Day 2, Mat385: Welcome, Let's Go!

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Last time:
- We tried out Lewis Carroll's The Game of Logic, and we tried our hands at playing the game.
  A couple of interesting observations:
  - Two types of "counters" -- grey (there's nothing in this cell) and red (there is -- or may be -- something in this -- or these -- cells). Frequently our goal is to force a red counter "off the fence" -- to make it choose sides.
  - It's important to keep the universe in mind: in the syllogisms that Carroll tackled on your handout, the first two involved animals (pigs and rabbits). In the first case, the universe might have been all creatures (say); in the second, Carroll declared that the universe was rabbits. Here's Carroll making that explicit.
    The structure of "the universe" makes a difference in how one treats a problem....
  Let's look at one example (the pigs) in detail: today we'll 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. one of those with multiple premises, found within this description of Lewis Carroll and his logic game:
  1. All pigs are fat.
  2. Nothing that is fed on barley-water is fat.
  We can actually arrive at two conclusions from this:
  1. Pigs are not fed on barley-water, and
  2. If something is fed on barley-water, it's not a pig.
  How do we arrive at those, the Carroll way?
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 highlight worksheet 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 highlight worksheet 1.2, so you can prepare for next Tuesday (and get started on those homework problems in section 1.2).
Links:
- Carroll's The Game of Logic -- a really nice version.
- Another Carrollinian example in detail: one of those with multiple premises, found within this description of Lewis Carroll and his logic game:
  1. No birds, except ostriches, are 9 feet high.
  2. There are no birds in this aviary that belong to anyone but me.
  3. No ostrich lives on mince pies.
  4. I have no birds less than 9 feet high.
  Clearly the conclusion is that "No bird in this aviary lives on mince pies....":)
  How do we arrive at that, the Carroll way?
  (you can do it with "algebra", as we will in these next few sections, but it's far less amusing!:) And, as you can see in the comments to this attempt, if you make a mistake in your formulation of the algebra, you won't arrive at the correct conclusion (except by accident, perhaps).
- 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.
- Some Fun, Motivating Examples from Propositional and Predicate Calculus
- Paradoxes are fun! Evidently Eubulides was the master of them.

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