Chapter 2: Syntax and Symbolization

1. Chapter 2: Syntax and Symbolization 80-210: Logic & Proofs July 2, 2009 Karin Howe

2. Syntax = grammar of a logical statement • Two different categories of basic expressions: • Atomic formulae • Sentences that have no logically relevant internal structure • Examples: • The cat is in the teapot. • The cat is all wet. • Logical connectives • Serve to connect formulae (atomic or otherwise) to create more complex formulas. • Examples: • Cats are a lot of trouble but they are also a lot of fun. • If the cat is in the teapot, then it is both mad and wet.

3. Types of logical connectives • Conjunction • "and" • Disjunction • "or" • Conditionals • "if … then" • Negation • "not"

4. Conjunctions • Recall that conjunctions usually involve the word "and" • However, conjunctions may also be expressed using any one of a number of (logical) synonyms for "and" • but, however, moreover, although, yet, even though, …

5. Symbolizing Conjunctions • Example: The cat is wetandmad. • Standardize: The cat is wetand the cat is mad. • Loglish: W and M • Symbolize: (W & M) • The two parts of a conjunction are called the right conjunct and the left conjunct

6. Practice With Conjunctions • My husband has many fine qualities, but he has one serious hangup. ~ letter to 'Dear Abby" • War is cruel and you cannot refine it. ~ General William Sherman • I May Be Fat, But You're Ugly–And I Can Diet! ~ bumper sticker • Santa Claus is alive and well and living in Argentina. ~ bumper sticker

7. The Two-Chunk Rule • The Two Chunk Rule says: Once more than one logical connective symbol is necessary to translate a statement, there must be punctuation that identifies the main connective of a symbolic statement. In addition, there cannot be any part of a statement in symbols that contains more than two statements, or chunks of statements, without punctuation • Examples: • A • (A & B) • (A & B  C) • A • Why do we care?? • WFFs • Finding the main connective • But why do we care about the main connective? • Truth tables (and truth trees) • Derivations

8. WFFs and Parse Trees • WFF = well formed formulae • Parse trees • Example: ((P & Q)  (R  S))

9. Disjunctions • Recall that disjunctions usually involve the word "or" • However, disjunctions may also be expressed using any one of a number of (logical) synonyms for "or" • either/or, and/or

10. Disjunctions: Exclusive vs. Inclusive • You can have either cherries or pickles on your ice cream. • Two ways you can interpret this: • Pick one - cherries or pickles • You can have both! (yuck!) • We will take "or" in the inclusive sense (thus and/or is just shorthand for inclusive "or") • If need be, we can represent exclusive "or" as follows: • You can have either chocolate sauce or pickles on your ice cream, but not both.

11. Symbolizing Disjunctions • Example: Either the cat is wetormad. • Standardize: Either the cat is wetor the cat is mad. • Loglish: W or M • Symbolize: (W  M) • The two parts of a disjunction are called the right disjunct and the left disjunct

12. Practice With Disjunctions • This woman must be either mad or drunk. ~ Plautus dialogue • Either that man's a fraud or he's your brother. ~ Plautus • I can either run the country or control Alice–not both. ~ Theodore Roosevelt • They'd better lost the attitude and listen to their dad, or they won't get diddly crap. ~ newspaper, lottery winner discussing her grandchildren

13. Conditionals • Recall that conditionals usually involve the phrase "if … then" • However, conditionals may also be expressed using any one of a number of (logical) synonyms for "if … then" • provided (that), given (that), should, will result in, only if, is a necessary condition for, is a sufficient condition for

14. Symbolizing Conditionals • Example: If the cat is wet, then it is mad. • Standardize: If the cat is wet, then the cat is mad. • Loglish: if W then M • Symbolize: (W  M) • The two parts of a conditional are called the antecedent and the consequent • antecedent appears before the  • consequent comes after the 

15. Tricky Conditionals • Necessary Conditions – P is a necessary condition for Q – Rewritten as: If Q, then P – Symbolized as: Q  P – Mnemonic: neceSSary conditions come second • Sufficient Conditions • P is a sufficient condition for Q • Rewritten as: If P, then Q • Symbolized as: P  Q • Mnemonic: suFFicient conditions come first

16. Tricky Conditionals, con't • "only if" • P only if Q • Rewritten as: If P, then Q • Symbolized as: P  Q • "unless" • P unless Q • Rewritten as: If not Q, then P • Symbolized as: Q  P

17. Practice With Conditionals • I am extraordinarily patient provided I get my own way in the end. ~ Margaret Thatcher • If 14-year-olds had the vote, I'd be President. ~ Evil Knievel • If the Grand Jury calls me back I will be glad to cooperate fully if my immunity is extended. ~ CREEP operative • If the axioms could be so selected that they were necessarily true, then, if the deductions were valid, the truth of the theorems would be guaranteed. ~ logician James Carney

18. Negations • Recall that negations usually involve the word "not" • However, negations may also be expressed using any one of a number of (logical) synonyms for "not" • It is not true that, it is false that, no, never, isn't (won't, didn't, etc.), it is not the case that, unless (equivalent to "if not"), without (equivalent to "but not"), neither/nor (equivalent to "it is false that either/or")

19. Symbolizing Negations • Example: The cat is notwet. • Standardize: It is not the case that the cat is wet. • Loglish: not W • Symbolize: W

20. Practice With Negations • [Read my lips], no new taxes ~ presidential candidate George Bush (senior) • I am not a crook. ~ Richard Nixon • Now we shall have duck eggs, unless it is a drake. ~ Hans Christian Anderson • If God didn't want them sheared, he wouldn't have made them sheep. ~ Eli Wallach in "The Magnificent Seven"

21. More Practice • Possession of a hot plate in the dorms is not illegal. (L = possession of a hot plate in the dorms is legal) ~ student newspaper • Kiss me, and a handsome prince will appear. ~ Wizard of Id • Marvin's being busted for "pot" possession is a sufficient condition for his being dropped from the team. • Nancy's scoring above 1,000 on the GRE is a necessary condition for her admission to graduate school. • It is illegal to feed or harass alligators. ~ Everglades sign

22. Final Note: Parentheses: To Drop, or Not To Drop • (P & Q)  (R  S) • ((P & Q)  (R  S)) • A & B  C  D • (((A & B)  C)  D) • A  B  C & D • (A  (B  (C & D))) • Procedure for re-inserting parentheses: • First, insert parentheses around every occurrence of & and its two conjuncts, starting with the rightmost & and ending with the leftmost • Next, insert parentheses in the same fashion for each  and its two disjuncts, from rightmost occurrence first, to the leftmost occurrence last • Finally, insert parentheses for each , and its antecedent and consequent, from rightmost occurrence first, to the leftmost occurrence last