Day 26 in Mat115

Today:

• Announcements:
  • Chaos homework is not yet graded - I had to get the tests graded.

  • The exam:
    • Curve: Add 8 to your raw score
    • Mean: 81.8
    • Standard Deviation: 17.9
    • The Key

  • In only a few weeks you're going to be describing your projects in class. Your final project due date is 12/9: by then, it should be in an electronic form that I can put on the web.

• Today: The Logic of Lewis Carroll
  • We begin a special section on logic, through the eyes and ideas of Lewis Carroll (as described in his book Lewis Carroll in Numberland), who thought that symbolic logic can be "an entertainment for children to develop their powers of logical thought."

  • Elements of The Game of Logic
    • An example:
      1. Babies are crazy;
      2. Crazy people should be in an asylum.

      What conclusion can we reach? If you reached a correct conclusion, then you've properly performed a logical calculation.

    • Fundamentals: tools to help us "calculate" logically:
      • Definition of the universe of discourse:
        the set of all things under discussion.

      • Definition of a statement (or proposition):
        A sentence with truth value (either true of false)

      • Definition of a syllogism:
        a group of statements (called premises) that lead to a conclusion

      • sorites (so-right-ease): A form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. An example is the "crazy babies" problem above.

      • The method: eliminating the common term

        So, in the baby example, the common term "crazy" is eliminated.

      • Three types of propositions:
        1. Some new cakes are nice
        2. No new cakes are nice
        3. All new cakes are nice

        Examples (from p. 175, Lewis Carroll in Numberland)

      • What is negation? x' = "not x". Let's try it in the case of each of the three types of propositions:
        1. (Some new cakes are nice)'
        2. (No new cakes are nice)'
        3. (All new cakes are nice)'

      • The board:
        • Bilateral
        • Trilateral
        • Counters:

          Let us agree that a Red Counter, placed within a Cell, shall mean "This Cell is occupied" (i.e. "There is at least one Thing in it").

          Let us also agree that a Red Counter, placed on the partition between two Cells, shall mean "The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are." Hence it may be understood to mean "At least one of these two Cells is occupied: possibly both are."

          Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind which of two political parties he will join: such a man is said to be "sitting on the fence." This phrase exactly describes the condition of the Red Counter.

          Let us also agree that a Grey Counter, placed within a Cell, shall mean "This Cell is empty" (i.e. "There is nothing in it").

    • Examples: let's use the board
      • Crazy babies
      • Here's a classic:
        1. Aristotle is a man.
        2. All men are mortal.
        3. Aristotle is mortal.

    • One of Lewis Carroll's favorite examples

    • Example:
      1. Babies are illogical
      2. Nobody is despised who can manage a crocodile
      3. Illogical persons are despised
      Solution:
      • Using Venn Diagrams (and their problems), and comparing
      • Carroll's method, and
      • A modification of the board

  • Further examples:
    • "No son of mine is dishonest; People always treat an honest man with respect."
    • "All cats understand French; Some chickens are cats".
    • Trios of Concrete Propositions (and a selection of Trios of Concrete Propositions for homework)