Intuitionistic Epistemic Logic

Abstract

We outline an intuitionistic view of knowledge which maintains the original Brouwer-Heyting-Kolmogorov semantics for intuitionism and is consistent with the wellknown approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view co-reflection A → KA is valid and the factivity of knowledge holds in the form KA → ¬¬A 'known propositions cannot be false'.We show that the traditional form of factivity KA → A is a distinctly classical principle which, like tertium non datur A ∨ ¬A, does not hold intuitionistically, but, along with the whole of classical epistemic logic, is intuitionistically valid in its double negation form ¬¬(KA → A).Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it.

Reasoning about proof and knowledge

“…The systems EL3 − and EL3-EL5 were introduced in [15] as epistemic/modal extensions of L3. These classical systems seem to reflect in a sense the basic principles of Intuitionistic Epistemic Logic introduced in [4]. The precise relationship between the S5-style modal systems of our hierarchy (i.e.…”

“…We are unable to find any kind of possible worlds semantics for weaker logics of our hierarchy. Interestingly, the presented semantic framework also describes the intuitionistic epistemic logics IEL − and IEL presented in [4], as we shall see in the next section. As in the preceding section, we work here with the full epistemic language F m. However, dropping the epistemic ingredients from Definition 4.1 below (more specifically: function E) results in (much simpler) frames for non-epistemic logic L5 over the modal sublanguage F m 1 ⊆ F m. In this way, relational semantics for L5, as well as corresponding soundness and completeness proofs, are implicitly contained in the following approach.…”

“…In this section, we work with the pure epistemic sublanguage F m e := {ϕ ∈ F m | symbol does not occur in ϕ}. As already mentioned above, the intuitionistic epistemic logics IEL − and IEL, introduced by Artemov and Protopopescu [4], can be axiomatized in language F m e by the axioms (INT), distribution of belief (KBel) K(ϕ → ψ) → (Kϕ → Kψ), intuitionistic co-reflection ϕ → Kϕ, and -only in case of IEL -intuitionistic reflection (IntRe) Kϕ → ¬¬ϕ. The only reference rule is Modus Ponens MP.…”

“…3 According to that clause, Kϕ can be read as "it is verified that ϕ has a proof". However, as pointed out in [4], IEL also captures the following possible reading: "it is verified that ϕ holds in some not specified constructive sense". We tend to the latter interpretation which seems to better harmonize with the informal and formal aspects of our approach.…”

“…Moreover, we propose in this paper a justification-based interpretation of the kind of belief and knowledge modeled by our epistemic extensions of L3-L5 originally introduced in [15], in contrast to the verification-based approach of IEL presented in [4]. The agent believes/knows a proposition for some reason or justification.…”