In this paper, we provide a semantic analysis of the well-known knowability paradox stemming from the Church–Fitch observation that the meaningful knowability principle /all truths are knowable/, when expressed as a bi-modal principle F --> K♢F, yields an unacceptable omniscience property /all truths are known/. We offer an alternative semantic proof of this fact independent of the Church–Fitch argument. This shows that the knowability paradox is not intrinsically related to the Church–Fitch proof, nor to the Moore sentence upon which it (...) relies, but rather to the knowability principle itself. Further, we show that, from a verifiability perspective, the knowability principle fails in the classical logic setting because it is missing the explicit incorporation of a hidden assumption of /stability/: ‘the proposition in question does not change from true to false in the process of discovery.’ Once stability is taken into account, the resulting /stable knowability principle/ and its nuanced versions more accurately represent verification-based knowability and do not yield omniscience. (shrink)

Intuitionistic epistemic logic by Artemov and Protopopescu (Rev Symb Log 9:266–298, 2016) accepts the axiom “if A, then A is known” (written $$A \supset K A$$ ) in terms of the Brouwer–Heyting–Kolmogorov interpretation. There are two variants of intuitionistic epistemic logic: one with the axiom “ $$KA \supset \lnot \lnot A$$ ” and one without it. The former is called $$\textbf{IEL}$$, and the latter is called $$\textbf{IEL}^{-}$$. The aim of this paper is to study first-order expansions (with equality and function (...) symbols) of these two intuitionistic epistemic logics. We define Hilbert systems with additional axioms called geometric axioms and sequent calculi with the corresponding rules to geometric axioms and prove that they are sound and complete for the intended semantics. We also prove the cut-elimination theorems for both sequent calculi. As a consequence, the disjunction property and existence property are established for the sequent calculi without geometric implications. Finally, we establish that our sequent calculi can also be formulated with admissible structural rules (i.e., in terms of a G3-style sequent calculus). (shrink)

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented (...) following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism. (shrink)