Tarski showed that it is impossible for any expressive formal language to contain its own truth predicate. We show that an expressive formal language can nevertheless contain its own “subjective probability” function. The assigned probabilities can be reflectively consistent in the sense of an appropriate analog of the reflection property. In practice, most meaningful assertions must already be treated probabilistically, and very little is lost by allowing some sentences to have probabilities intermediate between 0 and 1.
Robust Cooperation on the Prisoner’s Dilemma: Program Equilibrium via Provability Logic
Rational agents defect on the one-shot prisoner’s dilemma even though mutual cooperation would yield higher utility for both agents. Moshe Tennenholtz showed that if each program is allowed to pass its playing strategy to all other players, some programs can then cooperate on the one-shot prisoner’s dilemma. Program equilibria is Tennenholtz’s term for Nash equilibria in a context where programs can pass their playing strategies to the other players.
One weakness of this approach so far has been that any two programs which make different choices cannot “recognize” each other for mutual cooperation, even if they are functionally identical. In this paper, provability logic is used to enable a more flexible and secure form of mutual cooperation.