Outer Limits of Reason by Noson S. Yanofsky Page A

groundbreaking paper about the very nature of reason and beliefs. When playing a game of chess, you must play rationally and take into account the position of the pieces on the board. You also must take into account that your opponent is rational. Realize that just as you are going to make a rational move, so too will your opponent see what move you make and similarly make a rational move. Your opponent also takes into account that you are rational and she knows that you know she is rational. This goes back and forth and happens anytime there are strategies involved (as in figure 3.9 ). There are, however, problems with such scenarios. The ability of beliefs to deal with themselves will cause a self-referential paradox and hence a type of limitation.
    Figure 3.9
    Two people thinking about each other’s strategies
    A simple example has come to be known as the Brandenburger-Keisler paradox . It is a type of two-person liar paradox. Imagine Ann and Bob thinking about each other’s thoughts. Now consider the situation described by these two lines:
    Ann believes that Bob assumes that
    Ann believes that Bob’s assumption is wrong.
    Pose the following question:
    Does Ann believe that Bob’s assumption is wrong?
    If you answer yes, then you are agreeing with the second line. The first line says that Ann believes that this assumption is correct and not wrong . Hence the answer is no. Let us try the other way: the answer to the question is no. It is not the case that Ann believes that Bob’s assumption is wrong. Therefore Ann believes Bob’s assumption is correct. That is, the second line, which says Ann believes that Bob’s assumption is wrong , is true. So the answer must have been yes. This is a contradiction.
    Brandenburger and Keisler take such ideas and go much farther. Their revolutionary work proceeds to show that there will be limitations or “holes” in any type of game where two players reason about each other. That is, there will be situations where contradictions can happen.
    Further Reading
    Section 3.1
    The section on the ship of Theseus, the problem of identity, and the problem of personal identity was mostly motivated by very passionate classroom discussions with my students at Brooklyn College and by reading too much David Hume. Unger 1979 comes to similar conclusions from a slightly different perspective.
    Section 3.2
    Zeno’s motion paradoxes can be found in book VI of Aristotle’s Physics . My discussion benefited greatly from the following publications: Grünbaum 1955, Huggett 2010, Makim 1998, Vlastos 1972, and especially Glazebrook 2001. There are also many wonderful papers in Salmon 1972. Chapter 1 of Sainbury 2007 has a nice exposition as well. Mazur 2007 is a popular history book on Zeno’s paradoxes.
    The discussion of Gödel’s take on time travel can be found in Rucker 1982. In Yanofsky 2003, I show that the time-traveler paradoxes can be put into the same scheme as all other self-referential paradoxes.
    Section 3.3
    Sorensen 2001 is an important work on the general concept of vagueness. Chapter 2 of Sainsbury 2007 covers some of the same material. More on dialetheism and paraconsistent logic can be found in the works of Graham Priest, such as Priest 2003. Parikh 1994 is an interesting discussion of vague terms that is worth studying.
    Section 3.4
    The magazine article that made the Monty Hall problem famous was in Parade Magazine , September 9, 1990, 16. The front-page New York Times article (July 21, 1991) was by John Tierney: “Behind Monty Hall’s Doors: Puzzle, Debate and Answer?”
    You can read about the surprise-test paradox and many other epistemic paradoxes in Sorensen 2006. The Brandenburger-Keisler paradox and much more can be found in Brandenburger and Keisler 2006.

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    Infinity Puzzles
    The last function of reason is to recognize that there is an infinity of things which are beyond it. It is but feeble if it does not see so far