Exam 5: Full First-Order Logic

Rey received the highest grade on Test #1. Spencer, who is not Rey, received a grade on Test #1. So, Rey's grade is higher than Spencer's. (r: Rey; s: Spencer; t: Test #1; Gxy: x is a grade on y; Hxy: x is higher than y; Rxy: x received y) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

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 -The smartest Russian mocks Chloe and Bhavin.

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 -If one thing is smarter than another, then the two things aren't the same.

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. Fab • (?x)(Fax ? x=b) 2. ?Fac / b?c

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 -Megha's only brother is Bhavin.

[(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa) -Which of the following propositions is also derivable in F?

1. (?x){Px ? (?y)[Py • f(x)=y]} 2. Pa • (?x)f(x)?a / (?x)[(?y)f(x)=y • ?(?y)f(y)=x]

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 the successor of one 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 -Ahmed is Chloe's father and Delfia is Chloe's mother.

construct a derivation of each logical truth of -(∃x)[Px • (∃y)(Py • Qxy)] ⊃ (∃x)(∃y)[(Px • Py) • Qyx]

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 lions are larger than all cheetahs.

1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy 2. (∃x)(∀y)∼Bxy -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: 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 -No number is greater than the sum of that number and two.

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 -All thoughtful philosophers except Nietzsche are altruists.

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 -The smartest Russian mocks Chloe and Bhavin.

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 lion is faster than all tigers if, and only if, some cheetah is larger than any tiger faster than it.

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 -If one thing is smarter than another, then the two things aren't the same.

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 -There are exactly two philosophy majors on the Dean's list.

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (∀x)[(Px $\lor$ Qx)] ⊃ Rxx] 2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]} 3. Pn • Qn / Rnn • Snn

provide a conterexample in a finite domain to each given invalid argument. -1. (∃x)(Dxa • Ex) 2) (∃x)(Dxa • Fx) / (∃x)(Ex • Fx)