Solutions to Chapter 5 Connectives

A. Using the symbolization key given, symbolize each English sentence in TFL.

$M$ :: Those creatures are men in suits
$C$ :: Those creatures are chimpanzees
$G$ :: Those creatures are gorillas

1.

Those creatures are not men in suits.
$\neg M$
2.

Those creatures are men in suits, or they are not.
$(M\vee\neg M$ )
3.

Those creatures are either gorillas or chimpanzees.
$(G\vee C)$
4.

Those creatures are neither gorillas nor chimpanzees.
$\neg(C\vee G)$
5.

If those creatures are chimpanzees, then they are neither gorillas nor men in suits.
$(C\rightarrow\neg(G\vee M))$
6.

Unless those creatures are men in suits, they are either chimpanzees or they are gorillas.
$(M\vee(C\vee G))$

B. Using the symbolization key given, symbolize each English sentence in TFL.

$A$ :: Mister Ace was murdered
$B$ :: The butler did it
$C$ :: The cook did it
$D$ :: The Duchess is lying
$E$ :: Mister Edge was murdered
$F$ :: The murder weapon was a frying pan

1.

Either Mister Ace or Mister Edge was murdered.
$(A\vee E)$
2.

If Mister Ace was murdered, then the cook did it.
$(A\rightarrow C)$
3.

If Mister Edge was murdered, then the cook did not do it.
$(E\rightarrow\neg C)$
4.

Either the butler did it, or the Duchess is lying.
$(B\vee D)$
5.

The cook did it only if the Duchess is lying.
$(C\rightarrow D)$
6.

If the murder weapon was a frying pan, then the culprit must have been the cook.
$(F\rightarrow C)$
7.

If the murder weapon was not a frying pan, then the culprit was either the cook or the butler.
$(\neg F\rightarrow(C\vee B))$
8.

Mister Ace was murdered if and only if Mister Edge was not murdered.
$(A\leftrightarrow\neg E)$
9.

The Duchess is lying, unless it was Mister Edge who was murdered.
$(D\vee E)$
10.

If Mister Ace was murdered, he was done in with a frying pan.
$(A\rightarrow F)$
11.

Since the cook did it, the butler did not.
$(C\wedge\neg B)$
12.

Of course the Duchess is lying!
$D$

C. Using the symbolization key given, symbolize each English sentence in TFL.

$E_{1}$ :: Ava is an electrician
$E_{2}$ :: Harrison is an electrician
$F_{1}$ :: Ava is a firefighter
$F_{2}$ :: Harrison is a firefighter
$S_{1}$ :: Ava is satisfied with her career
$S_{2}$ :: Harrison is satisfied with his career

1.

Ava and Harrison are both electricians.
$(E_{1}\wedge E_{2})$
2.

If Ava is a firefighter, then she is satisfied with her career.
$(F_{1}\rightarrow S_{1})$
3.

Ava is a firefighter, unless she is an electrician.
$(F_{1}\vee E_{1})$
4.

Harrison is an unsatisfied electrician.
$(E_{2}\wedge\neg S_{2})$
5.

Neither Ava nor Harrison is an electrician.
$\neg(E_{1}\vee E_{2})$
6.

Both Ava and Harrison are electricians, but neither of them find it satisfying.
$((E_{1}\wedge E_{2})\wedge\neg(S_{1}\vee S_{2}))$
7.

Harrison is satisfied only if he is a firefighter.
$(S_{2}\rightarrow F_{2})$
8.

If Ava is not an electrician, then neither is Harrison, but if she is, then he is too.
$((\neg E_{1}\rightarrow\neg E_{2})\wedge(E_{1}\rightarrow E_{2}))$
9.

Ava is satisfied with her career if and only if Harrison is not satisfied with his.
$(S_{1}\leftrightarrow\neg S_{2})$
10.

If Harrison is both an electrician and a firefighter, then he must be satisfied with his work.
$((E_{2}\wedge F_{2})\rightarrow S_{2})$
11.

It cannot be that Harrison is both an electrician and a firefighter.
$\neg(E_{2}\wedge F_{2})$
12.

Harrison and Ava are both firefighters if and only if neither of them is an electrician.
$((F_{2}\wedge F_{1})\leftrightarrow\neg(E_{2}\vee E_{1}))$

D. Using the symbolization key given, translate each English-language sentence into TFL.

$J_{1}$ :: John Coltrane played tenor sax
$J_{2}$ :: John Coltrane played soprano sax
$J_{3}$ :: John Coltrane played tuba
$M_{1}$ :: Miles Davis played trumpet
$M_{2}$ :: Miles Davis played tuba

1.

John Coltrane played tenor and soprano sax.
$J_{1}\wedge J_{2}$
2.

Neither Miles Davis nor John Coltrane played tuba.
$\neg(M_{2}\vee J_{3})$ or $\neg M_{2}\wedge\neg J_{3}$
3.

John Coltrane did not play both tenor sax and tuba.
$\neg(J_{1}\wedge J_{3})$ or $\neg J_{1}\vee\neg J_{3}$
4.

John Coltrane did not play tenor sax unless he also played soprano sax.
$\neg J_{1}\vee J_{2}$
5.

John Coltrane did not play tuba, but Miles Davis did.
$\neg J_{3}\wedge M_{2}$
6.

Miles Davis played trumpet only if he also played tuba.
$M_{1}\rightarrow M_{2}$
7.

