Exam 5: Full First-Order Logic

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 -If a number is divisible by three, then its successor is not.

1. (∀x)[(Hx • x≠a) ⊃ Jax] 2. Hb • Ib • Mb • (∀x)[(Ix • Mx) ⊃ x=b] 3. Ha • ∼Ma -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

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 -No philosopher is more skeptical than Hume.

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 -Any number is divisible by two if, and only if, the successor of its successor is divisible by two.

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

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 -No cheetahs that are larger than some tiger are faster than all lions or tigers.

1. (∃x)(Ax • Bx • Cax) 2. (∀x)[(Ax • Cax) ⊃ x=d] -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: 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 -Exactly one student is a valedictorian.

1. (∃x)(∃y)(Dx • Exb • Dy • Eyb • x≠y) 2. (∀x)(Dx ⊃ Ebx) -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: 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 are faster than all lions.

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)] 2. (∀x)(Dx ⊃ Ex) / (∃y)(Ey • Fyy)

1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)] 2. (∀x)(Dx ⊃ Ex) -Which of the following propositions is derivable from the given premises in F?

1. Fab • (∀x)(Fax ⊃ x=b) 2. ∼Fac -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

construct a derivation of each logical truth of -(∀x)(∀y)(Pxy ≡ ∼Pyx) ⊃ [(∃x)(∃y)Pxy ⊃ ∼(∀x)(∀y)Pxy]

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 -The richest philosopher is smarter than any of Chloe's brothers.

determine whether the given argument is valid or invalid. If it is valid, derive the conclusion using our rules of inference and equivalence. If it is invalid, provide a counterexample. -1. (?x)[Lx ? (?y)(My • Nxy)] 2. (?x)[Lx • (?y)Nxy] / (?x)[Mx • (?y)Nxy]

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 -There is at most one queen in Britain.

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)] 2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)] -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

All philosophers respect each other. Some philosopher doesn't study some philosopher. Anything which respects something without studying it is open-minded, if ignorant. So something is open-minded and ignorant. (Ix: x is ignorant; Ox: x is open-minded; Px: x is a philosoper; Rxy: x respects y; Sxy: x studies y) -Which of the following is the best translation into F of this argument?

select the best translation into predicate logic, using the following translation key: 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 -No lion is faster than some tigers.