Exam 5: Full First-Order Logic

1. (∀x)(∀y)f(x,y)=f(y,x) 2. (∀x)f(x,o)=o -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: 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 that are not larger than any lion are faster than some tigers.

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?

1. (∀x){Ax ⊃ [Bx • Bf(x)]} 2. ∼Bf(f(e)) -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. (∃x)(∃y){Gx • Gy • x≠y • (∀z)[Gz ⊃ (z=x $\lor$ z=y)]} 2. Ga • Gb • Gc / a=b $\lor$ a=c $\lor$ b=c

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 philosophers more skeptical than Hume influenced all philosophers more skeptical than Locke.

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 tigers and all cheetahs are faster than all lions.

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 philosophy major on the Dean's list is a valedictorian.

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 -All philosophy majors except Sean are on the Dean's list.

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 -Any lion larger than all cheetahs is not faster than any tiger.

1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy 2. (∃x)(∀y)∼Bxy -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: 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 -Delfina loves both her mother and her father.

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 • Ox) ⊃ (∃y)[(Ey • Ny) • Gyx)]}

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 -The Queen of England is Elizabeth.

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 -Any lion larger than all cheetahs is not faster than any tiger.

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 three sentences, at least one of which uses an existential quantifier and at least one of which uses a universal quantifier.

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 derivable from the given premises in F?

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 an immediate (one-step) consequence in F of the given premises?

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 -There is at most one queen in Britain.

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