Day 24 in Mat115

Today:

• Announcements:
  • You have a homework due (chaos!)

  • Our second exam is this Thursday, 11/18. It will cover the material in our study guide (except for the material on chaos -- we need to have more time for that, and we have enough other material for the exam). Chaos will be covered on the final.

  • Election results (so far): because my precinct information was so bad (:(, about the best I can do is give you the compilation of our results.

    Totals:

    Candidate	Paul	Conway
    Votes	373	220
    Percentage	.63	.37

    The refusal percentage was 23%.

    Official results: (for comparison)
    In Campbell County, the results were 65% to 35%;
    In Boone County, the results were 74% to 26%;
    In Kenton County, the results were 67% to 33%;

  • Our new topic today is logic (as "played" by Lewis Carroll).

  • You should have received a reply if you sent me a project proposal revision. If not, please let me know after class, and I'll get back to you.

• Before we go on: do you have questions about the Game of Life?

• 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:
      • What is the universe of discourse?
        the set of all things under discussion.

      • What's a statement (or proposition)?
        A sentence with truth value (either true of false)

      • What's 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?

      • 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.

    • A modification of the board

    • Here's how a computer does it (in the applet -- check the box -- use Babies, Asylum, Crazy -- x, y, m)

      (Alternatively, show that Aristotle is mortal, if Aristotle is a man, and all men are mortal: use Aristotle, Mortal, Man -- x, y, m)

    • Example (In the applet: story, devoid, yawn)

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

  • 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)