Reasoning about proof and knowledge

Abstract

In previous work [Lewitzka, Log. J. IGPL 2017], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [Lewitzka, J. Log. Comp. 2015]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [Artemov and Protopopescu, Rev. Symb. Log. 2016]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer-Heyting-Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verificationbased intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.1 These natural axioms ensure that the identity connective ≡ is a congruence relation modulo any given theory. We read ϕ ≡ ψ as "ϕ and ψ have the same meaning (denotation, Bedeutung)" or "ϕ and ψ denote the same proposition". We strictly distinguish between formulas (syntactical objects) and propositions (semantic entities): a formula denotes a proposition. This is in accordance with our non-Fregean, intensional, view on logics: the semantics of a formula is, in general, more than its truth value. The Fregan Axiom (ϕ ↔ ψ) → (ϕ ≡ ψ) is not valid (see [18]).

Access-based Intuitionistic Knowledge

“…1.2 Adopting a justification-based view: Lewitzka , 2019 Logic L5 was introduced in [10] together with a hierarchy L L3 L4 L5 of classical Lewis-style modal logics for the reasoning about intuitionistic truth, i.e. proof.…”

“…Obviously, this embedding of IPC into classical modal systems is simpler than the well-known standard translation of IPC into modal logic S4 due to Gödel. We argued in [12] that the S5-style system L5 is an adequate system for reasoning about proof showing that it is complete w.r.t. extended BHK semantics, i.e.…”

“…proof, and a kind of intuitionistic knowledge based on an informal notion of justification (cf. [12]). In the present paper, we extend modal logic L5 with the purpose of formalizing a new concept of constructive knowledge which, in our multi-agent setting, relies on the intuition that agent i knows ϕ if i has gained access to a proof of ϕ.…”

“…Basic principles are distribution and factivity of knowledge as well as, where ϕ reads 'ϕ is proved'. The formalization extends a family of classical modal logics designed in [Lewitzka 2015[Lewitzka , 2019 as combinations of IPC and CPC and as systems for the reasoning about proof, i.e. intuitionistic truth.…”

“…We adopt a formalization of common knowledge from [Lewitzka 2011] and interpret it here as access-based common knowledge. We compare our proposal with recent approaches to intuitionistic knowledge [Artemov and Protopopescu 2016;Lewitzka , 2019 and bring together these different concepts in a unifying semantic framework based on Heyting algebra expansions.…”