There are several issues with the standard approach to the
relationship between conditionals and assertions, particularly when the
antecedent of a conditional is (or may be) false. One prominent alternative is to
say that conditionals do not express propositions, but rather make conditional
assertions that may generate categorical assertions of the consequent in certain
circumstances. However, this view has consequences that jar with standard
interpretations of the relationship between proofs and assertion. Here, I analyse
this relationship, and say that, on at least one understanding of proof,
conditional assertions may reflect the dynamics of proving, which (sometimes)
generate categorical assertions. In particular, when we think about the
relationship between assertion and proof as rooted in a dialogical approach to
both, the distinction between conditional and categorical assertions is quite
natural.