How Expected is the Unexpected Hanging

This is about a famous paradox introduced by philosophers in the 1940s. It was casually dismissed by Quine [1] in 1953. He cited five papers that preceded his in the prestigious British journal Mind. Martin Gardner wrote about unexpected quizzes, unexpected eggs, unexpected tigers, etc. in the 60s and 70s [2], [3], and his bibliography in [2] listed 23 references. Twenty years after Quine the paradox was dismissed again, in Mind, by A. J. Ayer [4]. In the 80s the paradox of the unexpected hanging was attacked with the methods of formal logic [5] by Professor R. M. Sainsbury, the current editor of Mind. It also reached the popular book stores [6]. Now we are into the 90s and Martin Gardner's book [2] has been reissued by the University of Chicago Press in a new edition [8] listing 57 references on the paradox! After a half-century of effort and nearly 60 papers it seems unlikely that any single article will resolve it. In fact, it is unclear which domain of human intelligence should take custody of it. Is it a problem in pure logic? semantics? psychology? probability theory? Is it a problem without a solution or with multiple solutions? One thing is clear: The paradox of the unexpected hanging has rarely been approached as a mathematical problem. Questions like "What is the numerical probability of the unexpected hanging?" or "What are necessary and sufficient conditions for the unexpected hanging?" have not been posed. That is what we do here.