3.1.xa. Exercise answers

1. a.

The soup was hot but not too hot ∧ the soup was thick but not too thick

(the soup was hot ∧ the soup was not too hot) ∧ (the soup was thick ∧ the soup was not too thick)

(the soup was hot ∧ ¬ the soup was too hot) ∧ (the soup was thick ∧ ¬ the soup was too thick)

(H ∧ ¬ T) ∧ (K ∧ ¬ O)
both both H and not T and both K and not O
H: the soup was hot; K: the soup was thick; O: the soup was too thick; T: the soup was too hot
  b.

The equipment isn’t here ∧ the equipment is unlikely to arrive soon

¬ the equipment is here ∧ ¬ the equipment is likely to arrive soon

¬ H ∧ ¬ S
both not H and not S
H: the equipment is here; S: the equipment is likely to arrive soon
  c.

No one answered the phone ∧ the phone rang 10 times

¬ someone answered the phone ∧ the phone rang 10 times

¬ A ∧ R
both not A and R
A: someone answered the phone; R: the phone rang 10 times
  d.

The alarm must have gone off ∧ Ted didn’t hear anything

The alarm must have gone off ∧ ¬ Ted heard something

A ∧ ¬ H
both A and not H
A: the alarm must have gone off; H: Ted heard something
  e.

¬ they will both meet the deadline and stay within the budget

¬ (they will meet the deadline ∧ they will stay within the budget)

¬ (D ∧ B)
not both D and B
B: they will stay within the budget; D: they will meet the deadline
  f.

They won’t meet the deadline ∧ they will stay within the budget

¬ they will meet the deadline ∧ they will stay within the budget

¬ D ∧ B
both not D and B
B: they will stay within the budget; D: they will meet the deadline
  g.

They won’t meet the deadline ∧ they won’t stay within the budget

¬ they will meet the deadline ∧ ¬ they will stay within the budget

¬ D ∧ ¬ B
both not D and not B
B: they will stay within the budget; D: they will meet the deadline
  h.

Tod shut off the alarm ∧ ¬ Tod woke up

A ∧ ¬ W
both A and not W
A: Tod shut off the alarm; W: Tod woke up
  i.

¬ they will meet the deadline without going over the budget

¬ (they will meet the deadline ∧ ¬ they will go over the budget)

¬ (D ∧ ¬ G)
not both D and not G
D: they will meet the deadline; G: they will go over the budget
  j.

Larry joined in ∧ Larry did not join in without being coaxed

Larry joined in ∧ ¬ Larry joined in without being coaxed

Larry joined in ∧ ¬ (Larry joined in ∧ ¬ Larry was coaxed)

J ∧ ¬ (J ∧ ¬ C)
both J and not both J and not C
C: Larry was coaxed; J: Larry joined in

This is equivalent to J ∧ ¬ ¬ C and also to J ∧ C, but the analysis shown is closer to the form of the English.

  k.

Ann liked the movie ∧ neither Bill nor Carol liked the movie

Ann liked the movie ∧ (¬ Bill liked the movie ∧ ¬ Carol liked the movie)

A ∧ (¬ B ∧ ¬ C)
both A and both not B and not C
A: Ann liked the movie; B: Bill liked the movie; C: Carol liked the movie

The alternative (and equivalent) analysis as A ∧ ¬ E (where E: either Bill or Carol liked the movie) is closer to the English but it is less satisfactory because it displays less structure. The next chapter will give us the means carry this sort of analysis further by analyzing E as a compound of B and C.

2. a.
not not both A and B
 
 
  b.
not both not A and B
 
 
  c.
¬ A ∧ ( ¬ B ∧ C)
   
 
  d.
¬ ( A ∧ B ) ∧ ¬ C
   
 
3. a.

It was cold ∧ ¬ there was frost

It was cold ∧ there was no frost

It was cold, but there was no frost

  b.

¬ someone saw the accident ∧ (Sue heard a crash ∧ Sue went to investigate)

No one saw the accident ∧ Sue heard a crash and went to investigate

No one saw the accident, but Sue heard a crash and went to investigate

  c.

(it was a design ∧ it was new) ∧ ¬ it pleased someone

It was a new design ∧ it pleased no one

It was a new design, and it pleased no one

  d.

¬ (we’ll win in Iowa ∧ we’ll win in New York)

¬ (we’ll win in both Iowa and New York)

We won’t win in both Iowa and New York

  e.

¬ we’ll win in Iowa ∧ we’ll win in New York

We won’t win in Iowa ∧ we’ll win in New York

We won’t win in Iowa, but we’ll win in New York

  f.

¬ (we’ll win in Iowa ∧ ¬ we’ll lose in New York)

¬ (we’ll win in Iowa without losing in New York)

We won’t win in Iowa without losing in New York

4. Numbers below the tables indicate the order in which values were computed.
  a.
A B C A ∧ ¬ (B ∧ C)
T F F  ⓉT F
       32 1
  b.
A B C A ∧ (¬ B ∧ C)
T F F Ⓕ T F
       31 2
[Note that, while in a, it is the value under the ¬ that is used in calculating the value of the main conjunction, in b it is the value under the second ∧; this is due to the change in relative scope of these two connectives.]
  c.
A B C D (¬ A ∧ ¬ B) ∧ (¬ (A ∧ C) ∧ D)
F T T T T F F Ⓕ T F T
        1 2 1 4 2 1 3
Glen Helman 16 Sep 2009