Exam 5: Full First-Order Logic

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 • Gxx)

1. (∃x)[Px • (∀y)(Py ⊃ y=x) • (∃y)(Qy • Rxy)] 2. (∃x)[Px • Sx • (∀y)∼Ryx] -Which of the following propositions is derivable from the given premises in F?

construct a derivation of each logical truth of identity theory. -(?x)(?y)(Axy • x=y) ? (?x)Axx

use the following translation key to write the sentences below in FF. 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.

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

select the best translation into predicate logic, using the following translation key: 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 philosophers mock Plato and Nietzsche.

1. (∃x){Ex • Fx • (∀y)[(Ey • y≠x) ⊃ Gxy]} 2. En • ∼Fn -Which of the following propositions is derivable from the given premises in F?

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. Ha • (∀x)[(Hx • x≠a) ⊃ Jax] 2. Hb • Ib • Mb • (∀x)[(Ix • Mx) ⊃ x=b] 3. ∼Ma / Jab

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} -Ma • Mc (∃x)Cxc ∼(∃x)(∼Cxc • Mx)

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

construct theories for which the following interpretation is a model (i.e. construct at least two sentences which are true under the given interpretation). Domain = {1, 2, 3, ..., 28, 29, 30} a = 1 e = 21 b = 2 f = 23 c = 4 g = 27 d = 19 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>, ... } -Construct a theory of at least two sentences, at least one of which uses a universal quantifier.

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 Locke.

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. (∃x)[Px • (∀y)(Py ⊃ y=x) • (∃y)(Qy • Rxy)] 2. (∃x)[Px • Sx • (∀y)∼Ryx] / (∃x)(∃y)(Rxy • ∼Ryx)

1. Ha • Ia 2. (∀x)[(Hx • Ix) ⊃ x=a] 3. Hb • Jb 4. (∀x)[(Hx • Jx) ⊃ x=b] 5. Ka • ∼Kb -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 -Some rationalists influenced some philosophers who were more skeptical than them.

translate the given paragraphs into arguments written in F, using the given translation key. Then, derive their conclusions using the rules of inference for -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)

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 has exactly three sons.

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 -Chloe is Irving's sister; they share a father.

1. (∃x){Ex • Fx • (∀y)[(Ey • y≠x) ⊃ Gxy]} 2. En • ∼Fn -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?