Exam 5: Full First-Order Logic

consider the following domain, assignment of objects in the domain, and assignments sets to predicates. Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto} a = Mercury b = Jupiter c = Saturn d = Pluto Mx = {Mercury, Mars} Px = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} Cxy = {, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } -Given the customary truth tables, which of the following theories is modeled by the above interpretation?

select the best English interpretation of the given statements of predicate logic, using the following translation key: t: two Ox: x is odd Ex: x is even Nx: x is a number Gxy: x is greater than y -(∀x)[Nx ⊃ (Ox ⊃ ∼Ex)]

1. (∀x){Px ⊃ (∃y)[Py • f(x)=y]} 2. Pa • (∀x)f(x)≠a -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

(∃x)(∃y)(Axy • x=y) ⊃ (∃x)Axx -Consider assuming '(∃x)(∃y)(Axy • x=y)' for a conditional proof of the above logical truth. Which of the Following propositions is a legitimate second step in that proof?

translate each sentence of predicate logic into natural, English sentences using the following translation key: t: two Ox: x is odd Ex: x is even Nx: x is a number Gxy: x is greater than y -∼(∃x)(Nx • Gxx)

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 -Some philosophers more skeptical than Hume influenced all philosophers more skeptical Than Locke.

Rey received the highest grade on Test #1. Spencer, who is not Rey, received a grade on Test #1. So, Rey's grade is higher than Spencer's. (r: Rey; s: Spencer; t: Test #1; Gxy: x is a grade on y; Hxy: x is higher than y; Rxy: x received y) -Which of the following is the best translation into F of this argument?

1. (∀x)(∀y)(∀z)[(Pxy • Pyz) ⊃ Pxz] 2. (∀x)Pxf(x) -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

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 -No one is the only son of Ahmed and Delfina.

The virtue ethicist who teaches Aristotle does not publish on moral theory. Smith is a virtue ethicist who teaches Aristotle. Either one publishes on moral theory or one publishes on applied ethics. So, Smith publishes on applied ethics. (s: Smith; Ax: x publishes on applied ethics; Mx: x publishes on moral theory; Tx: is teaches Aristotle; Vx: x is a virtue ethicist) -Which of the following claims can also be derived from the premises of this argument?

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. (∃x){Ex • Fx • (∀y)[(Ey • y≠x) ⊃ Gxy]} 2. En • ∼Fn / (∃x)(Ex • Gxn)

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 -Only Sean and Diego are both on the Dean's list and philosophy majors.

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)] 2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)] / (∃x)(∃y)[Pf(x,y) • Rxy • x≠y]

use: b: Britain c: Charles e: Elizabeth t: Tescos Px: x is a person Qx: x is a Queen of England Wx: x is a woman Exy: x is more exalted than y Ixy: x is in y Pxy: x shops at y Sxy: x is a son of y -Charles and his two sons are not the most exalted people in Britain.

1. (∀x)[(Cx • Exa) ⊃ Dx] 2. Cd • ∼Dd -Which of the following propositions is derivable from the given premises in F?

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 -There are exactly two philosophy majors on the Dean's list.

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 claims can also be derived from the premises of this argument?

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 -If some rationalist is more skeptical than Locke, then no philosopher influenced Hume.

select the best translation into predicate logic, using the following translation key: b: Britain c: Charles e: Elizabeth t: Tescos Px: x is a person Qx: x is a Queen of England Wx: x is a woman Exy: x is more exalted than y Ixy: x is in y Pxy: x shops at y Sxy: x is a son of y -Elizabeth is the most exalted woman in Britain.

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 -Exactly three Russian philosophers are joyful.