Everyone has seen a bear

Everyone is such that (he or she has seen a bear)

(∀x: x is a person) x has seen a bear

(∀x: Px) a bear is such that (x has seen it)

(∀x: Px) (∃y: y is a bear) x has seen y

(∀x: Px) (∃y: By) Sxy
∀x (Px → ∃y (By ∧ Sxy))

B: [ _ is a bear]; P: _ is a person; S: [ _ has seen _ ]

Everyone was talking about a certain movie

A certain movie is such that (everyone was talking about it)

(∃x: x is a movie) everyone was talking about x

(∃x: Mx) everyone is such that (he or she was talking about x)

(∃x: Mx) (∀y: y is a person) y was talking about x

(∃x: Mx) (∀y: Py) Tyx
∃x (Mx ∧ ∀y (Py → Tyx))

M: [ _ is a movie]; P: _ is a person; T: [ _ was talking about _ ]

A capital was chosen by each state

Each state is such that (a capital was chosen by it)

(∀x: x is a state) a capital was chosen by x

(∀x: Sx) a capital is such that (it was chosen by x)

(∀x: Sx) (∃y: y is a capital) y was chosen by x

(∀x: Sx) (∃y: Cy) Hyx
∀x (Sx → ∃y (Cy ∧ Hyx))

C: [ _ is a capital]; H: _ was chosen by _ ; S: [ _ is a state]

There is a capital that was chosen by each state

Something is a capital that was chosen by each state

Something is such that (it is a capital that was chosen by each state)

∃x x is a capital that was chosen by each state

∃x (x is a capital ∧ x was chosen by each state)

∃x (Cx ∧ each state is such that (x was chosen by it))

∃x (Cx ∧ (∀y: y is a state) x was chosen by y)

∃x (Cx ∧ (∀y: Sy) Hxy)
∃x (Cx ∧ ∀y (Sy → Hxy))

C: [ _ is a capital]; H: _ was chosen by _ ; S: [ _ is a state]

Someone who no reporter knew leaked the information

Someone who no reporter knew is such that (he or she leaked the information)

(∃x: x is a person who no reporter knew) x leaked the information

(∃x: x is a person ∧ no reporter knew x) Lxi

(∃x: Px ∧ no reporter is such that (he or she knew x)) Lxi

(∃x: Px ∧ (∀y: y is a reporter) ¬ y knew x) Lxi

(∃x: Px ∧ (∀y: Ry) ¬ Kyx) Lxi
∃x ((Px ∧ ∀y (Ry → ¬ Kyx)) ∧ Lxi)
or: (∃x: Px ∧ ¬ (∃y: Ry) Kyx) Lxi

K: [ _ knew _ ]; L: [ _ leaked y]; P: _ is a person; R: [ _ is a reporter]; i: the information

A head of a horse is the head of a mammal

Every head of a horse is such that (it is the head of a mammal)

(∀x: x is the head of a horse) x is the head of a mammal

(∀x: a horse is such that (x is the head of it)) a mammal is such that (x is the head of it)

(∀x: (∃y: y is a horse) x is the head of y) (∃z: z is a mammal) x is the head of z

(∀x: (∃y:Hy) x = the head of y) (∃z: Mz) x = the head of z

(∀x: (∃y: Hy) x = hy) (∃z: Mz) x = hz
∀x (∃y (Hy ∧ x = hy) → ∃z (Mz ∧ x = hz))
or: (∀x: (∃y: Hy) Dxy) (∃z: Mz) Dxz

D: [ _ is a head of _ ]; H: _ is a horse; M: [ _ is a mammal]; h: the head of _ 

In this interpretation, which seems most natural given the content of the sentence, a is understood to indicate a generalization rather than a claim of exemplification. That is, it amounts to any in a use that is equivalent to every rather than in contrast with it. It appears in a location where any would not contrast with every, so if the sentence were understood to make a claim of exemplification, substituting any for a would change the meaning.

Everyone who has seen a rainbow has seen a rainstorm

Everyone who has seen a rainbow is such that (he or she has seen a rainstorm)

(∀x: x is person who has seen a rainbow) x has seen a rainstorm

(∀x: x is person ∧ x has seen a rainbow) a rainstorm is such that (x has seen it)

(∀x: x is person ∧ a rainbow is such that (x has seen it)) (∃z: z is a rainstorm) x has seen z

(∀x: x is person ∧ (∃y: y is a rainbow) x has seen y) (∃z: Rz) Sxz

(∀x: Px ∧ (∃y: Ry) Sxy) (∃z: Tz) Sxz
∀x ( (Px ∧ ∃y (Ry ∧ Sxy)) → ∃z (Tz ∧ Sxz) )

P: [ _ is a person]; R: _ is a rainbow; S: [ _ has seen _ ]; T: _ is a rainstorm

Every child was given a toy by each Santa

Every child is such that (he or she was given a toy by each Santa)

(∀x: x is a child) x was given a toy by each Santa

(∀x: Cx) each Santa is such that (x was given a toy by him or her)

