 Lehmann and Magidor (1992) proposed a variation on Adams’s idea. Instead of using the delta-epsilon construction, they made use of nonstandard measure theory, that is, a theory of probability functions that can take values that are infinitesimals (infinitely small numbers). In addition, instead of defining the consequence relation by quantifying over all probability functions, Lehmann and Magidor assume that we can select a single probability function (representing something like the ideally rational, or objective probability). On their construction, a conclusion \(p\) follows from \(T\) just in case the probability of \(p\) is infinitely close to 1, on the assumption that the probabilities assigned to members of \(T\) are infinitely close to 1. Lehmann and Magidor proved that the resulting consequence relation is always not only preferential: it is also rational. The logic defined by Lehmann and Magidor also corresponds exactly to the theory of Popper functions, another extension of probability theory designed to handle cases of conditioning on propositions with infinitesimal probability (see Harper 1976; van Fraassen 1995; Hawthorne 1998). For a brief discussion of Popper functions, see the following supplementary document: