 Validity (relative to a class of frames) is defined as truth-preservation at all worlds of all models (based on any frame in the class). It is easy to see that the strict conditional invalidates the paradoxes of the material conditional, relative to models built on arbitrary Kripke frames. The strict conditional retains Modus Ponens and a number of classic properties, such as Monotonicity, Transitivity, and Contraposition (see below). Starting with Goodman (1955), however, such inferences have been regarded with suspicion, both for indicative conditionals (see Adams 1965) and for counterfactuals conditionals (see Stalnaker 1968, Lewis 1973). For instance, in the case of Transitivity, we have the following counterexamples: