the shattering is counterfactually dependent on X (if X hadn’t happened, the bottle would not have shattered). The crucial assumption is again that no backtracking is allowed, for we cannot accept that if Suzie’s stone hadn’t been flying, Billy would have thrown his stone). Lewis’s modification has one more advantage: it ensures that the causal relation is transitive (as we remember, the relation of counterfactual dependence is not transitive).

However, Lewis’s analysis faces more threats from modified cases of pre-emption. Suppose that in the Suzie and Billy case Billy has actually thrown his stone, but Suzie’s stone has reached the bottle first, thus pre-empting Billy’s throw. In this case, known as late pre-emption, Lewis’s improved analysis still gives the wrong answer, because there is no moment during the flight of Suzie’s stone at which we could say that if there had been no stone, the bottle would not have shattered. Another troublesome case is called “trumping pre-emption”. A major and a sergeant both shout the same order to a soldier. The soldier obeys, but given the military hierarchy it looks like it was the major’s order and not the sergeant’s which caused the soldier’s action. But there is no counterfactual dependence: if the major’s had not given the command, the soldier would have obeyed the sergeant’s order.

Lewis considered several possible corrections to his approach in order to deal with the problems of late pre-emption and trumping pre-emption. One possibility is to adopt a conception of events whose identity conditions are so strict that even a small modification produces a numerically distinct event (such events are called ‘fragile’). If the shattering of the bottle is a fragile event, then the shattering produced by Billy’s stone is numerically different from the shattering brought about by Suzie’s stone (the stones are flying from slightly different directions, with slightly different speeds, etc.). Thus it is true that if Suzie’s throw had not occurred, this particular shattering would not have occurred (although a similar one would have replaced it). Notice that the fragile character of events is supported in Kim’s conception, according to which events are differentiated by properties, and quantitative properties can be close in value and yet numerically distinct. But an undesirable consequence of this solution is that now plenty of insignificant background conditions will become causes of a given event. Even a gust of wind counts as a cause of the shattering, because it certainly, although minimally, affected the trajectory of the stone, so it is true that if there had been no gust of wind, there would have been no actual shattering, but a very similar yet numerically distinct one. On the other hand, the solution based on the assumption of fragility may be defended against this objection, if we observe that the purported cause (the gust of wind) is itself a fragile event. Hence, the counterfactual assumption that there was no gust of wind can be made true by assuming that the gust of wind was slightly different, and given that the dependence of the stone’s trajectory on the wind is very weak, it is natural to expect that a slight change of the strength of the wind would produce no discernible differences in the qualities of the shattering.

But even the fragility solution is unable to cope with the following counterexample. Suppose that at a point where the railway tracks split two terrorists plan an attack on a coming train. One of them operates the switch, sending the train on a dead end track and causing a train wreck. The second terrorist acts as a back-up, in case the first one does not carry out the sabotage. Clearly, there is no counterfactual dependence: if the first terrorist had not moved the switch, the second one would have acted and the train would have crashed. But the counterfactual dependence cannot be restored even if we assume that all events are fragile. The train wreck is identical regardless of which terrorist moves the switch, or when exactly the switch is moved, or how the switch is moved. The reason is that the characteristic of the train wreck depends solely on the properties of the train and its travel (speed, brakes, etc.), and not the manner in which the switch is moved.

Reading:

E.J. Lowe, Chapter 10, "Counterfactuals and event causation", A Survey of Metaphysics, pp. 174-191.

Causation and counterfactuals

Let us now consider the way Mackie characterizes sufficient and necessary conditions. Standard definitions of these notions are as follows:
A is a sufficient condition of B iff, if A occurs, B occurs
A is a necessary condition of B iff, if B occurs, A occurs (or, equivalently, if A doesn’t occur, B doesn’t occur).

But these definitions are correct only when A and B are general types of events, and not names of individual objects (in that case the right-hand sides of the equivalences have to be interpreted as general statements: “For all x, if x is A and x occurs, then there is a y such that y is B and y occurs”). If A and B are singular names, the aforementioned definitions wrongly imply that all actual events are sufficient and necessary conditions of each other. Mackie attempts to give a better analysis applicable to singular claims, not general ones. His analysis is presented with the help of following equivalences:

x is a sufficient condition of y iff since x occurred, y occurred
x is a necessary condition of y iff if x had not occurred, y would not have occurred.

Mackie stresses that the conditionals used in each explanans can’t be interpreted as material conditionals. But this creates an immediate problem for the regularity approach, as one of its main assumptions is that causation should be explicated without resorting to modal notions, such as necessity or possibility. Mackie suggests the following interpretations of the non-material conditionals used above. He treats them as “telescoped arguments” in which some premises are omitted. For instance, the statement “If the short circuit had not occurred, there would have been no fire” can be expanded into the statement that there are some true universal propositions which together with true statements about the conditions of the house and together with the supposition that the short circuit did not occur logically imply that there was no fire. A similar analysis can be given for the statement “Since x occurred, y occurred”.


One of the most serious challenges for any account of causation is presented by the so-called redundant causation. There are two main types of redundant causation: overdetermination and pre-emption. Overdetermination occurs when there is more than one acting cause, each of which is sufficient for the effect to occur. An example can be an execution by a firing squad, in which each bullet causes a lethal injury. Is each individual shot a cause of the death of the condemned man? In Mackie’s approach the answer is negative, because one of his characteristics of causation is that no alternative sufficient conditions are present (a cause is necessary post factum for the effect). Only the disjunction of all shots is necessary in this sense.

The case of pre-emption can be described using the following example. Two children, Billy and Suzie, are throwing stones at a bottle. Seeing that Suzie has thrown her stone and shattered the bottle, Billy does not hurl his stone, but if Suzie had not thrown, Billy would have thrown his stone. We call this pre-emption, because Suzie’s throw pre-empts Billy’s action which would otherwise have taken place. Any reasonable theory of causation should imply that Suzie’s throw was the actual cause of the shattering. But wasn’t her throw unnecessary, given that Billy was present as a backup? Are the conditions imposed by Mackie satisfied? It turns out that they are, in spite of our initial worries. According to Mackie’s definition, there has to be a set of conditions X such that together with Suzie’s throw (let’s call it A) they would constitute a sufficient condition for the shattering, and moreover no alternative sets of sufficient conditions can be present. We can reasonably believe that without A Bill’s throw together with its conditions would create a different sufficient condition for the shattering of the bottle, but this set is not present at the time of Suzie’s throw. Suzie’s throw is still a necessary part of its own set of conditions – without it this set would not be sufficient, although a different one would be. So Mackie’s definition gives the right answer in the case of pre-emption.

David Lewis has noted that all regularity theories of causation, including Mackie’s, have problems with distinguishing causes from effects, and direct causal links from correlations arising due to a common cause. If only events of type A can cause B, we can say that a given event B is a sufficient condition (or part of a sufficient condition) of A, and hence B is wrongly classified as a cause of A. Even if we artificially exclude this possibility by stipulating that a cause has to be earlier than its effect, still the problem remains. Suppose that an event A causes B and C in succession, and that B can be created only by events of type A. In such a case B is a sufficient condition (or, as in Mackie’s conception, a necessary part of a sufficient condition) of A, and A is in turn a sufficient condition of C, hence B comes out to be a cause of C.

Lewis suggests an alternative account of causation: a counterfactual analysis. The simplest version of such an analysis (the so-called naive counterfactual analysis) is as follows: x is a cause of y iff if x had not occurred, y would not have occurred. But this definition is in need of serious corrections. For instance, suppose that I shut the door by slamming it. If I hadn’t shut the door, I wouldn’t have slammed it (we assume that each slamming shuts the door), but the shutting of the door is not a cause of its slamming. Similarly, if I hadn’t written the letter “L”, I would not have written the word “Lewis”, but the first did not cause the second. To eliminate these counterexamples, we have to assume that x and y are distinct events (i.e. they are not identical, nor one is part of the other). But in order to further advance the counterfactual analysis, we have to understand better the meaning of counterfactual conditionals.

Counterfactual conditionals can be generally presented as statements of the form “If it were (had been) the case that p, then it would be (have been) the case that q”. It is well known that sentences of that form are not truth-functional, i.e. the truth value of the entire complex statement is not determined by the truth values of its components. To see this, it suffices to compare the following two statements: “If Rodin’s sculpture ‘The Thinker’ were made out of wood, it would float” and “If Rodin’s sculpture ‘The Thinker’ were made out of wood, it would fly”. In both cases the antecedent and the consequent are false, and yet the first conditional is true, while the second false. The counterfactual connective is considered to be a modal one, and it receives an interpretation in terms of possible worlds, similar to that of necessity and possibility. The most commonly accepted analysis stipulates that the conditional “If it were p, then it would be q” is true if and only if q is true in all possible worlds in which p is true and which are closest to the actual world of all p-worlds. Thus the statement “If I threw a stone at a window, it would shatter” is true if in the possible worlds in which I throw the stone and which otherwise are as similar to the actual world as the truth of the antecedent allows, the window shatters. And this is what we expect to get, because in such worlds the laws of nature and the properties of materials such as glass and rock will be the same as in our world. On the other hand, if we considered a far away world in which glass is tougher than rock, the consequent would not be true. This shows that for a counterfactual to be true, the consequent does not have to be true in all worlds in which the antecedent is true (this truth condition defines the so-called strict conditional).

Counterfactual conditionals follow a slightly different logic than material conditionals or strict conditionals. Let us consider the three following logical laws: the law of the strengthening of the antecedent, the law of transposition and the law of transitivity. The first one states that if p implies q, then the conjunction p and r also implies q. But consider the following example: If someone shot a gun pointed at my chest, I would be dead, but if someone shot at me and I was wearing a bulletproof vest, I would survive. Given that in the actual world I am not wearing a bulletproof vest now, both statements seem to be true, which shows that the law of the strengthening of the antecedent is violated for counterfactual conditionals. The reason behind this is that we choose different possible worlds to evaluate both statements: the first one is the world where I am being fired at, but I am not wearing any protection, and the second one is the one in which I am protected by a bulletproof vest.

The law of transposition states that if p implies q, not-q implies not-p. A counterexample to this law is as follows: it is true that if I didn’t come to my lecture, the building in which I am lecturing would still stand. But from this it does not follow that if the building collapsed (for instance because of an earthquake), I would still come to the lecture. Finally, the law of transitivity prescribes that if p implies q, and q implies r, then p implies r. We already presented a case which violates this law when we discussed the problem of the transitivity of the causal relation. In this case p = the bomb was not planted, q = the bomb was not defused, r = the politician was assassinated. We can also note that the violation of transitivity follows directly from the violation of the strengthening of the antecedent, given that it is a logical truth that if it were p and r, then it would be p. Now we can choose p, q and r such that it is true that if it were p, it would be q, but it’s false that if it were p and r, then it would be q, and the transitivity is violated.

Reading:

E.J. Lowe, Chapter 8 "Counterfactual conditionals", pp. 137-154, A Survey of Metaphysics.

Regularity theories of causation

Having rejected the postulate of the necessity of causal links, Hume replaces it with the condition of regularity. Paraphrasing his words, if x is a cause of y, each event that is similar to x is followed by an event similar to y. Thus, the complete definition of causation can look like this: x is a cause of y iff x is spatiotemporally contiguous with y, x temporally precedes y, and for all x’, if x’ is similar to x, then there is a y’ such that y’ is similar to y and y’ follows x’. One of the main difficulties with this definition is the notorious vagueness of the notion of similarity between events. If we interpret similarity as being identical in some respect, then it is plausible that every object is similar to every other object, and as a consequence the condition of regularity can never be satisfied. On the other hand, similarity conceived as identity in all respects (qualitative identity) collapses into numerical identity (given the PII), and therefore the condition of regularity reduces to the previous two conditions (contiguity and temporal precedence). The main challenge to Hume’s analysis is to characterize the notion of a relevant aspect with respect to which the similarity between events is interpreted. But even if this can be done, Hume’s analysis is open to serious criticism. Critics point out that there are cases of regular successions of events without a causal link. Days regularly follow nights, and yet there is no causal relation between the two. Similarly, if according to the timetable after the departure of train A train B regularly arrives at the station, this does not indicate that one causes the other. Note that typically the regular but not causal correlations can be explained with the help of a common cause (the succession of day and night is explained by the rotation of the earth, and the timetable acts as the common cause of both the departure of train A and the arrival of train B). So it can be concluded that Hume’s regularity approach has difficulties with distinguishing between direct causal regularities and regularities arising from a common cause.

Another group of counterexamples to the Humean analysis contains cases of causal links where no regularity is present. It is sometimes argued that causal links between unique events (for instance the Big Bang) don’t satisfy the requirement of regularity. Actually, they satisfy it but trivially, making all spatiotemporally conjoined unique events causally connected with each other. But it is unquestionable that in real life we make numerous causal claims where there is no underlying regularity. We say that the failure of brakes was a cause of the car crash, although not every failure of that sort leads to a crash. This clearly shows that Hume’s analysis is in need of serious corrections.

