Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)

TL;DR

This paper introduces LDiiP, a decidable modal logic for negation-complete, disjunctive interactive proofs that internalize agent-centric proof theories with unique properties like non-monotonicity and classical semantics.

Contribution

It develops LDiiP, a new logic that internalizes negation-complete proof theories with disjunction property, expanding the understanding of interactive proof systems and their semantic foundations.

Findings

LDiiP internalizes agent-centric proof theories with negation completeness.

LDiiP is a classical modal logic with standard Kripke semantics.

The logic supports disjunctive internalized proofs and relates to belief and provability modalities.

Abstract

We produce a decidable classical normal modal logic of internalised negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP) from an existing logical counterpart of non-monotonic or instant interactive proofs (LiiP). LDiiP internalises agent-centric proof theories that are negation-complete (maximal) and consistent (and hence strictly weaker than, for example, Peano Arithmetic) and enjoy the disjunction property (like Intuitionistic Logic). In other words, internalised proof theories are ultrafilters and all internalised proof goals are definite in the sense of being either provable or disprovable to an agent by means of disjunctive internalised proofs (thus also called epistemic deciders). Still, LDiiP itself is classical (monotonic, non-constructive), negation-incomplete, and does not have the disjunction property. The price to pay for the negation completeness of our interactive proofs is their non-monotonicity and non-communality (for singleton agent communities only). As a normal modal logic, LDiiP enjoys a standard Kripke-semantics, which we justify by invoking the Axiom of Choice on LiiP's and then construct in terms of a concrete oracle-computable function. LDiiP's agent-centric internalised notion of proof can also be viewed as a negation-complete disjunctive explicit refinement of standard KD45-belief, and yields a disjunctive but negation-incomplete explicit refinement of S4-provability.