This article examines Simpson's paradox as applied to the theory of probabilites and percentages. The author discusses possible flaws in the paradox and compares it to the Sure Thing Principle, statistical inference, causal inference and probabilistic analyses of causation.
The essay motivates and provides a semantics for pictorial representations. A taxonomy of pictorial denoting symbols is developed that determines a semantics which defines the following: S if true in picture Y, S is false in picture Y, S is neither true nor false in picture Y, Z is the content of Picture Y, Picture Y entails that S, Picture Y implies that S. The semantics is then applied to solve or resolve a number of puzzles concerning pictorial representation.
When decision makers have more to gain than to lose by changing their minds, and that is the only relevant fact, they thereby have a reason to change their minds. While this is sage advice, it is silent on when one stands more to gain than to lose. The two envelope paradox provides a case where the appearance of advantage in changing your mind is resilient despite being a chimera. Setups that are unproblematically modeled by decision tables that are used (...) in the formulation of the two envelope paradox are described, and variations on them are stipulated. The problems posed by the paradoxical modeling are then contrasted with the variations. The paper concludes with a brief explanation of why the paradoxical modeling does not gain support from the fact that one envelope has twice the amount that is in the other. (shrink)
Two-boxers in Newcomb Problems face the question: Why aren't you rich? The essay argues that one-boxers have a false sense of advantage. They fail to align their credences during deliberation with the credences they will have when they act. This puts them in violation of the so-called Principle of Reflection, and it exposes them to a dynamic Dutch Book that will leach any gains they achieve away.
Four variations on Two Envelope Paradox are stated and compared. The variations are employed to provide a diagnosis and an explanation of what has gone awry in the paradoxical modeling of the decision problem that the paradox poses. The canonical formulation of the paradox underdescribes the ways in which one envelope can have twice the amount that is in the other. Some ways one envelope can have twice the amount that is in the other make it rational to prefer the (...) envelope that was originally rejected. Some do not, and it is a mistake to treat them alike. The nature of the mistake is diagnosed by the different roles that rigid designators and definite descriptions play in unproblematic and in untoward formulations of decision tables that are employed in setting out the decision problem that gives rise to the paradox. The decision makerâs knowledge or ignorance of how one envelope came to have twice the amount that is in the other determines which of the different ways of modeling his decision problem is correct. Under this diagnosis, the paradoxical modeling of the Two Envelope problem is incoherent. (shrink)
A pair of interpreters of a speaker's sentences can disagree by assigning different truth-values to sentences in the speaker's language that the speaker neither accepts nor rejects. Alternately, they can assign different truth-values to some sentence that the speaker accepts as true. Neither source of disagreement is open to the speaker: on pain of inconsistency in the latter case, and ex hypothesis, the speaker neither accepts nor rejects the contested sentence in the former case. Arguably, interpreters possibilities of disagreement do (...) not undercut first-person semantic authority. (shrink)
Simpson's Paradox is introduced and analysed via the mishaps of a researcher who at first falls afoul of the traps Simpson-reversals can set, and then he learns to exploit those traps to advantage. (Note: An error in the treatment of the Sure Thing Principle is corrected in "Simpson's Paradox: A Logically Benign, Empirically Treacherous Hydra").