Exam 5: Full First-Order Logic

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 -There is exactly one philosophy major on the Dean's list.

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 • (∀y)(Py ⊃ y=x) • (∃y)(Qy • Rxy)] 2. (∃x)[Px • Sx • (∀y)∼Ryx] -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: 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 -If anyone teaches the only son of Chloe and Eduardo, then someone teaches both Irving and Bhavin.

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. (∃x)(∃y)(Dx • Exb • Dy • Eyb • x≠y) 2. (∀x)(Dx ⊃ Ebx) / (∃x)(∃y)(Ebx • Exb • Eby • Eyb • x≠y)

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

provide a conterexample in a finite domain to each given invalid argument. -1. (?x)[Px ? (?y)Qxy] 2. (?x)[(?y)Qxy ? (?y)Rxy] 3. Pa / Rab

1. (∃x)(∃y)(Dx • Exb • Dy • Eyb • x≠y) 2. (∀x)(Dx ⊃ Ebx) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

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 -Of the philosophy majors, only Diego 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?

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 -No one in Britain is more exalted than Charles, except Elizabeth.

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){(Ex • Nx) ⊃ (∀y)[(Oy • Ny) ⊃ ∼Gxy]}

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

1. (∀x)(Ecx ⊃ x=d) 2. (∀x){(Fx • Gx) ⊃ (∀y)[(Fy • Gy) ⊃ y=x]} 3. (∃x)(Fx • Gx • Ecx) 4. Fa • Ga -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x)[(Cx • Exa) ⊃ Dx] 2. Cd • ∼Dd -Which of the following propositions is an immediate (one-step) consequence in F of the given premise?

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 -Bhavin's maternal grandmother loves Ahmed, Irving, and Eduardo.

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 -Ahmed is Chloe's father and Delfia is Chloe's mother.

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 -The only son of Chloe and Eduardo is Bhavin.

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} -Pa • Pb Cab • ∼Cba

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 -Elizabeth is the most exalted woman in Britain.