Day 2, Mat385

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  • Roll
  • Important: no class Monday! It's our Martin Luther King, Jr. holiday.
  • You will have a quiz on Wednesday of next week, however, over this section. So get started on those problems!
  • You might consider moving up, so that you can see better: it's going to be a little hard to see the screen, I'm afraid....

• Last time:
  • We tried out Lewis Carroll's The Game of Logic, and we tried our hands at playing the game.

    A couple of observations:

    • Two types of "counters" -- grey (there's nothing in this cell) and red (there is something in this -- or these -- cells). Frequently our goal is to force a red counter "off the fence" -- to make it choose sides.
    • Whenever we're making logical arguments, it's important to keep the universe in mind: in the first of the syllogisms that Carroll tackled on your handout,
      1. All pigs are fat.
      2. Nothing that is fed on barley-water is fat.
      the universe might have been all creatures on earth (say).

      Here's Carroll making the universe explicit in his syllogisms.

      The structure of "the universe" makes a difference in how one treats a problem....

    Let's look at the pigs in detail: today we'll see how to translate that using a different notation, and we'll soon get the rules to reach the same conclusion that we reach using Carroll's game.

    We can actually arrive at two conclusions from this:

    • Pigs are not fed on barley-water, and
    • If something is fed on barley-water, it's not a pig.

    How do we arrive at those, the Carroll way?

  • Then we worked on another logic puzzle:

    Five delegates (A, B, C, D and E) participated in an international summit.... The following was observed.

    1. Three delegates could speak Portuguese.
    2. The most common language was Spanish.
    3. The only common language between C and E was Italian.
    4. The only common language among A, B and E was French.
    5. B and C spoke English, but when D joined them, they all changed to Spanish, the only common language among the three of them.
    6. One of the delegates spoke all five languages, another one spoke four languages, one spoke three languages, one spoke two languages and the other spoke only one language.
    What language(s) did each delegate speak?

    Some questions:

    • What is "the answer"?
    • What tools suggest themselves when trying to solve this puzzle?
    • In what order does one tackle the clues?
    • What if we interpret "Spanish is the most common" to mean that ties are permitted? Does this solution work? (Here is my reasoning to get to that solution.)

    Some comments:

    • Language causes problems: does "Three delegates could speak Portuguese." mean exactly three? Could all five speak Portuguese, and this still be true?
    • Ditto for #6.
    • Does "most common language" allow for ties?

  • Today we delve into section 1.1, and take a more "mundane" look at logic, without the pigs, barley-water, crazy babies, etc.
    • Section 1.1 (Statements, Symbolic Representations, and Tautologies), which we'll examine using the highlights from 1.1. If you have a copy, play along! Or just use a sheet of paper to work the problems along the way.

    • Next time we'll start drawing conclusions using the tools of section 1.1; check out (my potentially old, but perhaps useful) highlights from 1.2, so you can prepare for next Wednesday (and get started on those homework problems in section 1.2).

• Links:
  • Is the moon made of green cheese? (source)
  • Lewis Carroll Puzzles, illustrating the power of hypothetical syllogism.
  • More "historical logic": this is a fun one (the dilemma of Protagoras and Eualthus)

    The story reminds me of a poem by A. E. Housman: The Laws of God, The Laws of Man. The reason that the story of Protagoras and Eualthus is a dilemma, a paradox, or a quandary, is that there are two different systems of laws operating.

  • More detail on Lewis Carroll's logic game
  • A description with an applet to play the game yourself
  • Some Fun, Motivating Examples from Propositional and Predicate Calculus

  • Paradoxes are fun! Evidently Eubulides was the master of them.