To determine if the two first-order logic statements are logically equivalent, let’s analyze them individually.

Statement 1:

1. $\exists u (A(u) \rightarrow \exists n \, S(n, \text{available}))$

This statement means:

• There exists some $u$ such that, if $A(u)$ holds, then there exists an $n$ for which $S(n, \text{available})$ is true.

Interpretation: This is essentially saying that if there exists a $u$ such that $A(u)$ is true, then $S(n, \text{available})$ must hold for some $n$. This could mean that if $A(u)$ holds for some $u$, then "available" is true for at least one instance of $n$.

Statement 2:

2. $\exists x A(x) \rightarrow \exists y \exists z (S(y, z) \land (z = \text{Available}))$

This statement can be broken down as follows:

• If there exists some $x$ such that $A(x)$ holds, then there exists some $y$ and $z$ such that $S(y, z)$ is true and $z = \text{Available}$.

Interpretation: This implies that if there exists an $x$ with property $A(x)$, then there is a pair $(y, z)$ where $S(y, z)$ is true and $z$ equals "Available".

Comparison for Logical Equivalence

Both statements seem to represent conditional forms:

• In Statement 1, we have a condition over $u$ implying the existence of an $n$ such that $S(n, \text{available})$.
• In Statement 2, if there exists an $x$ such that $A(x)$, it implies the existence of a $y$ and $z$ such that $S(y, z)$ holds and $z = \text{Available}$.

The two statements are not logically equivalent because:

1. Scope of Quantifiers: Statement 1 has $\exists u$ applied only to $A(u) \rightarrow \exists n S(n, \text{available})$, whereas Statement 2 starts with $\exists x A(x) \rightarrow \ldots$, introducing a slight difference in structure.
2. Dependency of Variables: Statement 1 implies a direct relationship through $u$ to $n$ under the scope of $\exists u$, while in Statement 2, we only need $\exists x$ for $A(x)$, independent of how $y$ and $z$ are quantified later.

Thus, the two statements are not logically equivalent, as their structures and interpretations differ.

Would you like further details on this or have any questions?

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Related Questions

1. What other examples of logical equivalence in first-order logic exist?
2. How can quantifiers affect logical equivalence in statements?
3. Can conditional statements be logically equivalent even if they differ in quantifier order?
4. How do free variables impact logical equivalence in logical statements?
5. What are common methods for proving logical equivalence?

Tip:

When assessing logical equivalence in first-order logic, carefully analyze the placement and scope of quantifiers, as they can significantly alter the interpretation of a statement.