Sam guessed the winning number

the winning number is such that (Sam guessed it)

(∃x: x is a winning number ∧ only x is a winning number) Sam guessed x

(∃x: Wx ∧ (∀y: ¬ y = x) ¬ y is a winning number) Gsx

(∃x: Wx ∧ (∀y: ¬ y = x) ¬ Wy) Gsx
∃x (Wx ∧ ∀y (¬ y = x → ¬ Wy) ∧ Gsx)
or:
(∃x: Wx ∧ (∀y: Wy) x = y) Gsx
∃x (Wx ∧ ∀y (Wy → x = y) ∧ Gsx)

[G: λxy (x guessed y); W: λx (x is a winning number); s: Sam]
[Note: λx (x is a winning number) might be open to further analysis as λx (x is a number ∧ x won)]

Sam guessed the winning number

G Sam the winning number

Gs(Ix x is a winning number)

Gs(Ix Wx)

The winner who spoke to Tom was well-known

The winner who spoke to Tom is such that (he or she was well-known)

(∃x: x is a winner who spoke to Tom ∧ only x is a winner who spoke to Tom) x was well-known

(∃x: (x is a winner ∧ x spoke to Tom) ∧ (∀y: ¬ y = x) ¬ (y is a winner ∧ y spoke to Tom)) Kx

(∃x: (Wx ∧ Sxt) ∧ (∀y: ¬ y = x) ¬ (Wy ∧ Syt)) Kx
∃x ((Wx ∧ Sxt) ∧ ∀y (¬ y = x → ¬ (Wy ∧ Syt)) ∧ Kx)
or:
(∃x: (Wx ∧ Sxt) ∧ (∀y: Wy ∧ Syt) x = y) Kx
∃x ((Wx ∧ Sxt) ∧ ∀y ((Wy ∧ Syt) → x = y) ∧ Kx)

K: λx (x was well-known); S: λxy (x spoke to y); W: λx (x is a winner); t: Tom]

The winner who spoke to Tom was well-known

The winner who spoke to Tom was well-known

K the winner who spoke to Tom

K(Ix (x is a winner who spoke to Tom))

K(Ix (x is a winner ∧ x spoke to Tom))

K(Ix (Wx ∧ Sxt))

The winner, who spoke to Tom, was well-known.

The winner is such that (he or she, who spoke to Tom, was well-known).

(∃x: x is a winner ∧ only x is a winner) x, who spoke to Tom, was well-known)

(∃x: x is a winner ∧ (∀y: ¬ y = x) ¬ y is a winner) (x spoke to Tom ∧ x was well-known)

(∃x: Wx ∧ (∀y: ¬ y = x) ¬ Wy) (Sxt ∧ Kx)
∃x (Wx ∧ ∀y (¬ y = x → ¬ Wy) ∧ (Sxt ∧ Kx))
or:
(∃x: Wx ∧ (∀y: Wy) x = y) (Sxt ∧ Kx)
∃x (Wx ∧ ∀y (Wy → x = y) ∧ (Sxt ∧ Kx))

[K: λx (x was well-known); S: λxy (x spoke to y); W: λx (x is a winner); t: Tom]

The winner, who spoke to Tom, was well-known.

The winner spoke to Tom ∧ the winner was well-known

S the winner Tom ∧ K the winner

S(Ix x is a winner)t ∧ K(Ix x is a winner)

S(Ix Wx)t ∧ K(Ix Wx)

Every number greater than one is greater than its positive square root

(∀x: x is a number greater than one) x is greater than its positive square root

(∀x: x is a number ∧ x is greater than one) x is greater than the positive square root of x

(∀x: Nx ∧ Gxo) the positive square root of x is such that (x is greater than it)

(∀x: Nx ∧ Gxo) (∃y: y is a positive square root of x ∧ only y is a positive square root of x) x is greater than y

(∀x: Nx ∧ Gxo) (∃y: (y is positive ∧ y is a square root of x) ∧ (∀z: ¬ z = y) ¬ (z is positive ∧ z is a square root of x)) Gxy

(∀x: Nx ∧ Gxo) (∃y: (Py ∧ Syx) ∧ (∀z: ¬ z = y) ¬ (Pz ∧ Szx)) Gxy
∀x ( (Nx ∧ Gxo) → ∃y ((Py ∧ Syx) ∧ ∀z (¬ z = y → ¬ (Pz ∧ Szx)) ∧ Gxy) )

or:

(∀x: Nx ∧ Gxo) (∃y: (Py ∧ Syx) ∧ (∀z: Pz ∧ Szx) y = z) Gxy
∀x ( (Nx ∧ Gxo) → ∃y ((Py ∧ Syx) ∧ ∀z ((Pz ∧ Szx) → y = z) ∧ Gxy) )

[G: λx (x is greater than y); N: λx (x is a number); P: λx (x is positive); S: λxy (x is a square root of y)]

Every number greater than one is greater than its positive square root

(∀x: x is a number ∧ x is greater than one) x is greater than the positive square root of x

(∀x: Nx ∧ Gxo) G x the positive square root of x

(∀x: Nx ∧ Gxo) Gx(Iy y is a positive square root of x)

(∀x: Nx ∧ Gxo) Gx(Iy (y is a positive ∧ y is a square root of x))

(∀x: Nx ∧ Gxo) Gx(Iy (Py ∧ Syx))
∀x ( (Nx ∧ Gxo) → Gx(Iy (Py ∧ Syx)) )

(∃x: x owns Spot ∧ (∀y: ¬ y = x) ¬ y owns Spot) x called

(∃x: x owns Spot ∧ only x owns Spot) x called

The owner of Spot is such that (it called)

Spot’s owner called

John found (Ix (x is a photographer ∧ x enlarged (Iy y is a picture of John)))

John found (Ix (x is a photographer ∧ x enlarged the picture of John))

John found (Ix (x is a photographer who enlarged the picture of John))

John found the photographer who enlarged the picture of him