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.
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