# Attainable Knowledge and Omniscience

## TL;DR

This paper explores how different amounts of evidence influence knowledge properties, showing that infinite evidence aligns with S5 logic, finite evidence with S4, and providing a bi-modal system for their interaction.

## Contribution

It introduces a bi-modal logical framework capturing the interplay between finite and infinite evidence-based knowledge properties.

## Key findings

- Knowledge from infinite evidence follows S5 logic
- Finite evidence knowledge follows S4 logic
- A sound and complete bi-modal system describes their interaction

## Abstract

The paper investigates an evidence-based semantics for epistemic logics. It is shown that the properties of knowledge obtained from a potentially infinite body of evidence are described by modal logic S5. At the same time, the properties of knowledge obtained from only a finite subset of this body are described by modal logic S4. The main technical result is a sound and complete bi-modal logical system that describes properties of these two modalities and their interplay.