Have a thing? Reasoning around recursion with dynamic typing in grounded arithmetic

Abstract

Neither the classical nor intuitionistic logic traditions are perfectly-aligned with the purpose of reasoning about computation, in that neither tradition can permit unconstrained recursive definitions without inconsistency: recursive logical definitions must normally be proven terminating before admission and use. We introduce grounded arithmetic or GA, a formal-reasoning foundation allowing direct expression of arbitrary recursive definitions. GA adjusts traditional inference rules so that terms that express nonterminating computations harmlessly denote no semantic value (i.e., bottom) instead of yielding inconsistency. Recursive functions may be proven terminating in GA essentially by "dynamically typing" terms, or equivalently, symbolically reverse-executing the computations they denote via GA's inference rules. Once recursive functions have been proven terminating, logical reasoning about their results reduce to the familiar classical rules. A mechanically-checked formal development of basic grounded arithmetic or BGA - a minimal kernel for GA - shows that BGA simultaneously exhibits the useful metalogical properties of being (a) semantically and syntactically consistent, (b) semantically complete, and (c) sufficiently powerful to express and prove the termination of arbitrary closed Turing-complete computations. This combination is impossible in classical logic due to G\"oel's incompleteness theorems, but our results do not contradict G\"odel's theorems because BGA is paracomplete, not classical. Leveraging BGA's power of computation and reflection, we find that grounded logical operators including quantifiers are definable as non-primitive computations in BGA, despite not being included as primitives.