Distinguishability of States: Observation Conditions and the Reliability of Inference

This paper examines the relationship between differences between system states, the possibility of detecting those differences, and the procedure used to distinguish the states. Simple examples show why an observation requires interpretation, under what conditions a trace can serve as evidence, and what limits the reliability of an inference. A numerical example demonstrates how the overall error rate depends on the relative frequencies of the states presented, and separates limitations of the observations from shortcomings in the use of the data. The paper offers a systematic framework for formulating a discrimination problem, drawing on established principles of probabilistic and statistical inference. The paper is intended for readers interested in the foundations of scientific inquiry, including students and specialists whose interests extend beyond the narrow boundaries of their own disciplines. No specialized mathematical background is required: probabilistic reasoning is explained through numerical examples.