We would like to build story understanding systems which are transparent, modular, and extensible. To this end, we have been working on a new logical approach to narrative understanding that features a GPSG-style grammar and an episodic logic with probabilistic inference rules. The grammar represents phrase structure and the relationship between phrase structure and logical form in a modular, explicit form. The logical representation allows propositional attitudes, unreliable generalizations, and other non-standard constructs, providing a uniform, transparent knowledge representation for both the explicit content of stories and for the background knowledge needed to understand them. It makes systematic use of episodic variables in the representation of episodic sentences, using these to capture temporal and causal relationships. The rules of inference include probabilistic versions of deduction rules resembling forward and backward chaining rules in expert systems. These can be used for predictive, explanatory, and simulative inference. We illustrate our approach with nontrivial grammar fragments (including semantic rules), and with an extended example of forward-chaining inference based on a sentence from Little Red Riding Hood. A pilot implementation is able to make many (though not all) of the inferences we describe. (KR)