New Foundations for Imperative Logic Iv: Natural Deduction

I present sound and complete natural deduction systems for (quantified modal) imperative logic, in five steps. (1) Syntax: I introduce imperative formal languages by using the imperative operator ‘!’; e.g., if ‘A’ is a declarative sentence, ‘!A’ (“let A be true’’) is an imperative sentence. (2) Semantics: I introduce interpretations of imperative formal languages, and (building on previous work) I define what it is for a declarative sentence to sustain a (declarative or imperative) sentence on an interpretation. (3) Semantic validity: I define an argument to be semantically valid exactly if, on every interpretation, every declarative sentence that sustains its premises also sustains its conclusion. (4) Syntactic validity: I define an argument to be syntactically valid exactly if its conclusion can be derived from its premises by applying certain “natural” replacement and inference rules that I introduce. (5) Soundness and completeness: I prove that semantic and syntactic validity coincide.