If Miles Davis played trumpet, then John Coltrane played at least one of these three instruments: tenor sax, soprano sax, or tuba.
$M_{1}\rightarrow(J_{1}\vee(J_{2}\vee J_{3}))$
8.

If John Coltrane played tuba then Miles Davis played neither trumpet nor tuba.
$J_{3}\rightarrow\neg(M_{1}\vee M_{2})$ or $J_{3}\rightarrow(\neg M_{1}\wedge\neg M_{2})$
9.

Miles Davis and John Coltrane both played tuba if and only if Coltrane did not play tenor sax and Miles Davis did not play trumpet.
$(J_{3}\wedge M_{2})\leftrightarrow(\neg J_{1}\wedge\neg M_{1})$ or $(J_{3}\wedge M_{2})\leftrightarrow\neg(J_{1}\vee M_{1})$

E. Give a symbolization key and symbolize the following English sentences in TFL.

$A$ :: Alice is a spy
$B$ :: Bob is a spy
$C$ :: The code has been broken
$G$ :: The German embassy will be in an uproar

1.

Alice and Bob are both spies.
$(A\wedge B)$
2.

If either Alice or Bob is a spy, then the code has been broken.
$((A\vee B)\rightarrow C)$
3.

If neither Alice nor Bob is a spy, then the code remains unbroken.
$(\neg(A\vee B)\rightarrow\neg C)$
4.

The German embassy will be in an uproar, unless someone has broken the code.
$(G\vee C)$
5.

Either the code has been broken or it has not, but the German embassy will be in an uproar regardless.
$((C\vee\neg C)\wedge G)$
6.

Either Alice or Bob is a spy, but not both.
$((A\vee B)\wedge\neg(A\wedge B))$

F. Give a symbolization key and symbolize the following English sentences in TFL.

$F$ :: There is food to be found in the pridelands
$R$ :: Rafiki will talk about squashed bananas
$A$ :: Simba is alive
$K$ :: Scar will remain as king

1.

If there is food to be found in the pridelands, then Rafiki will talk about squashed bananas.
$(F\rightarrow R)$
2.

Rafiki will talk about squashed bananas unless Simba is alive.
$(R\vee A)$
3.

Rafiki will either talk about squashed bananas or he won’t, but there is food to be found in the pridelands regardless.
$((R\vee\neg R)\wedge F)$
4.

Scar will remain as king if and only if there is food to be found in the pridelands.
$(K\leftrightarrow F)$
5.

If Simba is alive, then Scar will not remain as king.
$(A\rightarrow\neg K)$

G. For each argument, write a symbolization key and symbolize all of the sentences of the argument in TFL.

1.

If Dorothy plays the piano in the morning, then Roger wakes up cranky. Dorothy plays piano in the morning unless she is distracted. So if Roger does not wake up cranky, then Dorothy must be distracted.

$P$ :

Dorothy plays the Piano in the morning

$C$ :

Roger wakes up cranky

$D$ :

Dorothy is distracted
$(P\rightarrow C)$ , $(P\vee D)$ , $(\neg C\rightarrow D)$
2.

It will either rain or snow on Tuesday. If it rains, Neville will be sad. If it snows, Neville will be cold. Therefore, Neville will either be sad or cold on Tuesday.

$T_{1}$ :

It rains on Tuesday

$T_{2}$ :

It snows on Tuesday

$S$ :

Neville is sad on Tuesday

$C$ :

Neville is cold on Tuesday
$(T_{1}\vee T_{2})$ , $(T_{1}\rightarrow S)$ , $(T_{2}\rightarrow C)$ , $(S\vee C)$
3.

If Zoog remembered to do his chores, then things are clean but not neat. If he forgot, then things are neat but not clean. Therefore, things are either neat or clean; but not both.

$Z$ :

Zoog remembered to do his chores

$C$ :

Things are clean

$N$ :

Things are neat
$(Z\rightarrow(C\wedge\neg N))$ , $(\neg Z\rightarrow(N\wedge\neg C))$ , $((N\vee C)\wedge\neg(N\wedge C))$ .

H. For each argument, write a symbolization key and translate the argument as well as possible into TFL. The part of the passage in italics is there to provide context for the argument, and doesn’t need to be symbolized.

1.

It is going to rain soon. I know because my leg is hurting, and my leg hurts if it’s going to rain.
2.

Spider-man tries to figure out the bad guy’s plan. If Doctor Octopus gets the uranium, he will blackmail the city. I am certain of this because if Doctor Octopus gets the uranium, he can make a dirty bomb, and if he can make a dirty bomb, he will blackmail the city.
3.

A westerner tries to predict the policies of the Chinese government. If the Chinese government cannot solve the water shortages in Beijing, they will have to move the capital. They don’t want to move the capital. Therefore they must solve the water shortage. But the only way to solve the water shortage is to divert almost all the water from the Yangzi river northward. Therefore the Chinese government will go with the project to divert water from the south to the north.

I. We symbolized an exclusive or using ‘ $\vee$ ’, ‘ $\wedge$ ’, and ‘ $\neg$ ’. How could you symbolize an exclusive or using only two connectives? Is there any way to symbolize an exclusive or using only one connective?
For two connectives, we could offer any of the following:

$\neg(\mathscr{A}\leftrightarrow\mathscr{B})$

$(\neg\mathscr{A}\leftrightarrow\mathscr{B})$

$(\neg(\neg\mathscr{A}\wedge\neg\mathscr{B})\wedge\neg(\mathscr{A}\wedge% \mathscr{B}))$

But if we wanted to symbolize it using only one connective, we would have to introduce a new primitive connective.