An Epistemic Interpretation of Tensor Disjunction

Abstract: This paper aims to give an epistemic interpretation to the tensor disjunction in dependence logic, through a rather surprising connection to the so-called weak disjunction in Medvedev's early work on intermediate logic under the Brouwer-Heyting-Kolmogorov (BHK)-interpretation. We expose this connection in the setting of inquisitive logic with tensor disjunction discussed by Ciardelli and Barbero (2019}, but from an epistemic perspective. More specifically, we translate the propositional formulae of inquisitive logic with tensor into modal formulae in a powerful epistemic language of "knowing how" following the proposal by Wang (2021). We give a complete axiomatization of the logic of our full language based on Fine's axiomatization of S5 modal logic with propositional quantifiers. Finally, we generalize the tensor operator with parameters $k$ and $n$, which intuitively captures the epistemic situation that one knows $n$ potential answers to $n$ questions and is sure $k$ answers of them must be correct. The original tensor disjunction is the special case when $k=1$ and $n=2$. We show that the generalized tensor operators do not increase the expressive power of our logic, the inquisitive logic, and propositional dependence logic, though most of these generalized tensors are not uniformly definable in these logics, except in our dynamic epistemic logic of knowing how.