How to Translating sentences from English into predicate logic

Question

I am learning the formal method, and I am not so sure if I have translated these statements correctly.

a) “Every state has exactly one head of state.”

Interpretation:

$D$ = the set of all states and persons,

$Sx$: $x$ is a state,

$Hxy$: $x$ is a person and is head of state of state y.

My answer: $$∀x (Sx \to ∃1y Hyx)$$

b) “Batman and nobody else but Batman can save the world.”

Interpretation:

$D$ = the set of all people and superheroes.

$b$:Batman.

$Sx$: x can save the world.

My answer: $$∃b (Sb \land ∃y (Sy \land y = ¬b))$$

c) “There are at least two concrete objects.”

Interpretation:

Any interpretation of predicate logic with identity that has as its domain D = the set of all concrete objects.

$Cx$: $x$ is a concrete object.

My answer: $$∃x ∃y (Cx \land Cy)$$

Answer 1

a) “Every state has exactly one head of state.”

my answer: ∀x (Sx → ∃1y Hyx)

There are several ways to formalise the given sentence; one is $$∀x \,\big(Sx → ∃p ∀q \,(Hqx ↔ q=p)\big).$$

b) “Batman and nobody else but Batman can save the world.

b: Batman.

My answer:∃b (Sb & ∃y (Sy & y = ¬ b) )

Since you have defined $b$ as a constant (rather than a variable), it doesn't make sense to write “$∃b$”.

Part (b) is strictly easier than part (a), so use the structure of my suggested formalisation above to figure out this answer. Note that here we don't need “$∃p$” since we are referring to a specific object $b.$

c) “There are at least two concrete objects.”

My answer: ∃x ∃y (Cx & Cy)

This is actually equivalent to $∃x \,Cx,$ since $x$ and $y$ can point to the same object.

Hint: try $$∀x∃p\,\big(Cp\ldots\big).$$