Exam 5: Full First-Order Logic

(∃x)[Px • (∃y)(Qy • y=x)] ⊃ (∃x)(Px • Qx) -Which of the following propositions is also derivable in F?

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 -Everything is joyful, except Megha and Bhavin.

1. (?x)(?y){Pf(x,y) ? [Pf(x,x) • Pf(y,y)]} 2. a=f(d,b) 3. Pa / Pf(b,b)

1. (∀x)[Ex ⊃ (∀y)(Fy • Gxy)] 2. (∃x)(Ex • Hxb) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

(∃x)[Px • (∃y)(Qy • y=x)] ⊃ (∃x)(Px • Qx) -Consider assuming '(∃x)[Px • (∃y)(Qy • y=x)]' for a conditional proof of the above logical truth. Which Of the following propositions is a legitimate second step in that proof?

1. (∃x)(∃y){Gx • Gy • x≠y • (∀z)[Gz ⊃ (z=x $\lor$ z=y)]} 2. Ga • Gb • Gc -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

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 -Locke influenced Hume, but is not more skeptical than him.

1. (?x)(?y)(?z)[(Pxy • Pyz) ? Pxz] 2. (?x)Pxf(x) / (?x)Pxf(f(x))

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 -If a number is divisible by three, then the sum of its successor with the successor of one is also divisible by three.

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

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?

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)[Hx ⊃ (∃y)(Ix • Jxy)] 2) Ha 3) Ib / Jab

(∃x)(∃y)(Axy • x=y) ⊃ (∃x)Axx -Which of the following propositions is also derivable in F?

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)] 2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)] -Which of the following propositions is derivable from the given premises in FF?

All philosophers respect each other. Some philosopher doesn't study some philosopher. Anything which respects something without studying it is open-minded, if ignorant. So something is open-minded and ignorant. (Ix: x is ignorant; Ox: x is open-minded; Px: x is a philosoper; Rxy: x respects y; Sxy: x studies y) -Which of the following claims can also be derived from the premises of this argument?

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 -All philosophers are more skeptical than some rationalists.

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 -The successor of a number is greater than itself.

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) • (Pc • Pd) (∃x)Cxd • ∼(∃x)Cxa

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 tigers and all cheetahs are faster than all lions.

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)[(∃y)(Fxy • Gy) ⊃ (∀y)(Gy ⊃ Fxy)] 2. Ga / (∀x)(Fxa ⊃ Fax)