5.2.xa. Exercise answers

1. a. Tom was late  ← ¬ Tom left early
L ← ¬ E
¬ E → L
if not E then L
E: Tom left early; L: Tom was late
  b. ¬ you’ll get a good picture  ← ¬ you’ll take the cap off the lens
¬ P ← ¬ C
¬ C → ¬ P
if not C then not P
C: you’ll take the cap off the lens; P: you’ll get a good picture
  c.

Neither Ann nor Bill knew of it  ← ¬ Ann and Bill both knew of it

¬ (Ann knew of it  ∨  Bill knew of it) ← ¬ (Ann knew of it  ∧  Bill knew of it)

¬ (A ∨ B) ← ¬ (A ∧ B)
¬ (A ∧ B) → ¬ (A ∨ B)
if not both A and B then not either A or B
A: Ann knew of it; B: Bill knew of it
  d.

The bill will pass if the chairman supports it  ← ¬ public opinion will run heavily against the bill

(the bill will pass  ←  the chairman will support the bill) ← ¬ A

(P ← S) ← ¬ A
¬ A → (S → P)
if not A then if S then P
A: public opinion will run heavily against the bill; P: the bill will pass; S: the chairman will support the bill
  e.

¬ Ed will be late  →  we’ll get started on time and finish early if there isn’t a lot of business

¬ L → (we’ll get started on time  ∧  we’ll finish early if there isn’t a lot of business)

¬ L → (T ∧ (we’ll finish early  ←  there won’t be a lot of business))

¬ L → (T ∧ (F ← ¬ there will be a lot of business))

¬ L → (T ∧ (F ← ¬ B))
¬ L → (T ∧ (¬ B → F))
if not L then both T and if not B then F
B: there will be a lot of business; F: we’ll finish early; L: Ed will be late; T: we’ll get started on time

It would be possible to understand the sentence to make the whole of we’ll get started on time and finish early conditional on there won’t be a lot of business. On that interpretation, the form would be ¬ L → (¬ B → (T ∧ F)). However, the interpretation used above fits better with common sense expectations concerning the content, and those are often the grounds on which ambiguous sentences are understood in a particular way.

  f.

Bob was under no obligation to help  →  Bob worked only if he was in a good mood and had nothing to do

¬ Bob was under an obligation to help  → (¬ Bob worked  ← ¬ Bob was in a good mood and had nothing to do)

¬ O→ (¬ W ← ¬ (Bob was in a good mood  ∧  Bob had nothing to do))

¬ O→ (¬ W ← ¬ (G ∧ ¬ Bob had something to do))

¬ O→ (¬ W ← ¬ (G ∧ ¬ S))
¬ O→ (¬ (G ∧ ¬ S) → ¬ W)
if not O then if not both G and not S then not W
O: Bob was under an obligation to help; G: Bob was in a good mood; S: Bob had something to do; W: Bob worked
2. a. ¬ I’ll see it  → ¬ I’ll believe it

Unless I see it, I won’t believe it

or: I’ll believe it only if I see it

  b.

¬ the set works  ← ¬ (the set is plugged in  ∧ ¬ the set is broken)

¬ the set works  ← ¬ (the set is plugged in  ∧  the set isn’t broken)

¬ the set works  ← ¬ (the set is plugged in and isn’t broken)

The set works only if it is plugged in and isn’t broken

or: The set doesn’t work unless it is plugged in and isn’t broken

  c.

¬ (Adams will back out  ∨  Brown will back out) → (the deal will go through  ← ¬ (Collins will have trouble with financing  ∨  Davis will have trouble with financing))

¬ Adams or Brown will back out  → (the deal will go through  ← ¬ Collins or Davis will have trouble with financing)

Unless Adams or Brown backs out, the deal will go through if neither Collins nor Davis has trouble with financing

or: If neither Adams nor Brown backs out, the deal will go through unless Collins or Davis has trouble with financing

or: Unless Adams nor Brown backs out, the deal will go through unless Collins or Davis has trouble with financing

Glen Helman 09 Oct 2009