Neo-Humean approaches to causation try to eliminate some of its weak points. One of them is the nomological conception. According to it, an event x of type A is a cause of an event y of type B iff x and y occur in conditions C and there is a law of nature according to which if an event of type A happens in conditions C, an event of type B occurs. There are two main differences between this approach and Hume’s regularity account: the presence of background conditions C and the reference to laws of nature. The role of laws of nature is to eliminate accidental regularities, while the presence of conditions C should take care of the problem of the apparent lack of regularity of some causal links (the failure of brakes leads to an accident only in certain specific circumstances). But these additions to the regularity approach bring new problems. What are laws? Hume himself insisted that laws are nothing over and above mere regularities, but if that’s the case the addition of laws to the definition of causation does not constitute an improvement with respect to the old, Humean version. The introduction of conditions C creates a different problem. By a crafty selection of appropriate “conditions” we can make virtually any succession of events a causal one. It may be claimed, for instance, that the snapping of my fingers causes the lights to go out if we include in the appropriate conditions that someone turns off the switch at the same moment. Yet another objection is that there are laws which are not causal. Pascal’s law states that a gas exerts equal pressure in all directions, but it is not correct to say that the fact that the pressure in one direction equals p is caused by the fact that the pressure in the opposite direction is also p. Finally, most laws of physics are symmetric in time, but from this it does not follow that backward causation is a common fact.

One of the most sophisticated versions of the regularity approach is the conception of causation proposed by J.L. Mackie. Mackie notes that the notion of a cause is closely related to sufficient and necessary conditions, but a cause of x cannot be simply defined as a necessary or a sufficient condition of x. Let us consider Mackie’s example with a fire of a house being caused by a short circuit. The short circuit by itself is not sufficient for the house to burn down; other conditions have to be present, such as the presence of inflammable materials, of oxygen, the absence of automatic sprinklers and smoke detectors, etc. Generally speaking the short circuit is not necessary for the fire either, for fires can start in many different ways, for instance after a strike of a lightning bolt. But in the actual conditions the short circuit was necessary, because without it the conditions themselves would not have created the fire. (Mackie speaks in this case about a necessary condition post factum.) The short circuit is an INUS condition for the occurrence of the fire, where the acronym INUS stands for an Insufficient but Necessary part of an Unnecessary but Sufficient condition. A more precise definition of an INUS condition is as follows: A is an INUS condition for B iff there are conditions X and Y such that (AX or Y) is a necessary and sufficient condition of B, but neither A nor X is a sufficient condition of B. In our example A is the short circuit, B is the fire, X refers to all the conditions which together with A were sufficient for the fire to occur, and Y stands for a disjunction of all alternatives ways of starting a fire. In most cases a cause of an occurrence B is an INUS condition of B such that it occurred, and no alternative conditions Y were present (however, Mackie admits the possibility that a cause itself may be sufficient for its effect, or even sufficient and necessary – he refers to all such cases jointly as “at least” INUS conditions of B).

It has to be noted that some further restrictions on the causal condition A have to be introduced, otherwise Mackie’s analysis will lead to obviously incorrect conclusions. To see this, let us use the letter C to abbreviate the complete sufficient condition of B that was actually present, and let us select any fact S irrelevant to the occurrence of B that happened simultaneously with C (for instance the fact that when the fire started somebody walked past the house whistling “Ode to joy”). The formula [(S or C) and (not-S or C)] is logically equivalent to C, and hence if there is a condition Y such that (C or Y) is a necessary and sufficient condition of B, then {[(S or C) and (not-S or C)] or Y} is a necessary and sufficient condition of B too. But now observe that the condition (S or C) satisfies the requirement for an INUS condition of B. (S or C) is not sufficient for B, nor is (not-S or C), and yet their conjunction is sufficient (as it is equivalent to C). But it is highly unintuitive to pick the disjunction S or C as a cause of B. To eliminate cases like this it may be suggested that causes should not have the form of disjunctions of simple events.

An interesting element of Mackie’s conception is that he admits that causal claims are always made in a context. In order to account for this fact, he introduces the notion of a causal field. Let us consider as an example the case of a person going down with flu. The answer to the question “What caused this man to contract the flu?” depends on the context. If we consider as the causal field the set of all moments in his life, and ask why he contracted the disease at this moment rather than any other, then the correct answer may be that he was infected by influenza viruses. But we can also select as the causal field the set of all people who came into contact with influenza viruses, and we may be interested in selecting the factor which is responsible for the fact that some of them contracted the disease, while the others did not. Mackie introduces the causal field to his definition of a cause, assuming that the conditions characterizing this field are present when the cause is present.

Reading:

M.J. Loux, Chapter 6 "Causation", pp. 187-203, Metaphysics: A Contemporary Introduction.

Causation and necessity

We can distinguish two types of causal statements: general causal statements and singular ones. General statements relate types of phenomena (for instance: smoking causes cancer), whereas singular causal statements connect individual occurrences (for example: the cause of the sinking of the Titanic was that it collided with an iceberg). While the two categories of causal claims are undoubtedly related, their relation is not straightforward. It may seem that general causal statements of the form “Phenomenon A causes phenomenon B” can be reduced to the following singular claim: “For all x, if x is of type A, then x causes some y of type B”. But this won’t work. From the fact that smoking causes cancer it does not follow that every smoker will suffer from cancer. General causal claims are very often statistical only, and their truth is typically hedged by the ceteris paribus condition. On the other hand, if we wanted to define singular causal claims of the type “x causes y” with the help of the general formula “The type of phenomena A to which x belongs causes the type of phenomena B containing y”, we would encounter an immediate problem connected with the fact that each individual event can be classified into many distinct types. In the following we will restrict our analysis to singular claims only, and therefore we will interpret causation as a relation between individual objects.
It may be useful to start an analysis of causation from the following questions:

(1) What are the relata of the causal relation?
(2) What are formal properties of the causal relation?
(3) What is the temporal relation between a cause and its effect?

(1) Typically three categories of objects are regarded as being capable of standing in the causal relation: things, events and facts. One natural way of speaking about causal links seems to identify causes as things. For instance, we can say that John smashed a window glass with a stone, and a car hit a pedestrian. This suggests that causes are things (John, car) while effects are events (shattering the window, hitting the pedestrian). But clearly this is an oversimplified way of speaking. If John is busy talking on the phone, there is no shattering, although the purported cause (John) is still present. If the car is parked in a garage, no pedestrian is in danger of being hit by it. Strictly speaking, it is not John but his throwing the stone that causes the breaking, and it is not the car but its particular movement that causes the hitting of the pedestrian. This observation leads to the most commonly accepted conception of causation, according to which both causes and effects are events (throwing the stone – shattering the glass, movement of the car – hitting the pedestrian).

