Nuclear deterrence is paradoxical. One paradox of nuclear deterrence we may call the rationality paradox:
(RP) (1) While it is a rational policy to threaten nuclear retaliation against an opponent armed with nuclear weapons, it would not be rational to carry out the retaliation should the threat fail to deter; and (2) what would not be rational to do is not, in the circumstances characteristic of nuclear deterrence, rational to threaten to do.
This is a paradox in the standard sense that it involves contradictory claims, for it implies that adopting a policy of nuclear deterrence is both rational and not rational, yet we have strong reason to believe that each of the claims is true. Claim (1) is a recognition that, though we believe nuclear deterrence works, there would seem to be no reason to carry out the threat if it were to fail. Claim (2) is part of the logic of all forms of deterrence, military and nonmilitary, and it relates to the important notion of credibility: if an opponent knows that one has no reason to carry out a threat, the threat would not be credible and so one would have no reason to make it. Further, it is characteristic of a state of nuclear deterrence that the opponent would recognize that one would have no reason to carry out the threat.
The rationality paradox is but one of the paradoxes raised by nuclear deterrence. Some other of the paradoxes of nuclear deterrence have the same form as the rationality paradox: for a certain set of predicates x,
(PND) (1) the act of threatening nuclear retaliation (against an opponent with nuclear weapons) is x, while the act of carrying out the threat would be not-x; and (2) if it is not-x to perform some action, then, in the circumstances generally characteristic of a situation of nuclear deterrence, it is not-x to threaten to perform that action.