We discuss O&C's probabilistic approach from a probability logical point of view. Specifically, we comment on subjective probability, the indispensability of logic, the Ramsey test, the consequence relation, human nonmonotonic reasoning, intervals, generalized quantifiers, and rational analysis.

The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite (...) satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic. (shrink)

In the first half of this paper, we present a fragment of relational syllogisms named RELSYLL consisting of quantified statements with a special set of numerical quantifiers, and introduce a number of concepts that are useful for the later sections, including indirect reduction, quantifier transformations and equivalence of syllogisms. After determining the valid and invalid syllogisms in RELSYLL, we then introduce two Derivation Methods which can be used to derive valid relational syllogisms based on known valid simple syllogisms. We also (...) show that the two Methods are sound and complete for RELSYLL. In the second half of this paper, we discuss ways to extend the Derivation Methods, including the use of more valid syllogisms and the use of existential assumptions. In this way, we are able to derive more relational syllogisms that contain other types of non-classical quantifiers, including “only” and proportional quantifiers. Finally, we state and prove a proposition concerning the relationship between the two Methods. (shrink)