One-Step Modal Logics, Intuitionistic and Classical, Part 1

This paper and its sequel “look under the hood” of usual sorts proof-theoretic systems for certain well-known intuitionistic classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be result prefixing in language with step-marker, this either 0 or 1. Think as indicating taking “one step away from 0.” Deductions constructed using formulas. 2 presents model-theoretic concepts, based on those [7], that guide rest paper. 3 Natural Deduction IK CK, formalizations one-step versions K. In these systems, occurrences step-markers allow deductions to display deductive structure covered over familiar “no step” such \(\square \) ♢ are governed by Introduction Elimination rules; K rule Necessitation derived (i.e. admissible) rules. CK adding 0-version Rule Excluded Middle rules which generate IK. Note: merely dropping generating without addition further axioms (as was needed [7]). These yield consequence relations obvious definition. 4 provides some examples what can deduced 5 defines concepts used 6 prove soundness relation (relative class models defined 2.) 7 proves completeness class). 8 extends results CK. (Looking ahead: Part investigate formalizing logics stronger than K.)