Infinity in ethics

3. Infinite time

The above puzzles involved infinitely many options for an agent in a given choice situation. Related puzzles can arise when there is only a finite number of options at a given time, but there are infinitely many future choice situations because time extends infinitely into the future. Here let us suppose for simplicity that the value of an action is the value (for example, happiness) that it produces in the world and that time is discrete (that is, for each time there is a well-defined ‘next’ time). Furthermore, suppose for simplicity that there is a first time and that at each time there is exactly one agent (either one agent exists forever or, when an agent dies, a replacement agent comes into being). At each time, the agent has two options. One option is to produce a certain amount of value immediately, in which case no further value will be produced in the world at later times. The other option is to produce no value immediately, in which case at the next time the agent will have a choice between (1) producing even more value immediately and nothing thereafter or (2) producing no value immediately but having a similarly structured choice at the next time. For example, the sequences of possible choices might look like this: <one unit immediately versus postpone>, <two units immediately versus postpone>, …, <n units immediately versus postpone> and so on. Assume here that any relevant discounting of value for temporal delays is already reflected in the numbers. An optimizing, or almost-optimizing, theory says that at each time the choice should be to postpone, but this will have the result that no value is ever produced! A satisficing theory using an absolute criterion of being good enough, however, will judge it permissible at some point to produce immediate value.

Even satisficing theories, however, confront a puzzle here. For they also claim that it is permissible at each point in time to postpone the production of value. For, if at some point producing n units of value is satisfactory, then postponing the production of value is also satisfactory (since the total amount of value that can be produced at will is even greater). One way of avoiding this problem (probably the only way) is to hold that morality is rule-based rather than act-based. The idea is that at a given time the agent faces infinitely many rules or strategies that they could adopt and then comply with in the future. In the problem situation, the possible strategies are: never produce immediate value (that is, postpone at each step), produce immediate value at time 1, produce immediate value at time 2 and so on. If n units are the minimal satisfactory level, then all strategies that produce at least n units of value are permissible. The strategy of never producing immediate value is clearly not satisfactory and thus not permissible. This solves the puzzle, but also raises questions about whether the moral permissibility of actions is indeed based on the consequences of compliance with rules rather than directly on their consequences.

The above puzzles arise when trying to determine what is morally permissible. Puzzles can also arise when trying to determine what is morally better than what. Suppose again that time extends infinitely into the future and that an agent has a choice between producing two units of value at each time or one unit of value at each time. Intuitively, it would seem that the former outcome is better than the latter outcome. The total value produced, however, is the same infinity in each case. Thus, if overall value is simply the sum of the values at each time, then it would seem that neither is better than the other. Of course, some theories of value reject the summative view (for example, egalitarian views – see Equality), but those (such as utilitarians) who endorse the summative view in the finite case face a puzzle. Either summation does not apply in certain infinite cases, or it does and the outcome of two units of value at each time is no better than the outcome of one unit of value at each time.

This puzzle, it turns out, was discovered first by economists and then twice by philosophers. In each case, principles were developed that preserve the commitment to additivity in the finite case, but reject it in the infinite case. These principles hold that more at each time is better than less at each time (in both finite and infinite cases). The core idea (ignoring some variations) is the principle that, if there is a time in the future such that at each later time the cumulative total value in one state of affairs is at least as great as the other, then the first state of affairs is at least as valuable as the second. In our example, the cumulative total of the two-at-each-time state of affairs is always greater than that of the one-at-each-time and hence it is judged more valuable (even though both have infinite totals). The principle also allows that a more valuable state of affairs may be worse in the short run (have a lower cumulative total initially) as long as it eventually prevails.

This principle (and other related ones) gives the seemingly correct judgement about the puzzle case. It also satisfies a kind of dominance condition that seems central to finitely additive theories of values: If one state of affairs is at least as valuable at each location (for example, time) as a second state of affairs and more valuable at some locations, then it is more valuable. On the other hand, it violates a kind of neutrality condition that seems also to be central: If one state of affairs is identical to a second except that the values at locations have been placed in a different order (permuted), then the two states of affairs are equally valuable. The above principle satisfies the condition when only finitely many locations have been reordered. It violates this condition, however, when an infinite number of locations may be affected. For example, the principle judges <1,1,1,0,1,0,1,0,1, … > (one unit of value for the first three times and alternating 0s and 1s thereafter) as better than <1,0,1,0,1,0, … > (alternating 0s and 1s starting immediately). For, starting with the second time, the cumulative total of the first sequence thereafter remains greater than that of the second. Hence it is judged more valuable. The first sequence, however, is simply an infinite reordering of the second (obtained by moving all 0s two positions to the right, the first 1 one position to the left and all remaining 1s two positions to the left). Hence, there is a genuine puzzle here as to what finitely additive theories of value should do in the infinite case. They cannot satisfy both the dominance condition and the infinite neutrality condition.