Exam 5: Full First-Order Logic

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)(Mx ⊃ Px) ∼(∀x)Px (∀x)[∼Px ⊃ (∀y)Cyx]

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 -Numbers that are not divisible by three have sums with themselves that are also not divisible by three.

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 the sum of the successors of any two numbers is divisible by three then the sum of two and the sum of the original two numbers is divisible by three.

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

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 propositions is an immediate (one-step) consequence in F of the Given premises?

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

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 -Locke influenced Hume, but is not more skeptical than him.

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 -At most two philosophy majors are on the Dean's list.

1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)] 2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)] 3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz] 4. (∀x)(Ex ⊃ Bx) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x)(∀y)(∃z)Sf(x)yz 2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz $\lor$ Mzyx)] -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: 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 -No one who shops at Tescos is more exalted than Charles.

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 -There are at most two Russian altruists

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 tiger is faster than all cheetahs.

1. (∀x)(∀y){Pf(x,y) ⊃ [Pf(x,x) • Pf(y,y)]} 2. a=f(d,b) 3. Pa -Which of the following propositions is derivable from the given premises in FF?

Translate each sentence into predicate logic, using the given translation keys. use: 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.

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

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

provide a conterexample in a finite domain to each given invalid argument. -1. (∃x)(Dxa • Ex) 2. (∃x)(Dxa • Fx) / (∃x)(Ex • Fx)

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 -The valedictorian is Diego.

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 has at least two sons, and so does Elizabeth.