Day 33: Heart of Mathematics

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.

• 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").

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

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

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