Chapter 2 Syntax and Symbolization PHIL 121 Methods

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

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.

Types of logical operators • Conjunction – "and" (&) • Disjunction – "or" ( ) • Conditionals – "if … then" ( ) • Negation – "not" ( ) • Biconditionals – "if and only if" ( )

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, …

Symbolizing Conjunctions • Example: The cat is WET and MAD. 1. Standardize: The cat is WET and the cat is MAD. 2. Loglish: W and M 3. Symbolize: (W & M) • The two parts of a conjunction are called the right conjunct and the left conjunct

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

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.

Symbolizing Disjunctions • Example: Either the cat is WET or MAD. 1. Standardize: Either the cat is WET or the cat is MAD. 2. Loglish: W or M 3. Symbolize: (W M) • The two parts of a disjunction are called the right disjunct and the left disjunct

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

Symbolizing Conditionals • Example: If the cat is WET, then it is MAD. 1. Standardize: If the cat is WET, then the cat is MAD. 2. Loglish: if W then M 3. Symbolize: (W M) • The two parts of a conditional are called the antecedent and the consequent – antecedent appears before the – consequent comes after the

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: nece. SSary conditions come second • Sufficient Conditions – P is a sufficient condition for Q – Standardized as: If P, then Q – Symbolized as: (P Q) – Mnemonic: su. FFicient conditions come first

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)

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")

Symbolizing Negations • Example: The cat is not WET. 1. Standardize: It is not the case that the cat is WET. 2. Loglish: not W 3. Symbolize: W

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

Symbolizing Biconditionals • Example: You can go to the MOVIES if and only you CLEAN up your room. 1. Standardize: You can go to the MOVIES if and only you CLEAN up your room. 2. Loglish: M iff C 3. Symbolize: (M C)

WFF = well formed formulae (pronounced "woof") Recursive definition of WFF in sentential logic: 1. Every atomic formula is a well-formed formula of sentential logic. 2. If is a well-formed formula of sentential logic then so is . 3. If and are well-formed formulae of sentential logic then so are each of the following: a) ( & ) b) ( ) c) ( ) d) ( ) 4. 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.

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) 1. 2. 3. 4. 5. 6. R R P Q (P & Q) ((P & Q) R) Rule 1 Rule 2, line 1 Rule 3 a, lines 3, 4 Rule 3 b, lines 5, 2

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 the formula can be "built up" step-by-step using the recursive definition Example: ((P & Q) & R) 1. 2. 3. 4. 5. R P Q (P & Q) Stuck! Rule 1 Rule 3 a, lines 2, 3

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

WFFs or Not? • Show whether the following formulas are WFFs or non-WFFs using the recursive definition: 1. 2. 3. 4. 5. A (A C) ( B A) (A & B C)

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

1. I will either get a PUPPY or a GUPPY, or possibly both. 2. You may get either a PUPPY or a GUPPY, but not both. 3. You may get neither a PUPPY nor a GUPPY. 4. Lina is SAD that she cannot get a guppy. 5. Lina thinks that a BETTA would make a better pet than a guppy anyway. 6. Neither TOM nor LINA nor KARIN are in the market for a new cat. 7. 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.

WFFs or Not? • Show whether the following formulas are WFFs or non-WFFs using the recursive definition: 1. 2. 3. 4. 5. A & (B C) X & (Y Z ) P P A (B C)

More Practice With WFFs 1. 2. 3. 4. 5. 6. 7. A (A B) C ( A ~C) A C (A & C) B & (A & C) A & (C D) 8. 9. 10. 11. 12. 13. 14. (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)

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

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

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

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"

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 Mc. Graw. ~ minutes of meeting