# Uncertainty About Evidence

## TL;DR

This paper introduces a logical framework for reasoning about knowledge and evidence where the interpretation of evidence can vary across possible worlds, capturing uncertainty and leading to a new bi-modal logic with extensions.

## Contribution

It develops a novel logical model allowing evidence interpretation to vary, generalizes topological spaces, and provides a complete axiomatization for reasoning about evidence and knowledge.

## Key findings

- The framework captures uncertainty in evidence interpretation.
- A sound and complete axiomatization is provided.
- Extensions include belief and knowability modalities.

## Abstract

We develop a logical framework for reasoning about knowledge and evidence in which the agent may be uncertain about how to interpret their evidence. Rather than representing an evidential state as a fixed subset of the state space, our models allow the set of possible worlds that a piece of evidence corresponds to to vary from one possible world to another, and therefore itself be the subject of uncertainty. Such structures can be viewed as (epistemically motivated) generalizations of topological spaces. In this context, there arises a natural distinction between what is actually entailed by the evidence and what the agent knows is entailed by the evidence -- with the latter, in general, being much weaker. We provide a sound and complete axiomatization of the corresponding bi-modal logic of knowledge and evidence entailment, and investigate some natural extensions of this core system, including the addition of a belief modality and its interaction with evidence interpretation and entailment, and the addition of a "knowability" modality interpreted via a (generalized) interior operator.