Exam 5: Full First-Order Logic

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 • ∼Ex) • Gxt]

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 any number and two is equal to the sum of the successor of that number and one.

1. (∀x)[Ax ⊃ (∃y)(By • Cxy)] 2. (∃x)(Ax • Dx) 3. (∀x)(Bx ⊃ Ex) -Which of the following propositions is derivable from the given premises in F?

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 -No philosopher is more skeptical than Hume.

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 -Exactly one student is a valedictorian.

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?

1. (∀x)[(Px • Qx) ⊃ Rf(x)] 2. (∀x)[Rx ⊃ (∃y)Pxy] 3. ∼(∀x)(Px ⊃ ∼Qx) -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

1. (?x)(?y)f(x,y)=f(y,x) 2. (?x)f(x,o)=o / (?x)f(o,x)=o

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)[Hx ⊃ (∃y)(Iy • Jxy)] 2. Ha 3. Ib / Jab

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 sum of one and two is three.

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 -Some Russian philosopher other than Chloe is an altruist.

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 is more skeptical than all rationalist philosophers.

(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px] -Which of the following propositions is also derivable in F?

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 -Any son of Elizabeth or Charles is more exalted than anyone who is not Charles or Elizabeth.

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. (?x)(Ax • Bx • Cax) 2. (?x)[(Ax • Cax) ? x=d] / Bd

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 and his two sons are not the most exalted people in Britain.

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 -The Queen of England is Elizabeth.

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 -All lions and tigers are larger than some cheetahs, but not faster than all cheetahs.

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)(∃y)(Pxy • ∼Pyx) 2. (∀x)[(∃y)Pxy ⊃ (∃y)Qxy] / (∀x)(∃y)(Qxy • ∼Pyx)

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (?x)[Lx ? (?y)(My • Nxy)] 2) (?x)[Lx • (?y)Nxy] / (?x)[Mx • (?y)Nyx]