What do people do when they comprehend language in discourse? According to many psychologists, they build and maintain cognitive representations of utterances in four complementary mental models for discourse that interact with each other: the surface text, the text base, the situation model, and the context model. When people encounter metaphors in these utterances, they need to incorporate them into each of these mental representations for the discourse. Since influential metaphor theories define metaphor as a form of (figurative) analogy, involving cross-domain mapping of a smaller or greater extent, the general expectation has been that metaphor comprehension is also based on analogizing. This expectation, however, has been partly borne out by the data, but not completely. There is no one-to-one relationship between metaphor as (conceptual) structure (analogy) and metaphor as (psychological) process (analogizing). According to Deliberate Metaphor Theory (DMT), only some metaphors are handled by analogy. Instead, most metaphors are presumably handled by lexical disambiguation. This is a hypothesis that brings together most metaphor research in a provocatively new way: it means that most metaphors are not processed metaphorically, which produces a paradox of metaphor. In this talk I will sketch out how this paradox arises and how it can be resolved by a new version of DMT, which I have described in my forthcoming book Slowing metaphor down: Updating Deliberate Metaphor Theory (currently under review). In this theory, the distinction between, but also the relation between, analogy in metaphorical structure versus analogy in metaphorical process is of central importance.