However, some philosophers insist that this account is too restrictive, as it does not make room for cases of negative causation. Sometimes it seems natural to single out absences of events rather than events themselves as causal factors contributing to a given effect. We say that the lack of attention of the driver was a cause of the crash, and that the absence of sprinklers contributed to the fire. But there are no negative events (in Kim’s conception, events are property attributions, but it is customary not to admit negative properties). In order to admit negative causation (sometimes also called causation by omission) it is proposed that causes and effects be facts, not events. Facts are just ontological counterparts of true statements, so there is no problem with the assumption that there are negative facts corresponding to negative statements. But critics point out that negative causation is really not necessary, and moreover that admitting it opens the door to many unintuitive cases of spurious causation. It may be claimed that underlying every case of apparent negative causation there is an instance of positive causation (for instance the driver’s lack of attention could have been actually his talking on the phone). And we tend to dismiss statements of the sort “The fact that I had not been struck by lightning caused me to survive” if there was no reason to expect that the lightning was imminent.

(2) It should be clear that the causal relation is not reflexive (there are events that don’t cause themselves). But is it irreflexive (no event is a cause of itself)? That depends. If we admit the possibility of causal loops (as in time travel), and we agree that causality is transitive, then there may be cases of self-causation (x causes y and y causes x, therefore x causes x). Similarly, causality is not symmetric, but it is open to debate whether it is asymmetric (if there are causal loops, clearly it is not asymmetric). The case for transitivity looks plausible enough, but recently this feature of causality came under attack. Some philosophers point out that there are cases which seem to violate the transitivity requirement, such as the following one. A bomb had been planted at the door of a politician’s house, but luckily it was spotted by the security and defused. It is natural to assume that the placing of the bomb was a cause of its defusing (if there hadn’t been a bomb, there wouldn’t have been the act of defusing), and the defusing of the bomb causes the politician to survive. But it is unnatural to say that the placing of the bomb was a cause of the politician’s survival (clearly the counterfactual “If the bomb had not been planted, the politician would not have survived” is false).

(3) It is typically assumed that a cause happens earlier (or, at least, not later) than its effect. But, again, if we want to admit that it is conceptually possible to have backward causation, we have to reject this requirement.

The main question we have to ask now is “What is causation?”. Answers to this question can be given in the form of a reductive analysis, explicating the causal relation in terms of some more fundamental concepts. We will start with the most famous reductive analysis of causation given by David Hume. Hume observes that it is an uncontroversial fact that causation displays the following two properties: the cause and the effect are contiguous in space and time (they “touch” each other), and the cause temporally precedes the effect. Actually, both claims can be questioned. The issue of temporal precedence has been already mentioned in point (3). As for the contiguity, at best it can be applied to direct causes only. Clearly there is a temporal and spatial gap between my act of hurling the stone and the smashing of the window. But it can be claimed that there has to be a chain of events contiguous in space and time leading from the throwing to the breaking. Still, this does not seem to be conceptually necessary. There is nothing inconsistent in considering causal links acting at a distance with no intermediate stages. Actually, this is how gravitational interaction between massive bodies can be assumed to work in Newtonian mechanics. So it looks like the two conditions proposed by Hume are not necessary for causation to occur. But we have to agree with Hume that they are not sufficient either, for there are plenty of events following one another which are not causally connected.

Hume then asks, what should be added in order to have a sufficient condition for the presence of a causal link. One typical response is that the cause has to be necessarily linked with its effect, or in other words, that if the cause occurs, the effect must occur. But Hume famously questions this. Firstly, he notices that the purported necessity cannot be of the logical kind, for no contradiction arises from the supposition that a given event does not produce its expected effect. I can imagine without contradiction the stone magically passing through the glass, or bouncing off it. But perhaps the necessity connecting causes and effects is of a different kind (nomological, or physical). Hume’s response is that no such necessity is given to us in sensory experience. We never perceive two events as connected, only as conjoined.

Clearly, Hume’s criticism of the necessary character of causation has its roots in his version of empiricism. Hume insists that every meaningful concept should be traced back to some sensory experience (‘impression’). But this requirement may be seen as overly restrictive. Hume’s radical empiricism does not square well with modern science which commonly postulates the existence of unobservable objects and properties. According to Hume’s criterion, along with the notion of necessity we should abandon such concepts as that of atoms, electrons, electromagnetic field, etc., as they cannot be supported by any direct sensory data either. On the other hand, more moderate versions of empiricism can in principle accommodate the notion of a necessary causal link, if it is treated as a theoretical concept used to make empirical predictions and explain observable facts.

Reading:

B. Garrett, "Causation", pp. 53-66, What is this thing called metaphysics?

How things persist

Things exist in time. More specifically, they persist. A thing, such as a tree, remains the same object throughout its existence, although it changes some of its properties, loses some of its parts and acquires new ones. The relation that holds between the same object at different times is called ‘diachronic identity’. But the question is: What is this new type of identity? How does it relate to numerical identity? Is diachronic identity reducible to numerical identity, or perhaps to qualitative identity? These questions are answered differently by two major conceptions of how things persist in time: endurantism and perdurantism. Endurantism can be characterised broadly as the position according to which things persist in time by being wholly and completely present at every single moment of their existence. On the other hand, perdurantism assumes that at a given moment only a small part of a thing is present. The whole thing is an object which extends in time as well as in space.

Let us look more closely at both views. Endurantism seems to be in agreement with our pre-philosophical intuitions regarding persistence. When I look at a table in front of me, I believe that no part of it is missing from my view. Things have only spatial parts and spatial dimensions, but no temporal ones. According to endurantists, the expressions “Napoleon at the time of the battle of Austerlitz” and “Napoleon at the time of the battle of Waterloo” refer to one and the same individual: Napoleon. Hence diachronic identity reduces to numerical identity. Things are three-dimensional objects taking up various spatial regions at different times.

Perdurantism, on the other hand, claims that things are four-dimensional objects taking up regions of space-time. The expressions “Napoleon at the time of the battle of Austerlitz” and “Napoleon at the time of the battle of Waterloo” refer to numerically different objects: temporal parts of the four-dimensional entity that we call “Napoleon” and whose temporal dimension stretches from the moment of Napoleon’s birth to his death. Consequently, the relation of diachronic identity is not defined as numerical identity, but instead can be explicated as the relation that holds between any two temporal parts of the same thing: x is diachronically identical with y iff x and y are different temporal parts of individual z. (An alternative interpretation of diachronic identity under perdurantism is that it is reducible to numerical identity after all, when we say that “Napoleon at t1 is identical with Napoleon at t2” means “The four-dimensional object whose temporal part at t1 is Napoleon at t1 is numerically the same as the four-dimensional object whose temporal part at t2 is Napoleon at t2”.) Temporal parts of an object can be divided into stages and slices. A stage of a thing x is a part of x that occupies a non-zero interval (it has a non-zero temporal “width”), whereas a slice of x is a part of x taken in a zero-length moment of time, and thus it has no temporal dimension. Slices are three-dimensional, and they represent what we would usually refer to as objects of our perception. It is noteworthy that the way things persist according to the perdurantist is analogous to the temporal existence of events. An event, such as the battle of Waterloo, is never fully present in an interval that is shorter than its entire duration.
The main motivation for perdurantism comes from the problem of change which threatens the endurantist approach. Things change their properties in time: for some moments t1 and t2 and a property P it is the case that x has P at t1 and x does not have P at t2. But according to endurantism x at t1 is numerically identical with x at t2. If we apply Leibniz’s law, which states that if x = y and Px, then Py, then we have to conclude that an object has P and doesn’t have P, which is obviously a contradiction. Perdurantism avoids this problem by assuming that the properties P and not-P are attributed to numerically different individuals: different temporal slices. But the endurantist is not without options with respect to the problem of property change. One solution is to relativise properties to time. Let us consider a poker which is cold at t1 and hot at t2. We may say that the poker possesses the property of being-cold-at-t1 and being-hot-at-t2, and these properties are not mutually exclusive as long as t1 is different from t2. But an objection can be raised that this approach treats properties as if they were relations between objects and moments, and consequently no property can be intrinsic. And, besides, isn’t it legitimate to speak about properties simpliciter, without any temporal relativisation? Another solution, available to the endurantist, is to relativise the relation of possession between the object and its properties. Objects don’t just possess properties, but they always possess them at certain moments. This position is known as adverbialism, as it amounts to the adverbial modification of the verb “be” (the poker is-at-t1 cold and the poker is-at-t2 hot). One consequence of this approach is that there is no single relation of exemplification between particulars and universals, but an infinite (even uncountable) number of different relations of exemplification. Finally, let us notice that all the above endurantist solutions seem to assume that moments exist independently, and therefore commit themselves to the substantivalist view.

We will now consider an argument against endurantists which employs the notion of change of parts. The argument is due to Peter van Inwagen, with some corrections added by Mark Heller. Suppose that a person X underwent an amputation of his left hand. Let t1 denote a moment before, and t2 after the amputation. Let us also denote by ‘X-minus’ the whole consisting of X’s body without the left hand (regardless of whether the hand is attached to it or not). The endurantist should accept the following identity statements:

(1) X at t1 = X at t2
(2) X-minus at t1 = X-minus at t2
(3) X-minus at t2 = X at t2

But from these three premises we can derive, using the assumption of the transitivity of identity, that

(4) X at t1 = X-minus at t1

This conclusion is clearly unacceptable. My body is not identical at this moment with my body minus my left hand. Now we will have to look closely at the justification of all the premises (1)-(3), to see which one should be rejected. Premise (1) follows from endurantism and the assumption that an object can lose its part without losing its identity. Premise (2) is a simple consequence of endurantism. Premise (3) is implied by the principle according to which two numerically distinct objects cannot occupy the same spatial region at the same time. Now it should be clear that rejecting endurantism and accepting perdurantism solves the problem. If we agree that the expressions “X (X-minus) at t1 (t2)” refer to temporal slices of appropriate four-dimensional objects, then premises (1) and (2) are evidently false, although (3) is unquestionably true. Another possible interpretation of (1)-(3) under perdurantism is that actually these identities are between appropriate four-dimensional objects, identified by their three-dimensional slices. In that case (1) and (2) are true, but (3) becomes false (two distinct four-dimensional objects can nevertheless share their three-dimensional slices).

But the endurantist has some viable strategies of defence. Firstly, he can claim that a thing cannot lose any of its parts without losing its numerical identity. But this is a highly unintuitive supposition, and if it’s true, then with each passing second our bodies are turned into new things, because they are constantly losing old parts and acquiring new ones. Secondly, the existence of X-minus can be called into question. For instance Van Inwagen rejects the doctrine which he calls “the doctrine of arbitrary and undetached parts”. X-minus before the amputation is not a separate, autonomous object, but an undetached part of X, and its existence is questionable. Thirdly, the assumption that two distinct things cannot occupy the same space at the same time can be rejected. It is argued that a sculpture, for instance “The Thinker” by Rodin, is numerically distinct from the lump of material it is made of (bronze) and yet throughout some period of time the two things occupy the same space. Finally, some authors question the transitivity of identity. According to Peter T. Geach, identity is a relative and contextual notion. We can say that X at t1 is the same person as X at t2, and that X-minus at t1 is the same body as X-minus at t2, but from this it doesn’t follow that X at t1 is the same body as X-minus at t1, nor that X at t1 is the same person as X-minus at t2.

One problem for perdurantism is that it does not offer clear criterions of how to distinguish four-dimensional wholes which are genuine things from arbitrary regions filled with matter. A given temporary slice of a four-dimensional object has an infinite numbers of future continuations. Which one is selected as the right one, and why? Yet another difficulty was noticed by van Inwagen. Four dimensional objects are often presented as collections of stages (slices). But a collection of objects possesses its elements necessarily. From this it follows that Napoleon could not have different stages from the ones he really had (for instance, he could not have been born earlier or later).

Endurantism is typically associated with presentism, and perdurantism with eternalism. But other combinations are also possible. Perdurantism can logically coexist with the theory of the growing (shrinking) universe. In such a case things would be four-dimensional wholes that grow or shrink as time passes. It is also possible to have both endurantism and eternalism. It seems that the only combination which is logically impossible is that of perdurantism and presentism (although some authors disagree with that). Perdurantism assumes that things have different temporal parts, so it is essential to admit that moments other than the present one exist. Also, presentism implies that the universe is three-dimensional (as time is not a dimension, because it is reduced to a point). But perdurantism identifies things with four-dimensional objects, and four-dimensional objects cannot exist in a three-dimensional world.

Readings:

E.J. Lowe, Chapter 3 "Qualitative change and the doctrine of temporal parts", pp. 41-58, A Survey of Metaphysics.
M.J. Loux, Chapter 8 "Concrete particulars II: persistence through time", pp. 230-256, Metaphysics: A Contemporary Introduction.

Time's arrow

Not only vicious causal loops can create a problem for time travel. Suppose that we have a situation in which a later event A causes an earlier event B, whereas B causes A. This does not seem to lead to any logical contradiction, yet it gives rise to a serious conceptual problem. Suppose that in the year 2011 you are visited by yourself from the year 2020, and your older self hands you in a complete blueprint for a time machine. You then build it, and in 2020 you enter your machine to visit yourself in 2011 and deliver the plans. The question is: where did the blueprint come from, and who invented the time machine? Yet another paradox goes as follows. The traveller from the future hands you in an empty notebook and instructs you to enter the time machine, go to the future and write an entry about your journey in the notebook. Then you are supposed to give the notebook to the traveller in the future, whereupon he enters the time machine, goes back in time and gives you in the past the instruction to write another entry in the notebook while travelling to the future. This creates an immediate problem: how many entries are there in the notebook? It looks like the number of entries should correspond to the number of ‘loops’ made, but actually there is only one loop, not many repeated ones.

