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.