Ambiguity in Logic: A Comparative Analysis of the Sleeping Beauty and Boy or Girl Paradox

Probability paradoxes occur when problem formulations allow for sev- eral interpretations, thus giving rise to conflicting conclusions that remain mathematically sound. This study considers two of the most famous para- doxes—the Sleeping Beauty Problem and the Boy or Girl Problem—and discovers a common essential ambiguity in them. Both paradoxes result in two broadly argued probability resolutions, 1/3 and 1/2, which are caused by the variation in the formulation of the respective probability space. The Sleeping Beauty Problem is the canonical case of a problem in- volving self-locating uncertainty. Sleeping Beauty is placed in a sleep state, and a fair coin is tossed. If the coin comes up heads, she is wo- ken and questioned once on Monday. If the coin comes up tails, she is woken and questioned twice on separate occasions—on Monday and Tues- day—without recalling the previous awakening. When she awakens, she is invited to estimate the chance that the coin has landed on heads. The Boy or Girl Problem addresses another but connected form of uncertainty in probability. In a two-child family, with the condition that at least one of the children is a boy, the probability that both children are boys hinges on the wording of the problem. The paradox is due to the distinction between conditioning probability on the family level versus the individual child level. This paper contends that the two paradoxes have the same structural ambiguity, viz. regarding the conditioning of probability and the reference class definition. In Sleeping Beauty, the issue is whether each awakening represents a separate event or if she has to rely on the pre-experimental probability. In Boy or Girl, the issue is whether we are sampling fami- lies or individual children as the foundation for determining probability. These findings point to broader concerns in probability theory, Bayesian inference, and the Reference Class Problem with important ramifications for how probability questions are framed to avoid misleading solutions. By concentrating on the common vagueness that defines these paradoxes, this study further advances the understanding of probability paradoxes and their implications for decision theory and applied probability. The study also hypothesizes that most so-called probability paradoxes stem from subtle misinterpretation of information construction and processing, rather than any intrinsic flaws in probability theory itself.