INTRO LOGIC DAY 16 Translations in PL 2

INTRO LOGIC DAY 16 Translations in PL 2 1

Overview + + Exam 1: Exam 2: Exam 3: Exam 4: Exam 5: Exam 6: Sentential Logic Translations (+) Sentential Logic Derivations Predicate Logic Translations Predicate Logic Derivations (finals) very similar to Exam 3 (finals) very similar to Exam 4 When computing your final grade, I count your four highest scores (A missed exam counts as a zero. ) zero 2

REVIEW of DAY 1 3

Existential Quantifier original sentence paraphrase someone is happy there is someone who is happy pronoun variable there is some x (s. t. ) x is happy formula x Hx 4

Universal Quantifier original sentence paraphrase pronoun variable formula everyone is happy no matter who you are happy no matter who x is happy x Hx 5

Negative-Existential Quantifier original sentence paraphrase pronoun variable formula no one is happy there is no one who is happy there is no x (s. t. ) x is happy x Hx 6

Negative-Universal Quantifier original sentence paraphrase pronoun variable formula not everyone is happy not: no matter who you are happy not: no matter who x is happy x Hx 7

Equivalences x. Hx not-everyone is happy someone is un-happy x Hx x. Hx no-one is happy everyone is un-happy x Hx = 8

new material for day 2 9

Quantifier Specification Generic Quantifier versus Specific Quantifier every one is H every F is H not-every one is H not-every F is H every one is un-H every F is un-H some one is H some F is H no one is H no F is H some one is un-H some F is un-H 10

Example 1 original sentence some Freshman is Happy paraphrase there is someone who … there is someone who is F and who is H there is some x x is F x ( Fx and & x is H Hx ) DON’T FORGET PARENTHESES 11

Example 2 original sentence no Freshman is Happy paraphrase there is no one who … there is no one who is F and who is H there is no x x is F x ( Fx and & x is H Hx ) DON’T FORGET PARENTHESES 12

Example 3 original sentence every Freshman is Happy paraphrase no matter who you are … you no matter who you are IF you are F THEN you are H no matter who x is IF x is F x ( Fx THEN x is H Hx ) DON’T FORGET PARENTHESES 13

Arrow versus Ampersand Rule of Thumb (not absolute) the connective immediately “beneath” a universal quantifier ( ) is usually a conditional ( ) ( ) the connective immediately “beneath” an existential quantifier ( ) is usually a conjunction (&) ( & ) 14

Summary of Quantifier Specification everyone is H x. Hx every F is H x(Fx Hx) not-everyone is H x. Hx not-every F is H x(Fx Hx) everyone is un-H x Hx every F is un-H x(Fx Hx) someone is H x. Hx some F is H x(Fx & Hx) no-one is H x. Hx no F is H x(Fx & Hx) someone is un-H x Hx some F is un-H x(Fx & Hx) 15

Conjunctive Predicate-Combinations x is an American Biker = x is an American, and x is a Biker x is an AB = [ Ax & Bx ] every AMERICAN BIKER is CLEVER every AB is C x ( [ Ax & Bx ] Cx ) some AMERICAN BIKER is CLEVER some AB is C x ( [ Ax & Bx ] & Cx ) no AMERICAN BIKER is CLEVER no AB is C x ( [ Ax & Bx ] & Cx ) 16

Non-Conjunctive Predicates Combinations alleged criminal imitation leather expectant mother experienced sailor small whale large shrimp deer hunter racecar driver woman racecar driver baby whale killer dandruff shampoo productivity software 17

Ambiguous Examples Bostonian driver Bostonian attorney cab 18

A Pitfall Compare the following: every Bostonian Attorney is Clevery BA is C vs. every Bostonian and Attorney is Clevery B and A is C 19

Zombie Logic every x {( for any IF thing Cat and Dog Cx & Dx it is a Cat and it is a Dog ) is a Pet Px THEN } it is a Pet in other words every CAT-DOG is a PET 20

WHAT EXACTLY IS A CAT-DOG? 21

Another Candidate 22

“Distributive” Use of ‘And’ every Cat and Dog is a Pet every Cat and every Dog is a Pet every Cat is a Pet, and every Dog is a Pet x ( Cx Px ) & x ( Dx Px ) 23

“Plural” Use of ‘And’ every member of the class Cats-and-Dogs is a Pet no matter who x is if x is a member of the class Cats-and-Dogs, then x is a Pet IS = = to be a member of the class Cats-and-Dogs to be a Cat or a Dog x is a member of the class Cats-and-Dogs, x is a Cat or x is a Dog [Cx Dx] x ( [ Cx Dx ] Px ) 24

‘Only’ as a Quantifier only are examples only Citizens are Voters only Men play NFL football employees only members only cars only right turn only Employees are Allowed only Members are Allowed only Cars are Allowed only Right turns are Allowed 25

Recall ‘only if’ only IF not IF not THEN not ‘only’ only is an implicit double-negative modifier 26

One Rendering of ‘only’ only are only if you are , are you (no matter who you are) you are only if you are (no matter who you are) x is only if x is (no matter who x is) x is not if x is not (no matter who x is) if x is not , then x is not (no matter who x is) x ( x x ) 27

Alternative Rendering of ‘only’ only = no non only are no non- are no one who is not is there is no one who is not but who is there is no x ( x is not but x is ) x ( x & x ) = x ( x x 28

THE END 29