Probability and Nonclassical Logic

Abstract

This chapter presents axioms for comparative conditional probability relations. The axioms presented here are more general than usual. Each comparative relation is a weak partial order on pairs of sentences but need not be a complete order relation. The axioms for these comparative relations are probabilistically sound for the broad class of conditional probability functions known as Popper functions. Furthermore, these axioms are probabilistically complete. Arguably, the notion of comparative conditional probability provides a foundation for Bayesian confirmation theory. Bayesian confirmation functions are overly precise probabilistic representations of the more fundamental logic of comparative support. The most important features of evidential support are captured by comparative relationships among argument strengths, realized by the comparative support relations and their logic.

Solving the Problem of Logical Omniscience

“…But, as Williams observes in another paper (Williams ), for some familiar ways of rejecting classical logic, or rejecting classical semantics, D3 * is not an attractive principle. There is the supervaluationist who thinks the disjunction, “Patchy is red or not red”, is true while each disjunct is not true, because Patchy is a borderline shade of red.…”

“…A better option for the doxastic probabilist might be to reformulate D3 * so that it isn't explicitly about disjunctions. An idea for a way to do that can be drawn from Williams (). Instead of talking of disjunctions as such, we can talk about propositions that are partitioned by other propositions, where the partitioning is not classical logical partitioning.…”

“…Instead of talking of disjunctions as such, we can talk about propositions that are partitioned by other propositions, where the partitioning is not classical logical partitioning. Here is a proposal—I'm drawing this from a more general technical discussion in Williams (, sec. 3)—for a rule for how doxastic probabilists might, in effect, handle disjunctions.…”

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

“…An entirely different view is taken in [12], where probabilistic semantics is given for a couple of many-valued paraconsistent logics. The connections between non-classical logics and probability are not circumscribed to paraconsistent logic; see, e.g., [13] for the case of infinite-valued logic and in particular [14] for a broader and more philosophical perspective.…”

“…Some specific problems connected to probabilities in non-classical logics are pointed out by J. R. G. Williams in [14]:…”