An arithmetic interpretation of intuitionistic verification

Intuitionistic epistemic logic introduces an epistemic operator to intuitionistic logic, which reflects the intended Brouwer–Heyting–Kolmogorov (BHK) semantics of intuitionism. The fundamental assumption concerning intuitionistic knowledge and belief is that it is the product of verification. The BHK interpretation of intuitionistic logic has a precise formulation in the logic of proofs and its arithmetical semantics. We show here that this interpretation can be extended to the notion of verification upon which intuitionistic knowledge is based, providing the systems of intuitionistic epistemic logic based on verification with an arithmetical semantics too. This confirms that the conception of verification incorporated in these systems reflects the BHK interpretation.

Truth, Demonstration and Knowledge. A Classical Solution to the Paradox of Knowability

After introducing semantic anti-realism and the paradox of knowability, the paper offers a reconstruction of the anti-realist argument from the theory of understanding. The proposed reconstruction validates an unrestricted principle to the effect that truth requires the existence of a certain kind of “demonstration”. The paper shows that the principle fails to imply the problematic instances of the original unrestricted knowability principle but that the overall view still has unrestricted epistemic consequences. Appealing precisely to the paradox of knowability, the paper also argues, against BHK semantics, for the non-constructive character of the demonstrations envisaged by anti-realists, and contends that, in such a setting, one of the most natural arguments in favour of a revision of classical logic loses all its force.