Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)
This work addresses foundational issues in logic and interactive proof theory, offering a novel framework for epistemic deciders, but it is incremental as it builds directly on prior logic (LiiP).
The paper tackles the problem of constructing a decidable classical modal logic (LDiiP) that internalizes negation-complete and disjunctive interactive proofs, achieving a system where all proof goals are either provable or disprovable to an agent, based on an existing logic (LiiP). The result is a logic with a standard Kripke semantics justified via the Axiom of Choice and an oracle-computable function.
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.