I want to prove that for a wff in CPL, an interpretation (truth assignment) is defined by its definition on the atomic sentences. It sounds obvious but I don't understand how to approach proving this formally.
For example proving by induction: As a basis, the truth value of the atomic sentence is of course defined only by the interpretation on itself, but then as a step i'm stuck. we can assume it for smaller wffs, then using the connectives that it is defined by the smaller functions from its truth table, which from the hypothesis are defined by the interpretation on the atomic sentences. Is this a correct proof? I'm not sure I can just make that transitive statement in the step.
Thanks.