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

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.

The first-order hypothetical logic of proofs

“…All logical conclusions of known facts are assumed to be known. Variants of this logic are developed in [Kra14]. 3) that this is required in order to prove all FOLPtheorems in FOHLP, it seems reasonable to explore truth dependent modalities [NPP08] at the possible cost of capturing a subset of LP-theorems.…”

Logic of Intuitionistic Interactive Proofs (Formal Theory of Perfect Knowledge Transfer)

“…• the monotonicity property becomes the similar property that Properties From the detailed reminder in Section 1.1.1 of [Kra12b,Kra13b], recall that first, the negation-completeness property implies the discussed disjunction property ("negation-complete implies disjunctive"); second, any internalised negation-complete notion of proof 2 is non-monotonic, that is,…”

“…The core ingredient of the concrete semantics of LIiP are so-called input histories, which were introduced in [Kra12b,Kra13b] and could also be used in an even more concrete semantics of LiP. Input histories are finite words of input events and serve as concrete states s ∈ S in the state space S, on which the concrete and abstract accessibility relation M R CM ⊆ S × S and M R CM ⊆ S × S for LIiP is defined, respectively.…”

“…• Like in [Kra12c,Kra13c] and [Kra12b,Kra13b], we endow the proof modality with a standard Kripke-semantics [BvB07], but first define its accessibility relation M R CM constructively in terms of elementary set-theoretic constructions, 3 namely as M R CM (cf. Section 2.2.1), and then match it to an abstract semantic interface in standard form (which abstractly stipulates the characteristic properties of the accessibility relation [Fit07]).…”

“…LIiP, the result of the construction, however is independent of LiP. Further note that like in [Kra12a], [Kra12c,Kra13c], and [Kra12b,Kra13b], we still understand interactive proofs as sufficient evidence for intended resource-unbounded proof-checking agents (who are though unable to guess), and leave probabilistic and polynomial-time resource bounded agents for future work. Finally note that we choose our meta-logic to be classical (singleton meta-universe or meta-world unicity, cf.…”