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
D: [ _ was disappointed]; E: _ was enthusiastic; P: [ _ is a person]
¬ ∀x (Px → Ex) ∧ ∀x (Px → ¬ Dx) |
| 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)
D: [ _ is defective]; L: _ will be replaced; P: [ _ will be repaired]; U: _ is a unit
∀x ((Ux ∧ Dx) → (Px ∨ Lx)) |
| 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
M: [ _ is a member of _ ]; P: _ will pass quickly; S: [ _ will support _ ]; b: the bill; c: the committee
(∀x: Mxc) Sxb → Pb ∀x (Mxc → Sxb) → Pb |
| 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: [ _ suited _ ]; 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
C: [ _ contacted _ ]; P: _ is a person; S: [ _ signed up _ ]
∀x (Px → ¬ Stx) ∧ ¬ ∀x (Px → Ctx) |
| 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)
A: [ _ will arrive]; B: _ is a bill; F: [ _ will be forwarded to _ ]; o: you
∀x (Bx → (Ax → Fxo)) |
| 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
A: [ _ will be added to _ ]; P: _ is a person; W: [ _ will be won by _ ]; n: the next drawing; p: the prize
∀x (Px → ¬ Wpx) → Apn |
| 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
B: [ _ is a cabinet]; L: _ looked in _ ; S: [ _ is a closet]; r: Ralph
∀x (Sx → Lrx) ∧ ∀x (Bx → Lrx) or: (∀x: Sx ∨ Bx) Lrx |
| 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)
H: [ _ has _ ]; P: _ is a person; S: [ _ will sound]; T: _ will try to open _ ; a: the alarm; c: the combination; d: the door
(∀x: Px ∧ ¬ Hxc) (Txd → Sa) ∀x ((Px ∧ ¬ Hxc) → (Txd → Sa)) |
| 2. | a. |
¬ (∀x: x glitters) x is gold ¬ everything that glitters is gold
Not everything that glitters is 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
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
|
| 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
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
|
| 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 |