Apparently this last example is akin to the grandfather paradox, as the traveller attempts to do something in the past (i.e. convince you to enter the time machine and write something in the notebook) which would cause some future event (the traveller’s entering the time machine with an empty notebook) to disappear. So perhaps the solution would be, again, that something should prevent you from writing an entry in the notebook, and thus the situation in the future will be exactly as it should: the time traveller, despite his attempts, will enter the time machine in the future with an empty notebook. On the other hand, if the notebook you receive in the past already contains an entry, the problem looks more like the blueprint paradox we discussed above. You try to write another entry, but you fail, so you hand in the notebook with the old entry to the future traveller. But the question remains: who wrote the entry?

Even if the notion of time travel can escape some of the most threatening conceptual difficulties, the question remains whether it is nomically possible. We will not attempt to answer this question generally, but only with respect to one physical theory, namely the special theory of relativity. First let us notice that the conceptual framework of STR seems to leave room for a realisation of a journey to the past. Let us consider three events A, B and C such that C is in the absolute past of A, whereas B is space-like separated from both A and C. We know that there is a frame of reference f in which event B looks later than event A, so at least in principle it seems possible to send somebody from A to B along the ordinary direction of time from the past to the future. But there is also a different frame of reference f’ in which C is later than B, hence another trip from B to C seems feasible. Combining these two together, we can see that the journey took us from A to its absolute past. But this scenario is not physically realizable, due to the fact that in order to reach B from A (and C from B) the traveller would have to accelerate beyond the speed of light, and this is prohibited by the laws of relativistic dynamics. Let us note in passing that STR does not exclude the possibility of the existence of superluminal particles (called tachyons) – particles that travel faster than light. But if tachyons exist, they cannot cross the light barrier by slowing down to a subluminal speed.

How can we explain the fact that time has an objective direction, in contrast with space? And what does it mean precisely that time has a direction? The directionality (or asymmetry) of time can be defined as the property of being ordered by the asymmetric relation “being earlier than”. But points on a spatial line can be similarly ordered in one direction. To distinguish between the two cases let us observe that the temporal relation of being earlier than is an intrinsic relation of two moments which does not depend on anything extrinsic from these moments. On the other hand, for a spatial line to be ordered in a similar fashion we have to choose some external “point of reference”. For instance, we can order points from left to right, but this ordering depends on the position of the external observer. Analogously, it is possible to order points on a meridian, but again we have to choose the South Pole and the North Pole to do that.

Is the directionality a primitive and irreducible property of time, or can it be grounded in some more fundamental property? There are three standard ways of grounding time’s arrow: in the psychological arrow, causal arrow and thermodynamic arrow. The psychological arrow is based on the observation that perceptions always precede memories. My perception of a given occurrence is designated as being earlier than my memory of the same occurrence. But clearly this criterion can be applied to a very narrow category of events only: the mental ones. How, then, can we use the criterion of such a limited scope to order all the events? The answer is that in some circumstances it may be sufficient to define a direction for only two points in time, and the rest will be ordered accordingly. Let us suppose that the set of events is equipped with a symmetric structure defined by the three-place relation of being between. This means that we can say that an event x lies between events y and z (in short, B(x, y, z)) without deciding whether y is earlier than z or vice versa. Now the question is: What has to be added to the structure defined by the relation B in order to have a full linear order? It is easy to notice that if we decide for a pair of selected events a and b which of them is earlier and which later, the rest of the events will be ordered thanks to the relation B. For instance, let’s suppose that a is earlier than b (E(a, b)), and consider two events x, y lying between a and b (i.e. such that B(x, a, b) and B(y, a, b)). In this case, E(x, y) iff B(x, a, y) (or, equivalently, B(y, x, b)). This definition can be easily extended for all possible distributions of x and y with respect to a and b.
From this it follows that in principle we need only one instance of a perception and its memory to give time its direction (under the condition that the relation of betweenness is already given). But in practice we obviously need more such instances. The psychological arrow suffers from obvious shortcomings. It is strongly anthropocentric, as it requires the existence of humans (sentient beings) in order for time to have a direction. But surely there would be earlier and later events even if there were no humans. Another difficulty is that this criterion excludes a priori the possibility of clairvoyance (which may be impossible physically, but does not look like a contradictory concept). Knowing the future requires that a ‘memory’ of a given event is earlier than its perception (by ‘perception’ we mean the ordinary way of seeing things, and not the one claimed by the clairvoyants), but this contradicts the criterion.

The causal arrow assumes that if x is a cause of y, x is earlier than y. This reduction of the directionality of time encounters the following two objections. First, it excludes the possibility of backward causation (we considered this possibility when analysing time travel). Second, grounding the direction of time in the causal relation requires that we define causality without referring to temporal precedence. And yet some conceptions of causality, famously including Hume’s analysis, rely on just that. If we insist that causality grounds time’s arrow, we have to find an alternative way of making sure that the causal relation will be asymmetric.

The most commonly accepted of all three arrows is the thermodynamic arrow. It relies on the second law of thermodynamics which states that for isolated systems their entropy (a measure of disorder) never decreases. The second law explains why we observe so many irreversible processes (heat transfer, dissolution, etc.). Thus the thermodynamic criterion states that if x and y are two macrostates of a given isolated system, and the entropy of x is smaller than the entropy of y, then x is earlier than y. But there is a well-known foundational problem associated with this account of time’s arrow. The theory which describes the interactions that underlie all thermodynamic processes is just classical mechanics of many particles, and this theory is known to be time-reversible (if a process is admissible by the laws of classical mechanics, so is its reversed version). The standard attempt to explain the observed thermodynamic asymmetry is due to L. Boltzmann, and it is based on the fact that one macrostate of a system can be realized by a great number of various microstates (defined by the positions and velocities of all individual molecules of the system). Boltzmann proved that for a given macrostate the vast majority of corresponding microstates are such that their dynamic evolution leads to the increase of entropy. Thus the second law can be seen as probabilistic only, but the probability that a system will actually violate it is extremely small.

However, one of the main problems with Boltzmann’s argument is that it is essentially symmetric, i.e. it can be used in support of the claim that the system at a given moment evolved from a state of higher entropy. One way of avoiding this difficulty is to assume that the initial state of the universe had extremely low entropy, and hence the overwhelming tendency of the systems is to increase their entropy. This means that the thermodynamic arrow has to be grounded ultimately in a singular fact about the origin of the universe, together with the probabilistic laws of statistical mechanics. One interesting consequence of this fact is that it is theoretically possible that when the universe reaches the state of maximal entropy, the tendency can be reversed and the majority of systems will follow the entropy-decreasing evolution. It is not clear whether this would mean that the direction of time got reversed.

