First, Pierce's rule, case, and result are equivalent to syllogistic logic's major premise, minor premise, and conclusion. If we look at the classic syllogism example:
All men are mortal
Socrates is a man
Socrates is mortal
The first sentence (the major premise) establishes the rule that
all members of the category 'men' have the property 'mortal.'
I would argue that the difference in how the deductive-inductive dichotomy is framed in terms of premises and conclusions and how Pierce's characterization function is a result of the differences in how both topics relate to logic and epistemology and the two senses of argument. In the narrow sense, an argument is an inference from two or three sentences, ...