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ABSTRACT

According to the co-occurrence test, q is (non-monotonically) inferrible from p if and only if q holds in all the reasonably plausible belief change outcomes in which p holds. A formal model is introduced that contains representations of both the co-occurrence test (for nonmonotonic inference) and the Ramsey test (for conditionals). In this model, (non-nested) conditionals and non-monotonic inference satisfy the same logical principles. However, in spite of this similarity the two notions do not coincide. They should be carefully distinguished from each other.

5. Discussion

The theorem shows that conditionality () and non-monotonic inferribility (|∼) have the same logic. But there are three important caveats. First, the theorem was proved in one specific framework. There may be other frameworks in which both the Ramsey test and the co-occurrence test can be represented. The corresponding theorem may not be obtainable in all such frameworks. Secondly, although the theorem provides us with a reconstruction of any co-occurrence test as a Ramsey test, this derived Ramsey test is based on another initial belief set and another operation of belief revision than those referred to in the co-occurrence test that we started with. Therefore, the logical properties that connect to K or to ∗ need not hold if we replace by |∼, and vice versa. (The property CS mentioned in Section 2 is an example of this.) Thirdly and most importantly, according to the theorem conditionality and non-monotonic inferribility obey the same logical principles but do not coincide. This is not an unusual situation. Logical necessity and physical necessity may both have the same (S5) logic, but that is no reason to conflate them [8, pp. 104–105], cf. [3] and [5]. In social choice theory, we usually assume that different persons’ preferences satisfy the same logical rules, but in all non-trivial cases they differ in substance. Similarly, the analysis provided here gives us reason to treat conditionality and (non-monotonic) inferribility as distinct notions that satisfy the same logical principles. Possibly, the similarity of their logics has blinded us to the differences between the two concepts. R