Reading:

E.J. Lowe, Chapter 18 "Causation and the direction of time", pp. 325-344, A Survey of Metaphysics.

Time in special relativity and time travel

We will now discuss the changes in the notion of time and space brought about by the development of the special theory of relativity. Let us start with a brief characteristic of the classical account of space and time as incorporated in the Galilean-invariant version of Newtonian mechanics (i.e. the version that dispenses with the concept of absolute motion and absolute position). The main assumption is that no inertial frame of reference is privileged, and uniform motion is relative. However, the notion of simultaneity remains absolute, i.e. frame-independent. For each moment of time the set of events occurring at that moment defines a three-dimensional space with the usual Euclidean metric (distance) attached to it. Hence, Galilean space-time foliates naturally into separate spaces defined at different times. However, there is no absolute connection between points in spaces at different times (no absolute co-location). The question of which point of space at t2 is a continuation of a point at t1 does not receive a frame-independent answer. If we think that spatial points (places) are objects which retain their identity over time, then no such objects are present in the Galilean version of classical mechanics.

A step towards special relativity is the realisation that the relation of simultaneity has no obvious empirical content, due to the fact that all signals (including light) travel at finite speeds. What we observe as our ‘present’ is actually already in the past (the farther, the more distant the observed event is). The standard, operationally defined notion of simultaneity is given as follows: two events x and y are simultaneous iff light signals sent from x and y meet exactly half way between x and y. But this definition is obviously not frame-independent. Suppose that the definition of simultaneity is satisfied in a frame f, and let us consider another frame f’ which moves with respect to f in the direction of the event y. The spatiotemporal point where the two beams of light meet will not be located in the middle of the distance between their locations x’ and y’, but rather closer to x', so from the perspective of f’ y happened earlier than x. We have to add that the signal definition of simultaneity presupposes that the speed of light is constant in all frames of reference.

In special relativity neither simultaneity nor co-location are invariant notions (independent of the frame of reference). Thus space-time cannot be absolutely divided into space and time. However, there is a relation between events (spatiotemporal points) which stays the same in all frames of reference. This relation is defined by the so-called spatiotemporal interval: cdt^2 – dx^2 – dy^2 – dz^2., where c – the speed of light, and dt, dx, dy and dz are temporal and spatial intervals between the two events. Two events for which the spatiotemporal interval is positive are called ‘time-like separated’. Such events can be connected by a signal travelling slower than light. If the interval equals zero, the events can be only connected by a beam of light. Events separated by a negative interval are called space-like separated. Such events cannot directly communicate by way of sending signals.

The basic structure of relativistic space-time (so-called Minkowski space-time) can be given with the help of light cones. For a given event x, its forward light cone consists of all events reachable from x by beams of light. Similarly, x’s backward light cone contains all events which can reach x using beams of light. The area within x’s backward light cone is called its absolute past, and within the forward light cone its absolute future. The events outside of both light cones are neither past nor future with respect to x, but they cannot be interpreted as being simultaneous with x. Their temporal relation with x is frame-dependent: for every event y space-like separated from x there is a frame of reference f in which x and y are simultaneous. But if we choose a different event y’ also space-like separated from x, the frame of reference in which y’ is simultaneous with x will be generally different from f.

Now we will discuss some issues related to the asymmetry of time. Let us start with the problem of time travel. Is time travel conceptually possible, or does it involve logical contradiction? First we have to decide what process can be called time travel. For a given object we can say that it travels in time if there is a difference between its own time and the external time of the world. If the interval measured with the object’s time is shorter than the external interval, we can speak about travel into the future. Such travel is not only possible but actually happens, according to the special theory of relativity. Due to the effect known as time dilation, if an object moves, its own time measures shorter intervals than the external time. If a traveller embarks on a journey and then comes back, his clock will show that his journey was shorter than when measured by the external clocks (this is the basis of the so-called twin paradox).

The most radical type of time travel is when the traveller goes into the past, i.e. the duration of his journey measured according to the external time is actually negative. Some philosophers claim that travel into the past involves contradiction, because a time traveller could change the past, and this is impossible. More specifically, the concept of changing the past is applied to states of affairs. In order to change a given past state of affairs – for instance, by scratching an inscription on a rock a thousand years ago – this state of affairs (the rock being unscratched) has to be both real (‘before’ the change) and unreal (‘after’ the change). But of course the time of the occurrence of these two contradictory states of affairs is the same, so the contradiction seems unavoidable. But it can be observed that the same problem applies to the apparently uncontroversial case of a change in the future. To literally change a state of affairs at a future time t requires that this state of affairs exist before but not after my action. But, again, this leads to logical contradiction. This difficulty can be avoided, though, when we apply the notion of change to things, not states of affairs. I certainly changed the past rock: before my intervention it wasn’t scratched, and afterwards it bore an inscription.

The most celebrated grandfather paradox exploits the possibility of vicious causal loops that seems to be opened by admitting time travel. The time traveller goes back to the times of his grandfather’s youth and kills him in the past. If his grandfather dies before he can have any children, the traveller will not be born in the future, and a contradiction ensues: the traveller both exists in the future (because he came from there to kill the grandfather) and does not exist (because his grandfather dies childless). This paradox has the following general form: there are two events A (the beginning of the journey to the past) and B (the killing of the grandfather) such that A is later than B, A causes B and B causes non-A. In order to avoid the problem, serious restrictions have to be imposed on the possible interactions of the time traveller with the past. Speaking loosely, each time the traveller tries to kill his grandfather something must get in the way to prevent him from accomplishing this task. Note that this extends to any action of the traveller (even seemingly innocent, such as leaving footprints on the grass) which might lead to the consequences threatening the entire future travel to the past as it precisely occurred. One general solution may be to assume that causal links directed from the future to the past do not ‘couple’ with the causal links leading in the ordinary temporal direction. But this would effectively imply that the traveller could only observe the past and not interact with it the normal way.

