continuum hypothesis See CANTOR, CONTINUUM. PROBLE. continuum problem, an open question that arose in Cantor’s theory of infinite cardinal numbers. By definition, two sets have the same cardinal number if there is a one-to-one correspondence between them. For example, the function that sends 0 to 0, 1 to 2, 2 to 4, etc., shows that the set of even natural numbers has the same cardinal number as the set of all natural numbers, namely F0. That F0 is not the only infinite cardinal follows from Cantor’s theorem: the power set of any set (i.e., the set of all its subsets) has a greater cardinality than the set itself. So, e.g., the power set of the natural numbers, i.e., the set of all sets of natural numbers, has a cardinal number greater than F0. The first infinite number greater than F0 is F1; the next after that is F2, and so on. When arithmetical operations are extended into the infinite, the cardinal number of the power set of the natural numbers turns out to be 2F0. By Cantor’s theorem, 2F0 must be greater than F0; the conjecture that it is equal to F1 is Cantor’s continuum hypothesis (in symbols, CH or 2F0 % F1). Since 2F0 is also the cardinality of the set of points on a continuous line, CH can also be stated in this form: any infinite set of points on a line can be brought into one-to-one correspondence either with the set of natural numbers or with the set of all points on the line. Cantor and others attempted to prove CH, without success. It later became clear, due to the work of Gödel and Cohen, that their failure was inevitable: the continuum hypothesis can neither be proved nor disproved from the axioms of set theory (ZFC). The question of its truth or falsehood – the continuum problem – remains open. See also CANTOR, INFINITY, SET THEORY. P.Mad. contractarianism, a family of moral and political theories that make use of the idea of a social contract. Traditionally philosophers (such as Hobbes and Locke) used the social contract idea to justify certain conceptions of the state. In the twentieth century philosophers such as John Rawls have used the social contract notion to define and defend moral conceptions (both conceptions of political justice and individual morality), often (but not always) doing so in addition to developing social contract theories of the state. The term ‘contractarian’ most often applies to this second type of theory. There are two kinds of moral argument that the contract image has spawned, the first rooted in Hobbes and the second rooted in Kant. Hobbesians start by insisting that what is valuable is what a person desires or prefers, not what he ought to desire or prefer (for no such prescriptively powerful object exists); and rational action is action that achieves or maximizes the satisfaction of desires or preferences. They go on to insist that moral action is rational for a person to perform if and only if such action advances the satisfaction of his desires or preferences. And they argue that because moral action leads to peaceful and harmonious living conducive to the satisfaction of almost everyone’s desires or preferences, moral actions are rational for almost everyone and thus ‘mutually agreeable.’ But Hobbesians believe that, to ensure that no cooperative person becomes the prey of immoral aggressors, moral actions must be the conventional norms in a community, so that each person can expect that if she behaves cooperatively, others will do so too. These conventions constitute the institution of morality in a society.
So the Hobbesian moral theory is committed to the idea that morality is a human-made institution, which is justified only to the extent that it effectively furthers human interests. Hobbesians explain the existence of morality in society by appealing to the convention-creating activities of human beings, while arguing that the justification of morality in any human society depends upon how well its moral conventions serve individuals’ desires or preferences. By considering ‘what we could agree to’ if we reappraised and redid the cooperative conventions in our society, we can determine the extent to which our present conventions are ‘mutually agreeable’ and so rational for us to accept and act on. Thus, Hobbesians invoke both actual agreements (or rather, conventions) and hypothetical agreements (which involve considering what conventions would be ‘mutually agreeable’) at different points in their theory; the former are what they believe our moral life consists in; the latter are what they believe our moral life should consist in – i.e., what our actual moral life should model. So the notion of the contract does not do justificational work by itself in the Hobbesian moral theory: this term is used only metaphorically. What we ‘could agree to’ has moral force for the Hobbesians not because make-believe promises in hypothetical worlds have any binding force but because this sort of agreement is a device that (merely) reveals how the agreed-upon outcome is rational for all of us. In particular, thinking about ‘what we could all agree to’ allows us to construct a deduction of practical reason to determine what policies are mutually advantageous.
The second kind of contractarian theory is derived from the moral theorizing of Kant. In his later writings Kant proposed that the ‘idea’ of the ‘Original Contract’ could be used to determine what policies for a society would be just. When Kant asks ‘What could people agree to?,’ he is not trying to justify actions or policies by invoking, in any literal sense, the consent of the people. Only the consent of real people can be legitimating, and Kant talks about hypothetical agreements made by hypothetical people. But he does believe these make-believe agreements have moral force for us because the process by which these people reach agreement is morally revealing. Kant’s contracting process has been further developed by subsequent philosophers, such as Rawls, who concentrates on defining the hypothetical people who are supposed to make this agreement so that their reasoning will not be tarnished by immorality, injustice, or prejudice, thus ensuring that the outcome of their joint deliberations will be morally sound. Those contractarians who disagree with Rawls define the contracting parties in different ways, thereby getting different results. The Kantians’ social contract is therefore a device used in their theorizing to reveal what is just or what is moral. So like Hobbesians, their contract talk is really just a way of reasoning that allows us to work out conceptual answers to moral problems. But whereas the Hobbesians’ use of contract language expresses the fact that, on their view, morality is a human invention which (if it is well invented) ought to be mutually advantageous, the Kantians’ use of the contract language is meant to show that moral principles and conceptions are provable theorems derived from a morally revealing and authoritative reasoning process or ‘moral proof procedure’ that makes use of the social contract idea. Both kinds of contractarian theory are individualistic, in the sense that they assume that moral and political policies must be justified with respect to, and answer the needs of, individuals. Accordingly, these theories have been criticized by communitarian philosophers, who argue that moral and political policies can and should be decided on the basis of what is best for a community. They are also attacked by utilitarian theorists, whose criterion of morality is the maximization of the utility of the community, and not the mutual satisfaction of the needs or preferences of individuals. Contractarians respond that whereas utilitarianism fails to take seriously the distinction between persons, contractarian theories make moral and political policies answerable to the legitimate interests and needs of individuals, which, contra the communitarians, they take to be the starting point of moral theorizing. See also KANT , POLITICAL PHILOSOPHY, SOCIAL CONTRACT, SOCIAL PHILOSOPHY.
J.Ham.