The logical expression ∼∃𝑃(𝑥)[𝑄(𝑥)] ∧ ∀𝑥[𝑄(𝑥)] is not equivalent to any constant $\mathbb{c}$ because it represents a more complex logical structure than a simple constant, even though $P(x)$ restricts the existential quantifier. Let's break this down:

1. Understanding the Components

• $\sim \exists P(x)[Q(x)]$: This is the negation of the statement "there exists an $x$ such that $P(x)$ and $Q(x)$ hold."

  In other words, it asserts that for all $x$, either $P(x)$ is false or $Q(x)$ is false.

• $\forall x[Q(x)]$: This states that for all $x$, $Q(x)$ holds.

2. What Does $P(x)$ Restrict?

The term $P(x)$ restricts the scope of the existential quantifier, meaning the existential statement $\exists P(x)[Q(x)]$ is concerned only with those $x$ for which $P(x)$ is true. This restricts the domain of the existential quantification.

3. Why is this Not a Constant?

A constant $\mathbb{c}$ represents a single, unchanging truth value, either true ($\top$) or false ($\bot$).

However, the expression $\sim \exists P(x)[Q(x)] \wedge \forall x[Q(x)]$ is not simply true or false but a combination of conditions:

• The first part asserts that no $x$ exists such that both $P(x)$ and $Q(x)$ are true.
• The second part asserts that $Q(x)$ is true for all $x$.

The truth value of this combination depends on the structure of $P(x)$ and $Q(x)$, and how they interact over the domain of $x$. It is not a straightforward true/false constant because the interaction between $P(x)$ and $Q(x)$ may lead to a variety of results based on the logical model.

4. Breakdown of the Non-equivalence

• $P(x)$ restricts the domain of existential quantification, but this does not simplify the expression into a constant, because the overall truth of the expression depends on the relationships between $P(x)$ and $Q(x)$, and not just on a uniform true or false result.
• For example, if $P(x)$ is true for some $x$ and $Q(x)$ is also true for all $x$, the first part of the conjunction ($\sim \exists P(x)[Q(x)]$) might be false, while the second part ($\forall x[Q(x)]$) might be true.

This complexity prevents the expression from collapsing into a simple constant $\mathbb{c}$.

5. Conclusion

The expression $\sim \exists P(x)[Q(x)] \wedge \forall x[Q(x)]$ depends on the interaction between the formulas $P(x)$ and $Q(x)$, and it can't be reduced to a simple constant truth value because its truth value varies based on these conditions.

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Do you need more details or clarification?

Here are some follow-up questions:

1. How does negation of an existential quantifier work logically?
2. What happens if $P(x)$ is always true? Does this simplify the expression?
3. Can $\sim \exists P(x)[Q(x)]$ ever be equivalent to $\bot$?
4. How would changing $\forall x[Q(x)]$ to $\exists x[Q(x)]$ affect the expression?
5. What would the truth table for this expression look like with different truth values for $P(x)$ and $Q(x)$?

Tip: Always check if the domain restrictions imposed by quantifiers affect the overall logical structure when simplifying expressions!