Reading:

B. Garrett, "Time travel", pp. 94-99, What is this thing called metaphysics?

Absolutism and relationism

Now we will consider the question of the ontological status of time itself, and its relation to the material world. The problem can be stated as follows: is time a fundamental substance, capable of independent existence, or is it ontically dependent on things/events? One particular way of cashing out this general question is to ask whether it is possible for time to exist without any change. Of course the possibility in question has to be considered as metaphysical (stronger than logical but weaker than physical). The situation in which time exists but there is no change can be described as follows: there is a non-zero interval (t, t’) such that for any two moments t1 and t2 from this interval, all objects have exactly the same properties at t1 and at t2 (we can call the world in the interval a “frozen universe”). But now it can be argued that because all situations within the interval are indistinguishable, the statement that the length of the interval is non-zero has no empirical meaning. We could also appeal to the Principle of the Identity of Indiscernibles to argue that the moments t and t’ should be identified. However, Sydney Shoemaker has proposed an argument showing that under certain circumstances the hypothesis of the frozen universe can offer some advantages even to an empiricist. Suppose that the universe consists of three parts A, B, and C, and that the data gathered shows that region A goes through the period of a freeze every three years, region B freezes every four years, and region C freezes every five years. From these, empirically confirmed hypotheses it follows that the entire universe will freeze every 60 years, but of course this consequence can never be empirically confirmed or disconfirmed. Facing the choice between two empirically equivalent hypotheses we should choose the simpler one, and this is the one which assumes that there are no gaps in the regular patterns of freezing for regions A, B, C. So methodological postulates accepted by empiricists favour the hypothesis that the entire universe can freeze.

The issue of the dependence between time and the material world can be considered in an even more radical way. We may ask whether it is possible for time to exist without any events taking place. Can there be a period of time consisting of “empty” moments? Notice that this would be a case of time without change, but not all cases of time without change are cases of empty time. The view that it is fundamentally, metaphysically possible for such a situation to occur is known as absolutism, or substantivalism, with respect to time. On the other hand, those who believe that moments cannot exist without participating events (whether they are changes or not) are called relationists. Relationists believe that only spatiotemporal objects (things, events) and their temporal and spatial relations exist in the fundamental sense. Temporal objects (moments) are derived from the fundamental temporal relations. We should notice that absolutism and relationism can be formulated with respect to space as well as time. Relationists with respect to space believe that spatial points and the relations between them are mere reflections of events and their spatial relations (in particular, the relation of co-location).

Leibniz gave a strong argument against absolutism and in favour of relationism. Suppose that absolutism with respect to space is right and that space and spatial points exist independently of the material objects occupying them. Then shifting the entire world 5 metres in one direction would produce a distinct state of affairs (different points would be occupied by different objects) which nevertheless is indiscernible from the original one. Leibniz points out that such a possibility violates the principle of the identity of indiscernibles, and the principle of sufficient reason. We may also add that this argument shows that absolutism violates the principle of ontological parsimony, because it postulates the existence of objects (spatial points) which are not necessary to explain observable phenomena. In addition to this argument, known as the static shift argument, Leibniz also produced another one, based on a dynamic shift. The absolutist should consider the following two states of affairs as distinct: one in which the entire universe is stationary, and the other, in which it moves at a constant speed in a particular direction. But again, there is no observable effect that could distinguish between the two.

Newton was a proponent of absolutism. In support of his view, he pointed out that it is possible to distinguish between being at rest and moving, but this possibility applies only to a certain category of motions, namely accelerated motions. One example of such motions is rotation, which produces observable effects in the form of the centrifugal forces. Newton used this fact in his famous bucket argument. Consider a bucket full of water, suspended on a rope. The rope is twisted, and then released. In the first stage the bucket will start rotating, but the water will for some time remain stationary. The surface of the water will be flat. In the second stage the water being dragged by the sides of the bucket begins its rotational motion. This stage is characterised by a concave surface of the water, due to the centrifugal forces. Finally, the bucket is stopped, but the water inside it will continue spinning for some time. The concavity of the surface is still visible. Newton compared the first and the third stages, arguing that they are perfectly symmetrical with respect to the relative motions of the bucket and water. And yet only in the third stage we observe the concave surface. This can be only explained by postulating the existence of absolute space, against which the water rotates in the third, but not the first stage. But later critics, including Ernest Mach, pointed out that the situation is not exactly symmetrical. In the first case the water is stationary with respect to the rest of the universe, whereas in the second the water moves with respect to the fixed stars. Mach observed that Newton’s argument would be valid if we somehow managed to make the entire universe spin around the water inside the bucket. But this is impossible to achieve. Mach’s position is sometimes interpreted as suggesting that the inertial effects (e.g. centrifugal forces) are a result of the influence of all the masses in the universe on a given system. But Mach himself was sceptical of such a hypothesis, due to its apparent unverifiability.

We can distinguish several variants of the relationist account of space and time. The most radical version of relationism claims that space and time do not exist – the only entities are events and things which enter spatial and temporal relations. More moderate version of relationism accepts that time and space can be defined by abstractions from events, using relations of simultaneity and co-location. Moderate relationism rejects the existence of empty points of time and space, but this fact gives rise to a conceptual problem. Suppose that the universe consists entirely of three equidistant things A, B and C. In spite of the fact that no physical thing exists between A and B, or A and C, we would like to be able to say that there are points of space on line joining A, B and C. One solution could be to assume that a point exists if it is possible for a physical object (an event) to exist at this point. Such a position can be called “modal relationism”. But we have to note that modal relationism comes dangerously close to absolutism. The key is of course the notion of possibility, which has to be defined in a way that saves the distinction between relationism and absolutism.

The historical development of the debate between absolutism and relationism in the context of physical theories has followed a rather twisted path. Newtonian mechanics was originally founded on the idea of absolute space and time, whose mere reflections are temporal and spatial relations given to us in experience. But soon it became clear that Newtonian mechanics can be formulated in a relationist-friendly way, in the so-called Galilean invariant form. In this formulation, spatial and temporal coordinates of objects are defined not as positions in absolute space and time, but rather relatively to some frame of reference of a particular type, known as an inertial frame. All laws of classical mechanics have the same form in all inertial frames of reference, so no particular frame is privileged. The only types of frames which are physically distinguishable from the inertial frames are the ones that accelerate. In the special theory of relativity the absolute character of acceleration is retained, therefore the theory is not fully relationistic. Einstein attempted to include the principle of relationism in his general theory of relativity. Thanks to the principle of equivalence, accelerating frames of reference are locally indistinguishable from inertial frames of reference in a gravitational field. Although the general theory of relativity has certain mathematical features that make it well suited for a physical expression of relationism (mostly its covariant character), it turns out that some consequences of its fundamental equations seem to support substantivalism. In particular, there are solutions to Einstein’s field equations which describe an empty space-time (with no distribution of masses and energy). Another possible solution describes a solitary and yet rotating object, which goes against the idea that all motions are relative.

Readings:

B. Garrett, "Time: Three Puzzles", pp. 87-94, What is this thing caled metaphysics?
E.J. Lowe, Chapter 14 "Absolutism versus relationism", pp. 253-270, A survey of metaphyics.