7.3.xa. Exercise answers

1. a.

Not everyone was enthusiastic but no one was disappointed

Not everyone was enthusiastic ∧ no one was disappointed

¬ everyone was enthusiastic ∧ (∀x: x is a person) ¬ x was disappointed

¬ (∀x: x is a person) x was enthusiastic ∧ (∀x: x is a person) ¬ x was disappointed

¬ (∀x: Px) Ex ∧ (∀x: Px) ¬ Dx
¬ ∀x (Px → Ex) ∧ ∀x (Px → ¬ Dx)

[D: λx (x was disappointed); E: λx (x was enthusiastic); P: λx (x is a person)]
  b.

Any defective unit will be repaired or replaced

(∀x: x is a defective unit) x will be repaired or replaced

(∀x: x is a unit ∧ x is defective) (x will be repaired ∨ x will be replaced)

(∀x: Ux ∧ Dx) (Px ∨ Lx)
∀x ((Ux ∧ Dx) → (Px ∨ Lx))

[D: λx (x is defective); L: λx (x will be replaced); P: λx (x will be repaired); U: λx (x is a unit)]
  c.

The bill will pass quickly if every member of the committee supports it

The bill will pass quickly ← every member of the committee will support the bill

Pb ← (∀x: x is a member of the committee) x will support the bill

Pb ← (∀x: Mxc) Sxb
(∀x: Mxc) Sxb → Pb
∀x (Mxc → Sxb) → Pb

[M: λxy (x is a member of y); P: λx (x will pass quickly); S: λxy (x will support y); b: the bill; c: the committee]
  d.

Nothing suited both Ann and Bill.

∀x ¬ x suited both Ann and Bill

∀x ¬ (x suited Ann ∧ x suited Bill)

∀x ¬ (Sxa ∧ Sxb)

[S: λxy (x suited y); a: Ann; b: Bill]
  e.

Tom didn’t sign up anyone; however, he didn’t contact everyone

Tom didn’t sign up anyone ∧ Tom didn’t contact everyone

everyone is such that (Tom didn’t sign up him or her) ∧ ¬ Tom contacted everyone

(∀x: x is a person) ¬ Tom signed up x ∧ ¬ (∀x: x is a person) Tom contacted x

(∀x: Px) ¬ Stx ∧ ¬ (∀x: Px) Ctx
∀x (Px → ¬ Stx) ∧ ¬ ∀x (Px → Ctx)

[C: λxy (x contacted y); P: λx (x is a person); S: λxy (x signed up y)]
  f.

If a bill arrives, it will be forwarded to you

Every bill is such that (if it arrives, it will be forwarded to you)

(∀x: x is a bill) (if x arrives, x will be forwarded to you)

(∀x: Bx) (x will arrive → x will be forwarded to you)

(∀x: Bx) (Ax → Fxo)
∀x (Bx → (Ax → Fxo))

[A: λx (x will arrive); B: λx (x is a bill); F: λxy (x will be forwarded to y); o: you]
  g.

If the prize isn’t won by anyone, it will be added to the next drawing

the prize won’t be won by anyone → the prize will be added to the next drawing

everyone is such that (the prize won’t be won by him or her) → Apn

(∀x: x is a person) the prize won’t be won by x → Apn

(∀x: Px) ¬ the prize will be won by x → Apn

(∀x: Px) ¬ Wpx → Apn
∀x (Px → ¬ Wpx) → Apn

[A: λxy (x will be added to y); P: λx (x is a person); W: λxy (x will be won by y); n: the next drawing; p: the prize]
  h.

Ralph looked in every closet and cabinet

Ralph looked in every closet ∧ Ralph looked in every cabinet

(∀x: x is a closet) Ralph looked in x ∧ (∀x: x is a cabinet) Ralph looked in x

(∀x: Sx) Lrx ∧ (∀x: Bx) Lrx
∀x (Sx → Lrx) ∧ ∀x (Bx → Lrx)
or: (∀x: Sx ∨ Bx) Lrx

[B: λx (x is a cabinet); L: λxy (x looked in y); S: λx (x is a closet); r: Ralph]
  i.

The alarm will sound if anyone who doesn’t have the combination tries to open the door

everyone who doesn’t have the combination is such that (the alarm will sound if he or she tries to open the door)

(∀x: x is a person who doesn’t have the combination) the alarm will sound if x tries to open the door

(∀x: x is a person ∧ x doesn’t have the combination) (the alarm will sound ← x will try to open the door)

(∀x: x is a person ∧ ¬ x has the combination) (Sa ← Txd)

(∀x: Px ∧ ¬ Hxc) (Sa ← Txd)
(∀x: Px ∧ ¬ Hxc) (Txd → Sa)
∀x ((Px ∧ ¬ Hxc) → (Txd → Sa))

[H: λxy (x has y); P: λx (x is a person); S: λx (x will sound); T: λxy (x will try to open y); a: the alarm; c: the combination; d: the door]
2. a.

¬ (∀x: x glitters) x is gold

¬ everything that glitters is gold

Not everything that glitters is gold
or: All that glitters is not gold
[However, negating the main the verb is not always the clearest way of denying a generalization; for example, Everyone was not in the best of moods could be understood either as saying that not everyone was in the best of moods or as saying that no one was.]

Note also that we here treat the restricting predicate x glitters as if it were x is a thing that glitters; this sort of use of the class indicator thing is always possible when the restricting predicate does not already provide a common noun.
  b.

(∀x: x is a dog ∧ x was in the cage) x barked ∧ (∀x: x is a dog ∧ x was in the cage) x wagged x’s tail

(∀x: x is a dog that was in the cage) x barked ∧ (∀x: x is a dog that was in the cage) x wagged x’s tail

Every dog that was in the cage barked ∧ every dog that was in the cage wagged it’s tail

Every dog in the cage barked, and each wagged it’s tail
or: Every dog in the cage barked and wagged it’s tail
[However, the latter sentence would be more naturally analyzed as having the form (∀x: Dx ∧ Nxc) (Bx ∧ Wx).]

  c.

∀x ¬ Tom let x stop Tom

∀x Tom didn’t let x stop him

Tom didn’t let anything stop him
or: Tom let nothing stop him

  d.

(∀x: x is a federal project ∧ ¬ x is a road) ¬ x is finished

(∀x: x is a federal project that is not a road) x is unfinished

Every federal project that is not a road is unfinished
or: No federal projects except roads are finished
[The latter approach—capturing the negation by a negative generalization—helps to avoid ambiguity in cases were an explicit negation would have to apply to the main verb, as in (∀x: x is a federal project ∧ ¬ x is a road) ¬ x is under way]

  e.

∀x (x was left on the roof → x is gone for good)

∀x (if x was left on the roof then x is gone for good)

If anything was left on the roof then it is gone for good
or: Anything that was left on the roof is gone for good

  f.

(∀x: x is a person ∧ Tom met x) (Tom knew x ∨ x knew Tom)

(∀x: x is a person Tom met) Tom knew or was known by x

Tom knew or was known by everyone he met

Glen Helman 25 Aug 2005