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How do quantified/unquantified variables work when considering conditional statements? The conditional is a statement, but the variables in the individual hypothesis/conclusion don't have to be quantified? i.e., can the hypothesis/conclusion be predicates and not have to be statements?

For example, "If $x$ is a real number, then $x^2 \geq 0$" is a conditional statement, hypothesis = "$x$ is a real number", conclusion = "$x^2 \geq 0$". The conclusion here is a predicate, not a statement (since $x$ is not quantified). (Or is there some implicit universal set based on the hypothesis?)

Also, in the contrapositive "If $x^2 < 0$, then $x$ is not a real number," now is $x$ not quantified in both the hypothesis and the conclusion?

curiousconfused
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    I addressed your question here; the short of it is that informally, $P(x){\implies}Q(x)$ is shorthand for $\forall x,\big(P(x){\implies}Q(x)\big).\quad$ However do note that $\forall x,P(x){\implies}\forall x,Q(x)$ is equivalent to $\exists x,\forall y,\big(P(x){\implies}Q(y)\big)$ instead. – ryang Feb 21 '22 at 07:26

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There is an implicit "for all" in statements of that form. We'd formalize "if $x$ is a real number, then $x^2\ge0$" as $\forall x(x\in\Bbb R\implies x^2\ge0)$. This way every well-formed statement has no free (i.e. unquantified) variables.

I'd say that the hypothesis and conclusion appear "in a quantified context" in which $x$ is bound, so in this context they represent statements and not predicates.

Notice that the variable names matter: $P(x,y)\implies Q(x,y)$ is very different from $P(x,y)\implies Q(y,x)$. Thinking of the conditional as operating on predicates doesn't make sense unless we associate extra "variable coordination information" with the operation.

Karl
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    I think this confirms that indeed the hypothesis and conclusion, standing each on their own, are predicates and not statements. We get a statement only after the two predicates are combined in the conditional and the quantifier is applied to the conditional. – David K Feb 20 '22 at 17:22
  • I think you could say they're open formulas but not predicates, since variable name coordination matters. I edited my answer to elaborate on this. – Karl Feb 20 '22 at 18:54