TITLE: Translating from English into predicate logic
QUESTION [0 upvotes]: 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: $$∃x ∃y (Cx \land Cy)$$

REPLY [0 votes]: a) “Every state has exactly one head of state.”
my answer: ∀x (Sx → ∃1y Hyx)

There is 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 $$∀x∃p\,\big(Cp\ldots\big).$$
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