Exam 5: Full First-Order Logic

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 -Charles has at least two sons, and so does Elizabeth.

[(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa) -Consider assuming '(∀x)Pax • ∼(∃x)Pxa' 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 • Ox) ⊃ (∃y)[(Ey • Ny) • Gyx)]}

consider the following domain, assignment of objects in the domain, and assignments sets to predicates. Domain = {1, 2, 3, ..., 28, 29, 30} a = 1 e = 21 b = 2 f = 23 c = 4 g = 27 d = 20 h = 29 Ex = {2, 4, 6, ..., 28, 30} Ox = {1, 3, 5, ..., 27, 29} Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29} Sxyz = The set of all triples such that the first is the sum of the second and third {<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... } -Given the customary truth tables, which of the following theories is modeled by the above interpretation?

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 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 cheetahs that are larger than some tiger are faster than all lions or tigers.

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 -The valedictorian is Diego.

1. (∀x)(∀y)(∀z)[(Ix • Jx • Iy • Jy • Iz • Jz) ⊃ (x=y $\lor$ y=z $\lor$ x=z)] 2. Ia • Ja • Ka • Ib • Jb • Kb • a≠b -Which of the following propositions is derivable from the given premises in F?

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)(Ax ⊃ Dex) 2. (∃x)(Bx • Dxe) 3. (∀x)(Bx ⊃ Ax) / (∃x)[Ax • (Dxe • Dex)]

construct a derivation of each logical truth of -(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px]

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

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

1. (∀x)[(Px $\lor$ Qx)] ⊃ Rxx] 2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]} 3. Pn • Qn -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: 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 in Britain is more exalted than Charles, except Elizabeth.

1. (∀x)[(Px • Qx) ⊃ Rf(x)] 2. (∀x)[Rx ⊃ (∃y)Pxy] 3. ∼(∀x)(Px ⊃ ∼Qx) / (∃x)(∃y)Pxy

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 who loves Chloe's mother is taught by Bhavin's maternal grandfather.

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 lion is faster than all tigers if, and only if, some cheetah is larger than any tiger faster than it.

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 is more skeptical than all rationalist philosophers.

1. (∀x){Ax ⊃ [Bx • Bf(x)]} 2. ∼Bf(f(e)) -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 -Hume is not more skeptical than some philosophers.