Four Formal Versions of the Two-Envelope Paradox

Philosophical discussion of the two-envelope paradox has suffered from a lack of formal precision. I discuss various versions of the paradoxical argument using formal probability theory, which helps to make diagnoses that are simpler, more insightful, and provably correct. Paradoxical arguments are revealed to be fallacious for one of three reasons: (1) the argument makes a formal mistake such as an equivocation fallacy; (2) the argument disregards relevant uncertainty about or variability in a unit of measurement; (3) the argument uses an invalid decision rule. I improve upon various existing diagnoses and discuss what kind of philosophical and decision-theoretic import the paradox has.