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Mastering Logic & Symbolization Techniques

Learn the syntax, symbolization, and methods of reasoning in logic with examples and explanations. Understand logical operators and connectives. Explore conjunctions, disjunctions, and conditionals in detail.

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Mastering Logic & Symbolization Techniques

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  1. Chapter 2: Syntax and Symbolization PHIL 121: Methods of Reasoning March 11, 2013 Instructor:Karin Howe Binghamton University

  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 operators • Conjunction • "and" (&) • Disjunction • "or" () • Conditionals • "if … then" () • Negation • "not" () • Biconditionals • "if and only if" ()

  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 WET and MAD. • Standardize: The cat is WET and 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. 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

  7. 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 CHERRIES or PICKLES on your ice cream, but not both.

  8. Symbolizing Disjunctions • Example: Either the cat is WET or MAD. • Standardize: Either the cat is WET or 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

  9. 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

  10. 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 

  11. Tricky Conditionals • "only if" • P only if Q • Standardized as: If P, then Q • Symbolized as: (P  Q) • Necessary Conditions – P is a necessary condition for Q – Standardized as: If Q, then P – Symbolized as: (Q  P) – Mnemonic: neceSSary conditions come second • Sufficient Conditions • P is a sufficient condition for Q • Standardized as: If P, then Q • Symbolized as: (P  Q) • Mnemonic: suFFicient conditions come first

  12. Tricky Conditionals, con't • "unless" • P unless Q • Standardized as: If not Q, then P • Symbolized as: (Q  P) • Note: the text considers the possibility that there could be an inclusive and an exclusive reading of "unless" • Consider the following: John will pick up Henry at the airport, unless Mary does it. • Inclusive reading: (M  J) • Exclusive reading: ((M  J) & (M  J)) • Text leaves this question unresolved -- are we using the "inclusive" interpretation or the "exclusive" interpretation? • We will use the inclusive interpretation ("unless" will be interpreted as a single conditional)

  13. 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")

  14. Symbolizing Negations • Example: The cat is not WET. • Standardize: It is not the case that the cat is WET. • Loglish: not W • Symbolize: W

  15. Biconditionals • Recall that biconditionals usually involve the phrase "if and only if" • However, biconditionals may also be expressed using any one of a number of (logical) synonyms for "if and only if" • just in case …, then and only then, it is a necessary and sufficient condition for • Logicians also sometimes abbreviate "if and only if" as iff

  16. Symbolizing Biconditionals • Example: You can go to the MOVIES if and only you CLEAN up your room. • Standardize: You can go to the MOVIES if and only you CLEAN up your room. • Loglish: M iff C • Symbolize: (M  C)

  17. WFF = well formed formulae (pronounced "woof") Recursive definition of WFF in sentential logic: • Every atomic formula  is a well-formed formula of sentential logic. • If  is a well-formed formula of sentential logic then so is . • If  and  are well-formed formulae of sentential logic then so are each of the following: • ( & ) • (  ) • (  ) • (  ) • An expression of sentential logic is a well-formed formula if and only if it can be formed through one or more applications of rules 1-3.

  18. Building up WFFs using the Recursive Definition • We can use the recursive definition to show that a formula is a WFF by showing how the formula can be "built up" step-by-step using the recursive definition • Example: ((P & Q)  R) • R Rule 1 • R Rule 2, line 1 • P Rule 1 • Q Rule 1 • (P & Q) Rule 3a, lines 3, 4 • ((P & Q)  R) Rule 3b, lines 5, 2

  19. Building up WFFs using the Recursive Definition • We can also use the recursive definition to show that a formula is a non-WFF by showing how we get "stuck" in trying to show how the formula can be "built up" step-by-step using the recursive definition • Example: ((P & Q) & R) • R Rule 1 • P Rule 1 • Q Rule 1 • (P & Q) Rule 3a, lines 2, 3 • Stuck!

  20. 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 major operator 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 • Some comments about punctuation • Why do we care about punctuation and the Two-Chunk Rule?? • WFFs • Finding the major operator • But why do we care about the major operator? • Truth tables (and truth trees) • Derivations

  21. WFFs or Not? • Show whether the following formulas are WFFs or non-WFFs using the recursive definition: • A • (A C) • (B  A) • (B  A) • (A & B  C)

  22. Practice With Symbolization • 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

  23. I will either get a PUPPY or a GUPPY, or possibly both. • You may get either a PUPPY or a GUPPY, but not both. • You may get neither a PUPPY nor a GUPPY. • Lina is SAD that she cannot get a guppy. • Lina thinks that a BETTA would make a better pet than a guppy anyway. • Neither TOM nor LINA nor KARIN are in the market for a new cat. • Lina wants to get a new HAMSETER if Rowan won't stay out of the TEAPOT, unless she thinks that a GERBIL would be better behaved.

  24. WFFs or Not? • Show whether the following formulas are WFFs or non-WFFs using the recursive definition: • A & (B  C) • X & (Y  Z ) • P  P • A  (B  C) • A  (B  C)

  25. A (A  B) C (A  ~C) AC (A &C) B &(A & C) A &(C D) (B & C) D B (B & A) A (B &D) (C  B)  A C (C  B)  (A & C) C  [B  (A  C)] (C & B) (A &C) More Practice With WFFs

  26. Practice With Symbolizing 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

  27. Practice With Symbolizing 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

  28. Practice With Symbolizing 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

  29. Practice With Symbolizing 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"

  30. Practice With Symbolizing Biconditionals • Our JUSTICE system works if, and only if, witnesses are willing to come FORWARD. ~ newspaper column • If and only if it [a motion on racial research] is APPROVED by a majority of the AAA membership will it become an official POSITION of the American Anthropological Association) ~ bulletin • The assistantship will be offered to MCGRAW if and only if he does not get a tuition WAIVER; and it will be offered to SPIEGELMAN if not offered to McGraw. ~ minutes of meeting

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