Someone is missing

(∃x: x is a person) x is missing

(∃x: Px) Mx
∃x (Px ∧ Mx)

M: [ _ is missing]; P: [ _ is a person]

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
¬ ∃x (Px ∧ Fxl)

F: [ _ found _ ]; P: [ _ is a person]; l: the loot

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.

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
∃x (Wxl ∧ ¬ Fx)

F: [ _ has come forward]; W: [ _ is a winner of _ ]; l: the lottery

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
∃x ((Dx ∧ Fxi) ∧ Wtx)

D: [ _ is a dance troop]; F: [ _ is from _ ]; W: [ _ watched _ ]; i: India]; t: Tod

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
¬ ∃x ((Cx ∧ Fxd) ∧ Tsx)

C: [ _ is a car]; F: [ _ fit _ ]; T: [ _ turned up _ ]; d: the description]; s: the search

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.

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

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
∃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

∃x x is burning

something is such that (it is burning)

Something is burning
or: There is something burning

(∃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

(∃x: x is a fire) Tamara reported x

Some fire is such that (Tamara reported it)

Tamara reported a fire

¬ (∃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

(∃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

∃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

∃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
or: There is something that Ann forgot and Bill remembered

(∃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
or: The instrument detected something fast and heavy