Someone is missing

(∃x: x is a person) x is missing

(∃x: Px) Mx
∃x (Px ∧ Mx)

[M: λx (x is missing); P: λx (x 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: λxy (x found y); P: λx (x 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: λxy (x is in y); T: λx (x 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: λx (x has come forward); W: λxy (x is a winner of y); 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: λx (x is a dance troop); F: λxy (x is from y); W: λxy (x watched y); 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: λx (x is a car); F: λxy (x fit y); T: λxy (x turned up y); 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: λx (x is a button); H: λxy (x is behind y); O: λxy (x will open y); 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 the 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: λx (x will be disappointed); F: λxy (x will find y); 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: λxy (x had heard of y); K: λxy (x knew y); P: λx (x is a person); R: λx (x is a restaurant); W: λxy (x went to y); 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