Assignment 1 (40 points) 1. Indicate which of the following expressions are formulas, and which of those are sentences of PL. For each formula, underline its main connective. For those expressions that are not formulas, or that are formulas but not sentences, explain why not. (2 points each) 1. (x)P x _ P y 2. (8z)(E x)(F z x&F x z) 3. (8x)F aa 4. (9x)[F x&(8x)(P x G x)] 5. ⇠ (8x))(G x ⌘ (9z)F z x) 6. (8x)[(9y)F y x (9y)F x y] 7. (9a)F a 8. (8z)(H za (9y)Ga z) 2. For each of the following formulas, list all of its subformulas and identify its main logical operator. (2 points each) 1. (9x) ⇠ (8y)B y x 2. (8y)[(8z) ⇠ B y z _ B y y] 3. Ra x ⇠ (8y)Ry x 4. (8x)(8y)(8z)M xy z&(8z)(8x)(8y)M y z x 5. (8z)[F z (9w)(⇠ F w&Gwa z)] 6. ⇠ [(8x)F x _ (8x) ⇠ F x] 3. For each of the following sentences, give the substitution instance in which ‘a’ is the instantiating term. (2 points each) 1. (9y)(M b y M ya) 2. (8x)[(Laa&La b ) La x] 3. (9w)[F w&(8y)(C yw 4. (8y)[(F y&H y) C wa)] [(9z)(F z&G z) Gy]] 1 4. Symbolize the following sentences of English into PL, using the given symbolization key. (2 points each) - UD: The jellybeans in the jar on the coffee table Bx: x is black Rx: x is red Lx: x is licorice-flavored Sx: x is sweet Tx: x is tasty 1. Some of the jellybeans are black. 2. If all jellybeans are black then none is red. 3. Some of the jellybeans are not black. 4. At least one jellybean is black. 5. If none is black all are red. 6. Some licorice-flavored jellybeans are sweet and some are not. 7. Either all licorice-flavored jellybeans are black or all sweet jellybeans are red. 8. No jellybean is tasty and not licorice-flavored. 9. Some jellybeans are sweet and some are red but none is sweet and red. 10. All jellybeans that are not licorice-flavored are not tasty unless they are sweet. 5. Symbolize the following sentences of English into PL, using the given symbolization key. (2 points each) - UD: People Sx: x is a sailor Lx: x is lucky Axy: x admires y. Dxy: x is a daughter of y. Cx: x is careless. 1. Some sailors are both careless and admired. 2. Some careless sailors aren’t lucky. 3. Not all lucky sailors are careless. 4. Not all daughters of sailors are sailors. 5. Every daughter admires some sailor. 6. Every sailor is admired by some daughter. 2 6. Use the given symbolization key to translate the following sentences into fluent English. (2 points each) - UD: Positive integers - Lxy: x is larger than y - Ex: x is even - Ox: x is odd - Px: x is prime - a: 1 - b: 2 - c: 3 - d: 4 1. ⇠ (Pa _ P d ) 2. ⇠ (9y)(E y&O y) 3. (9y) ⇠ Lb y 4. (8z)(P z Lza) 5. ⇠ (9x)(P x& ⇠ Lxa) 6. (9x)(8y)(P y ⇠ x = y) 3