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Welcome to. Gödel. Escher. Bach. Tuesday edition, February 27, 2001. News. Seminar Wed Feb 28, 4:00 PM Jepson 109 The Poincaré Conjecture Mike Kerckhove discusses whether we can determine whether our universe is in the shape of a sphere or a doughnut.

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  1. Welcome to Gödel Escher Bach Tuesday edition, February 27, 2001 News Seminar Wed Feb 28, 4:00 PM Jepson 109 The Poincaré ConjectureMike Kerckhove discusses whether we can determine whether our universe is in the shape of a sphere or a doughnut Exam II Almost everyone has taken it. Unlikely to be returned before next Tuesday.

  2. Symbols of the Propositional Calculus (Symbols of previous systems) MIU-System: M I U pq-System: - p q Atom particle: - Atoms: - or - - or - - - or ...

  3. Symbols of the Propositional Calculus (Symbols of previous systems) MIU-System: M I U pq-System: - p q Operation: p atoms - - - p - - Example:

  4. p p Symbols of the Propositional Calculus (Symbols of previous systems) MIU-System: M I U pq-System: - p q Operation: q atom - - - p - - q - - Example:

  5. Symbols of the Propositional Calculus (Symbols of previous systems) MIU-System: M I U pq-System: - p q tq-System: - t q C-System: - t q C Prefix operation: C C - - - Example:

  6. Symbols of the Propositional Calculus How to classify them? < > P Q R ’ ^ v ~ [ ] Atom particles Example: P All atom particles are atoms

  7. New atom Symbols of the Propositional Calculus How to classify them? < > P Q R ’ ^ v ~ [ ] Atom particles Atom suffix ’ Example: P All atoms are well-formed

  8. Symbols of the Propositional Calculus How to classify them? < > P Q R ’ ^ v ~ [ ] Atom particles Suffix Prefix op Formation Rule #1: If x is well-formed, then so is ~x Example: ~ x P

  9. Symbols of the Propositional Calculus How to classify them? < > P Q R ’ ^ v ~ [ ] Atom particles Suffix Infix ops Prefix op Formation Rules #2: If x and y are well-formed, then so is <x^y> Example: <x^y> <P^~Q>

  10. Symbols of the Propositional Calculus How to classify them? < > P Q R ’ ^ v ~ [ ] Infix op start Infix op stop Atom particles Suffix Infix ops Prefix op Procedure: Push Procedure: Pop Fantasy Rule Example: <P^~Q>

  11. (Study Question 3)What’s so recursive about the Formation Rules? Formation Rule #1: If x is well-formed, then so is ~x Formation Rule #1: If is well-formed, then so is ~

  12. ~ ~ (Study Question 3)What’s so recursive about the Formation Rules? Formation Rule #1: If x is well-formed, then so is ~x Formation Rule #1: If is well-formed, then so is ~ Formation Rule #1: If is well-formed, then so is ~

  13. (Study Question 3)What’s so recursive about the Formation Rules? Formation Rule #1: If x is well-formed, then so is ~x Formation Rule #1 P ~P ~~P

  14. Study Question 7Which of the strings are well-formed? Formation Rule #1: If x is well-formed, then so is ~xFormation Rules #2: If x and y are well-formed, then so is <x^y>Formation Rules #3: If x and y are well-formed, then so is <xvy>Formation Rules #4: If x and y are well-formed, then so is <x y> Formation Rule #0: All atoms are well-formed atoms <~P> < x > NOT well-formed <P^Q> <x^y> WELL-formed <x^y^z> <P^Q^R> NOT well-formed

  15. Study Question 7Which of the strings are well-formed? Formation Rule #1: If x is well-formed, then so is ~xFormation Rules #2: If x and y are well-formed, then so is <x^y>Formation Rules #3: If x and y are well-formed, then so is <xvy>Formation Rules #4: If x and y are well-formed, then so is <xy> Formation Rule #0: All atoms are well-formed atoms <<P v<QR>>^<~Pv~R’>> <<xv<xy>>^<~xv~y>> <<xv y>^< xv y >> < x^ y > WELL-formed

  16. Pause

  17. English to Propositional Calculus 1. I think; therefore I am.- Rene Descartes 2. Good judgment comes from experience, and often experience comes from bad judgment.” [Hint: ignore “often”]- Rita Mae Brown 3. Like it or not, here it comes.- Anonymous 4. If you are a retiree, use your age on the annuity starting date,... but if your annuity starting date was after 1997 and the payments are for your life and that of your beneficiary, use your combined ages on the annuity starting date. - One of the more comprehensible sections of the US tax code

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