8.1.xa. Exercise answers
| 1. | a. |
Someone is missing (∃x: x is a person) x is missing
(∃x: Px) Mx
M: [ _ is missing]; P: [ _ is a person] |
| b. |
No one found the loot. ¬ someone found the loot ¬ someone is such that (he or she found the loot) ¬ (∃x: x is a person) x found the loot
¬ (∃x: Px) Fxl
F: [ _ found _ ]; P: [ _ is a person]; l: the loot |
| c. |
There is a tavern in the town Something is a tavern in the town Something is such that (it is a tavern in the town) ∃x x is a tavern in the town ∃x (x is a tavern ∧ x is in the town) ∃x (Tx ∧ Ixt) I: [ _ is in _ ]; T: [ _ is a tavern]; t: the town
It would also be possible to understand in the town to modify the verb is rather the noun tavern. In that case, the sentence could be restated as A tavern is in the town and be analyzed using a restricted existential.
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| d. |
Some winner of the lottery has not come forward Some winner of the lottery is such that (he or she has not come forward) (∃x: x is a winner of the lottery) x has not come forward (∃x: x is a winner of the lottery) ¬ x has come forward
(∃x: Wxl) ¬ Fx
F: [ _ has come forward]; W: [ _ is a winner of _ ]; l: the lottery |
| e. |
Tod watched a dance troop from India A dance troop from India is such that (Tod watched it) (∃x: x is a dance troop from India) Tod watched x (∃x: x is a dance troop ∧ x is from India) Wtx
(∃x: Dx ∧ Fxi) Wtx
D: [ _ is a dance troop]; F: [ _ is from _ ]; W: [ _ watched _ ]; i: India]; t: Tod |
| f. |
The search turned up no car fitting the description ¬ the search turned up a car fitting the description ¬ a car fitting the description is such that (the search turned it up) ¬ (∃x: x is a car fitting the description) the search turned up x ¬ (∃x: x is a car ∧ x fit the description) Tsx
¬ (∃x: Cx ∧ Fxd) Tsx
C: [ _ is a car]; F: [ _ fit _ ]; T: [ _ turned up _ ]; d: the description]; s: the search |
| g. |
There is a button behind you that will open the door Something is a button behind you that will open the door Something is such that (it is a button behind you that will open the door) ∃x x is a button behind you that will open the door ∃x (x is a button behind you ∧ x will open the door) ∃x ((x is a button ∧ x is behind you) ∧ Oxd) ∃x ((Bx ∧ Hxo) ∧ Oxd) B: [ _ is a button]; H: [ _ is behind _ ]; O: [ _ will open _ ]; d: the door; o: you
If the prepositional phrase behind you is understood to modify is instead of button, the sentence could be restated as A button that will open the door is behind you. This sentence would be analyzed by the restricted existential (∃x: Bx ∧ Oxd) Hxo, in which two of the conjuncts from the quantified predicate in the analysis above appear instead in a restriction of the quantifier.
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| h. |
If Tom doesn’t find anything, he’ll be disappointed Tom won’t find anything → Tom will be disappointed ¬ Tom will find something → Tom will be disappointed ¬ something is such that (Tom will find it) → Dt ¬ ∃x Tom will find x → Dt ¬ ∃x Ftx → Dt D: [ _ will be disappointed]; F: [ _ will find _ ]; t: Tom |
| i. |
Al went to a restaurant no one he knew had heard of A restaurant no one Al knew had heard of is such that (Al went to it) (∃x: x is a restaurant no one Al knew had heard of) Al went to x (∃x: x is a restaurant ∧ no one Al knew had heard of x) Wax (∃x: Rx ∧ ¬ someone Al knew had heard of x) Wax (∃x: Rx ∧ ¬ someone Al knew is such that (he or she had heard of x)) Wax (∃x: Rx ∧ ¬ (∃y: y is a person Al knew) y had heard of x) Wax (∃x: Rx ∧ ¬ (∃y: y is a person ∧ Al knew y) Hyx) Wax
(∃x: Rx ∧ ¬ (∃y: Py ∧ Kay) Hyx) Wax
H: [ _ had heard of _ ]; K: [ _ knew _ ]; P: [ _ is a person]; R: [ _ is a restaurant]; W: [ _ went to _ ]; a: Al |
| 2. | a. |
∃x x is burning something is such that (it is burning)
Something is burning
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| b. |
(∃x: x is a person) x is at the door someone is such that (he or she is at the door) Someone is at the door |
| c. |
(∃x: x is a fire) Tamara reported x Some fire is such that (Tamara reported it) Tamara reported a fire |
| d. |
¬ (∃x: x is a person ∧ x was in the room) x knew Sam ¬ (∃x: x was a person in the room) x knew Sam ¬ someone in the room is such that (he or she knew Sam) ¬ someone in the room knew Sam No one in the room knew Sam |
| e. |
(∃x: x is a vase) (Vic touched x ∧ x shattered) (∃x: x is a vase) (Vic touched x and x shattered) A vase is such that (Vic touched it and it shattered) Vic touched a vase and it shattered |
| f. |
∃x (x had happened ∧ Jane left to deal with x) ∃x x had happened and Jane left to deal with x something is such that (it had happened and Jane left to deal with it) Something had happened and Jane left to deal with it |
| g. |
∃x (Ann forgot x ∧ Bill remembered x) ∃x (Ann forgot x and Bill remembered x) something is such that (Ann forgot it and Bill remembered it)
Ann forgot something and Bill remembered it
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| h. |
(∃x: x was fast ∧ x was heavy) the instrument detected x (∃x: x was fast and heavy) the instrument detected x (∃x: x is a thing that was fast and heavy) the instrument detected x Something that was fast and heavy was such that (the instrument detected it)
The instrument detected something that was fast and heavy
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