(∀x: Cx) (∀y: y is a Santa) x was given a toy by y

(∀x: Cx) (∀y: Sy) a toy is such that (x was given it by y)

(∀x: Cx) (∀y: Sy) (∃z: z is a toy) x was given z by y

(∀x: Cx) (∀y: Sy) (∃z: Tz) Gxzy
∀x (Cx → ∀y (Sy → ∃z (Tz ∧ Gxzy)))

C: [ _ is a child]; G: _ was given _ by _ ; S: [ _ is a Santa]; T: _ is a toy

Notice that, in spite of the capitalization, Santa is not used here as a proper name but instead as a sort of job title. As a result it is represented not by an individual term but instead by a predicate. For representation by an individual term to be appropriate, it would have to be possible to paraphrase the sentence using each thing that is Santa rather than each thing that is a Santa.

There is a toy that every child was given by each Santa

Something is a toy that every child was given by each Santa

∃x x is a toy that every child was given by each Santa

∃x (x is a toy ∧ every child was given x by each Santa)

∃x (Tx ∧ every child is such that (he or she was given x by each Santa))

∃x (Tx ∧ (∀y: y is a child) y was given x by each Santa)

∃x (Tx ∧ (∀y: Cy) each Santa is such that (y was given x by him or her))

∃x (Tx ∧ (∀y: Cy) (∀z: z is a Santa) y was given x by z)

∃x (Tx ∧ (∀y: Cy) (∀z: Sz) Gyxz)
∃x (Tx ∧ ∀y (Cy → ∀z (Sz → Gyxz)))

C: [ _ is a child]; G: _ was given _ by _ ; S: [ _ is a Santa]; T: _ is a toy

∀x ∃y x depends on y

∀x something is such that ( x depends on it)

∀x x depends on something

Everything is such that (it depends on something)

Everything depends on something

∃x ∀y x depends on y

∃x everything is such that ( x depends on it)

∃x x depends on everything

Something is such that (it depends on everything)

Something depends on everything

∀x ∃y y depends on x

∀x something is such that (it depends on x)

∀x something depends on x

Everything is such that something depends on it
or: Everything has something depending on it
or (perhaps): Something or other depends on each thing

∃x ∀y y depends on x

∃x everything is such that (it depends on x)

∃x everything depends on x

Something is such that everything depends on it
or: Something has everything depending on it
or: There is something that everything depends on
or (perhaps): All things depend on a certain thing

(∀x: x is a person ∧ x is humble) (∃y: y is a person) x admires y

(∀x: x is a humble person) someone is such that (x admires him or her)

(∀x: x is a humble person) x admires someone

Every humble person is such that (he or she admires someone)

Every humble person admires someone
or: Everyone who is humble admires someone

(∃y: y is a person) (∀x: x is a person ∧ x is humble) x admires y

(∃y: y is a person) (∀x: x is a humble person) x admires y

(∃y: y is a person) every humble person is such that (he or she admires y)

(∃y: y is a person) every humble person admires y

Someone is such that every humble person [or: everyone who is humble] admires him or her
or: Someone has every humble person admiring him or her
or: There is someone [or: a person] who every humble person admires
or (perhaps): All who are humble admire a certain person

¬ (∀x: x is a person ∧ (∃y: y is a person) x admires y) x is humble

¬ (∀x: x is a person ∧ someone is such that (x admires him or her)) x is humble

¬ (∀x: x is a person ∧ x admires someone) x is humble

¬ (∀x: x is a person who admires someone) x is humble

¬ everyone who admires someone is such that (he or she is humble)

¬ everyone who admires someone is humble

Not everyone who admires someone is humble
or: Not everyone who admires anyone is humble

¬ (∃x: x is a person) (∀y: y is a person ∧ y has seen x) x has seen y

¬ (∃x: x is a person) (∀y: y is a person who has seen x) x has seen y

¬ (∃x: x is a person) everyone who has seen x is such that (x has seen him or her)

¬ (∃x: x is a person) x has seen everyone who has seen x

¬ someone is such that (he or she has seen everyone who has seen him or her)

¬ someone has seen everyone who has seen him or her

No one has seen everyone who has seen him or her

¬ (∃x: x is a person ∧ (∀y: ¬ (y is a person ∧ y has spoken to x)) ¬ x has spoken to y) x is an extrovert

¬ (∃x: x is a person ∧ (∀y: ¬ y is a person who has spoken to x) ¬ x has spoken to y) x is an extrovert

¬ (∃x: x is a person ∧ only people who have spoken to x are such that(x has spoken to them)) x is an extrovert

¬ (∃x: x is a person ∧ x has spoken only to people who have spoken to x) x is an extrovert

¬ (∃x: x is a person who has spoken only to people who have spoken to him or her) x is an extrovert

¬ someone who has spoken only to people who have spoken to him or her is such that (he or she is an extrovert)

¬ someone who has spoken only to people who have spoken to him or her is an extrovert

No one who has spoken only to people who have spoken to him or her is an extrovert