Question 1

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.

Accepted Answer

The answer of Translate each sentence into predicate logic, using...

Question 2

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.

Accepted Answer

The answer of use the following translation key to write...

Question 3

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
{, , , , , , , ... }
-Given the customary truth tables, which of the following theories is modeled by the above interpretation?

Accepted Answer

A)  (∃x)(∃y)[(Ex • Ey) • Sfxy] (∃x)(∃y)[(Ox • Oy) • Sfxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] $ \lor $ (∃y)(∃z)[(Ox • Oy) • Sxyz]}} 
B)  (∃x)(∃y)[(Ex • Ey) • Sdxy]
(∃x)(∃y)[(Ox • Oy) • Sdxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] $ \lor $ (∃y)(∃z)[(Ox • Oy) • Sxyz]}} 
C)  (∃x)(∃y)[(Ex • Ey) • Sdxy]
(∃x)(∃y)[(Ox • Oy) • Sdxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] • (∃y)(∃z)[(Ox • Oy) • Sxyz]}} 
D)  (∃x)(∃y)[(Ex • Ey) • Sfxy]
(∃x)(∃y)[(Ox • Oy) • Sfxy]
(∀x){Ex ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] • (∃y)(∃z)[(Ox • Oy) • Sxyz]}} 
E)  (∃x)(∃y)[(Ex • Ey) • Sdxy]
(∃x)(∃y)[(Ox • Oy) • Sdxy]
(∀x){Ox ⊃ {(∃y)(∃z)[(Ey • Ez) • Sxyz)] $ \lor $ (∃y)(∃z)[(Ox • Oy) • Sxyz]}}

Question 4

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.

Accepted Answer

The answer of use the following translation key to write...

Question 5

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.

Accepted Answer

The answer of construct theories for which the following interpretation...

Question 6

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.

Accepted Answer

The answer of use the following translation key to write...

Question 7

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.

Accepted Answer

The answer of use the following translation key to write...

Question 8

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?

Accepted Answer

A)  Vs • Ts • (∀y)[(Vy • Ty) ⊃ y=s] • ~Ms 
B)  Ms ≡ ~As 
C)  Ts • ~Ms 
D)  Vx • Tx 
E)  Ve • Te • (∀y)[(Vy • Ty) ⊃ y=e] • ∼Me

Question 9

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.

Accepted Answer

A)  (∃x)(Bxm • x=b • ~(∃y)Bmy) 
B)  (∀x)(∀y)[(Bxm • Bym) ⊃ x=b] 
C)  Bbm • (∀x)(Bxm ⊃ x=b) 
D)  ~(∃x)(Bxm • x≠b) 
E)  Bbm • (∃x)(Bxm • x=b)

Question 10

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

Accepted Answer

A)  (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [(∀x)Px $ \lor $ ~(∃x)(∃y)Qxy] 
B)  (∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∀x)∼Px $ \lor $ (∃x)(∃y)Qxy] 
C)  (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [~(∀x)Px $ \lor $ ~(∃x)(∃y)Qxy] 
D)  (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ ~[(∀x)Px $ \lor $ ~(∃x)(∃y)Qxy] 
E)  (∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∀x)∼Px $ \lor $ (∃x)(∃y)~Qxy]

Question 11

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?

Accepted Answer

A)  (∃x)(∃y)(∃z)∼Cxf(y)g(f(y),z) 
B)  (∃x)(∃y)(∃z)∼Cxf(x)f(f(x)) 
C)  (∃x)(∃y)(∃z)∼Mf(x)yz 
D)  (∃x)(∃y)(∃z)∼Mzg(y)f(g(x)) 
E)  (∃x)(∃y)(∃z)∼Mzg(x)f(g(x))

Question 12

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

Accepted Answer

A)  Qax 
B)  Qay 
C)  (∀x)Qax 
D)  (∃x)f(b)=x 
E)  ∼(∃x)f(b)=x

Question 13

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.

Accepted Answer

The answer of use:
d: Diego 
s: Sean
Dx: x is on...

Question 14

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.

Accepted Answer

The answer of Translate each sentence into predicate logic, using...

Question 15

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]

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample in a domain of one member, in which: Ta: True Sa: False Waa: True
Ra: False Va: True 
C)  Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{lll}
\text { Ta: False } & \text { Sa: False } & \text { Waa: True } \
\text { Tb: True } & \text { Sb: True } & \text { Wab: False } \
\text { Ra: True } & \text { Va: False } & \text { Wba: False } \
\text { Rb: False } & \text { Vb: False } & \text { Wbb:True }
\end{array}$ 
D)  Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{lll}
\text { Ta: False } & \text { Sa: True } & \text { Waa: True } \
\text { Tb: True } & \text { Sb: True } & \text { Wab: True } \
\text { Ra: False } & \text { Va: False } & \text { Wba: False } \
\text { Rb: False } & \text { Vb:True } & \text { Wbb: False }
\end{array}$ 
E)  Invalid. Counterexample in a domain of two members, in which:
$\begin{array}{llll}
\text { Ta: True } & \text { Sa: True } & \text { Waa: True } & \
 \text { Tb: False } & \text { Sb: True } & \text { Wab: True } \
 \text { Ra: True } & \text { Va: False } & \text { Wba: True } \
\text { Rb: False } & \text { Vb:True } & \text { Wbb: True }
\end{array}$

Question 16

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?

Accepted Answer

A)  Rnn • Snn 
B)  (∀x)Rnx 
C)  (∀x)Rxn 
D)  (∀x)(Px ⊃ Sxx) 
E)  Rnn ≡ ~Snn

Question 17

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?

Accepted Answer

A)  Ma • ~Mc
(∃x)Cxc
∼(∃x)(∼Cxc • Mx) 
B)  Ma • Mc (∃x)Ccx (∃x)(~Cxc • Mx) 
C)  ~(Ma • Mc)
(∃x)Ccx
~(∃x)(Cxc • Mx) 
D)  ~(Ma $ \lor $ Mc) (∃x)Ccx
~(∃x)(Ccx • Mx) 
E)  Ma ≡ ~Mc
(∃x)Cax
~(∃x)(Cxc • ~Mx)

Question 18

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?

Accepted Answer

A)  (∃x)Baa 
B)  (∃x)Bf(x)f(x) 
C)  Baa 
D)  Bf(x)f(x) 
E)  Baf(a)

Question 19

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?

Accepted Answer

A)  f(o,o)=o 
B)  f(o)=o 
C)  f(o,x)=o 
D)  f(o,x)=x 
E)  f(x,o)=x

Question 20

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.

Accepted Answer

A)  Lg(g(b))aie 
B)  Lg(g(b))a • i • e 
C)  Lg(g(b))a • Lg(g(b))i • Lg(g(b))e 
D)  Lg(a) • Lg(i) • Lg(e) 
E)  Lg(a • i • e)

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.

(∀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?

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?

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.

(∀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?

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?

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) [(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?