Exam 5: Full First-Order Logic

1. (?x){Ax ? [Bx • Bf(x)]} 2. ?Bf(f(e)) / ?Af€

use: b: Bhavin c: Chloe m: Megha n: Nietzsche p: Plato Ax: x is an altruist Jx: x is joyful Px: x is a philosopher Rx: x is Russian Tx: x is thoughtful Bxy: x is a brother of y Mxy: x mocks y Rxy: x is richer than y Sxy: x is smarter than y -Everything is joyful, except Megha and Bhavin.

select the best translation into predicate logic, using the following translation key: b: Bhavin c: Chloe m: Megha n: Nietzsche p: Plato Ax: x is an altruist Jx: x is joyful Px: x is a philosopher Rx: x is Russian Tx: x is thoughtful Bxy: x is a brother of y Mxy: x mocks y Rxy: x is richer than y Sxy: x is smarter than y -All thoughtful philosophers except Nietzsche are altruists.

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. Ha • Ia • (?x)[(Hx • Ix) ? x=a] 2. Hb • Jb • (?x)[(Hx • Jx) ? x=b] 3. Ka • ?Kb / ?(?x)(Hx • Ix • Jx)

All cheetahs are faster than some tigers. Everything is striped if, and only if, it is a tiger. So, if some things are cheetahs, then some things have stripes. (Cx: x is a cheetah; Tx: x is a tiger; Sx: x has stripes; Fxy: x is faster than y) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

construct a model for each of the given theories in the following domain by assigning members of the domain to the constants used in the theory and sets of members of the domain to the predicates used in the theory. Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} -(∀x){Px ⊃ [(∃y)Cxy $\lor$ (∃y)Cyx]} Cab • ∼Cba Cac •∼Cca Cad • ∼Cda

Translate each sentence into predicate logic, using the given translation keys. use: Cx: x is a cheetah Lx: x is a lion Tx: x is a tiger Fxy: x is faster than y Lxy: x is larger than y -Some lions are larger than all cheetahs.

select the best translation into predicate logic, using the following translation key: d: Diego s: Sean Dx: x is on the Dean's list Px: x is a philosophy major Sx: x is a student Vx: x is a valedictorian -There is exactly one philosophy major on the Dean's list.

use the following translation key to write the sentences below in FF. a: Ahmed b: Bhavin c: Chloe d: Delfina e: Eduardo i: Irving f(x): the father of x g(x): the mother of x f(x,y): the only son of x and y Lxy: x loves y Sxy: x is a sister of y Txy: x teaches y -Delfina loves both her mother and her father.

provide a conterexample in a finite domain to each given invalid argument. -1. Ad ⊃ (∀x)Fdx / (∃x)Fdx

1. (∀x)(Ecx ⊃ x=d) 2. (∀x){(Fx • Gx) ⊃ (∀y)[(Fy • Gy) ⊃ y=x]} 3. (∃x)(Fx • Gx • Ecx) 4. Fa • Ga -Which of the following propositions is derivable from the given premises in F?

select the best translation into predicate logic, using the following translation key: a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The sum of one and the successor of one is divisible by three.

use: d: Diego s: Sean Dx: x is on the Dean's list Px: x is a philosophy major Sx: x is a student Vx: x is a valedictorian -At least four philosophy majors are on the Dean's list.

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)[(∃y)(Fxy • Gy) ⊃ (∀y)(Gy ⊃ Fxy)] 2) Ga / (∀x)(Fxa ⊃ Fax)

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The successor of a number is greater than itself.

use the following translation key to write the sentences below in FF. a: Ahmed b: Bhavin c: Chloe d: Delfina e: Eduardo i: Irving f(x): the father of x g(x): the mother of x f(x,y): the only son of x and y Lxy: x loves y Sxy: x is a sister of y Txy: x teaches y -Bhavin's paternal grandparents love each other.

select the best translation into predicate logic, using the following translation key: a: Ahmed b: Bhavin c: Chloe d: Delfina e: Eduardo i: Irving f(x): the father of x g(x): the mother of x f(x,y): the only son of x and y Lxy: x loves y Sxy: x is a sister of y Txy: x teaches y -Irving's father teaches Bhavin and Eduardo.

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)] 2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)] 3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz] 4. (∀x)(Ex ⊃ Bx) / (∀x)[Ax ⊃ (∀y)(Hy ⊃ Cxy)]

select the best translation into predicate logic, using the following translation key: h: Hume l: Locke Px: x is a philosopher Rx: x is a rationalist Ixy: x influenced y Sxy: x is more skeptical than y -Hume influenced some philosophers more skeptical than Locke.

Translate each sentence into predicate logic, using the given translation keys. use: Cx: x is a cheetah Lx: x is a lion Tx: x is a tiger Fxy: x is faster than y Lxy: x is larger than y -Some cheetahs that are not larger than any lion are faster than some tigers.