Exam 5: Full First-Order Logic

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 -No lion is faster than some tigers.

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 -The only son of Chloe and Eduardo is Bhavin.

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 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 -Any number is divisible by two if, and only if, the successor of its successor is divisible by two.

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 an existential quantifier.

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 -If the sum of the successors of any two numbers is divisible by three then the sum of two and the sum of those two numbers is divisible by three.

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

The virtue ethicist who teaches Aristotle does not publish on moral theory. Smith is a virtue ethicist who teaches Aristotle. Either one publishes on moral theory or one publishes on applied ethics. So, Smith publishes on applied ethics. (s: Smith; Ax: x publishes on applied ethics; Mx: x publishes on moral theory; Tx: is teaches Aristotle; Vx: x is a virtue ethicist) -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: 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 -Megha's only brother is Bhavin.

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

1. (∀x)(∀y)(∃z)Sf(x)yz 2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz $\lor$ Mzyx)] -Which of the following propositions is derivable from the given premises in FF?

1. (∀x)(∀y)[f(x)=y ⊃ (Pax • Qay)] 2. ∼Pab -Which of the following propositions is derivable from the given premises in FF?

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

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 -If some rationalist is more skeptical than Locke, then no philosopher influenced Hume.

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (?x)[Tx ? (?y)(Sy • Wxy)] 2) (?x)(Sx • Vx) 3) (?x)(Tx • Rx) / (?x)[(Rx • Vx) • (?y)Wxy]

1. (∀x)[(Px $\lor$ Qx)] ⊃ Rxx] 2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]} 3. Pn • Qn -Which of the following propositions is derivable from the given premises in F?

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)(∀y)(Bxy ≡ Dyx) 2. (∃x)Bxf(x) -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 -Which of the following propositions is an immediate (one-step) consequence in FF 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 -Bhavin's maternal grandmother loves Ahmed, Irving, and Eduardo.