Implication between propositions vs implication between predicates

Question

Based on what I read, I know that an implication $p\rightarrow q$ between two propositions p and q means “if p then q”. Since p and q are propositions, why, given a domain $D$, I often see sentences like $\forall x(p(x)\rightarrow q(x))$, where p(x) and q(x) are predicates, and thus not propositions? Is it an abuse of notation to express the proposition $\forall x (q(x))$, where the domain is now {$x|x\in D$ and $p(x)$}? Moreover, I know that $p\rightarrow q \equiv \neg q \rightarrow \neg p$. By analogy I suppose that $\forall x(p(x)\rightarrow q(x))$ is a synonym of $\forall x(q’(x)\rightarrow p’(x))$, where p’(x) means “p(x) is not true” and q’(x) means “q(x) is not true”. Am I right?

Answer 1

You seem to be conflating propositional logic with predicate logic. These are two different logics with a different language.

In propositional logic, we have propositional variables $p, q, ..$ and connectives $\neg, \to, ...$ between them, and that's it. All the formulas of propositional logic are made up of only propositional variables and connectives. Propositional variables directly stand for propositions, so something like $p$ will evaluate as true or false.

Predicate logic is a different language. Here we do not have propositional variables: Something like $p \to q$ is not a formula of predicate logic.
Instead, we have predicate symbols, which express properties and relations between individuals. These predicate symbols are also often called $p, q, ...$, but they do not stand for propositions. Instead, they stand for properties which are then applied to individuals (written $p(x)$) to produce a proposition. For example, $\text{even}$ could a predicate that expresses the property that some number is even, and $\text{even}(x)$ is the proposition "$x$ is even", which evaluates as true or false. Predicates are not themselves propositions (something that has a truth value), but can be seen as functions where you put in an individual (or a tuple of individuals) and you get out a truth value, depending on whether that individual has the property expressed by the predicate/the individuals stand in the relation expressed by the predicate.

By the definition of formulas in predicate logic (a term is an expression which stands for an individual):

If $p$ is an $n$-place predicate and $t_1, ..., t_n$ are terms, then $p(t_1, ..., t_n)$ is a formula.

$p$ is a predicate. $p(x)$ is not a predicate, but a formula. In predicate logic, any formula will evaluate as true or false, so given that $p(x)$ is a formula, $p(x)$ is a proposition -- more precisely an expression that expresses the proposition "$x$ has the property $p$" which evaluates to a truth value. In predicate logic, so-called atomic formulas of the form $p(x)$ or $r(x,y)$ (that is, a single predicate applied to individuals without connectives) take the place of propositional variables $p, q, ...$ in propositional logic.
Furthermore, we have that

If $\phi$ and $\psi$ are formulas, then $\phi \to \psi$ is a formula

and

If $\phi$ is a formula and $x$ is a variable, then $\forall x \phi$ is a formula.

So not only $p(x)$ and $q(x)$, but also $p(x) \to q(x)$ and $\forall x(p(x) \to q(x))$ are formulas/propositions.

(Sometimes a more fine-grained distinction is made between an expression (a string of symbols) that stands for something which will later evaluate to a truth value -- which is be what we called "formulas" here -- and a "proposition" in the narrower sense, which is the propositional semantic content that the expression stands for. But the term "proposition" is very often used to mean "a string of symbols that expresses a proposition, i.e. something that is true or false", so we can adapt that sloppy terminology here.)

Again: Propositional logic and predicate logics are two different languages. A symbol p in propositional logic means something different than a symbol p in predicate logic. In propositional logic, p is a propositional variable that stands for a proposition. In predicate logic, there are no propositional variables. Instead, p stands for a predicate, which produces a proposition when applied to an individual, $p(x)$.

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If you want to define a notaton $p'$ to mean "$p$ is not the case", then yes, $p(x) \to q(x)$ would be equivalent to $q'(x) \to p'(x)$, and hence also $\forall x(p(x) \to q(x))$ is equivalent to $\forall x(q'(x) \to p'(x))$. But there is no need to introduce a special notation that expresses the negation of each predication, because we already have a symbol that expresses exactly that: $\neg$. By the definition of formulas of predicate logic,

If $\phi$ is a formula, then $\neg \phi$ is a formula

-- and since by the first definition $p(x)$ and $q(x)$ are formulas, so are $\neg p(x)$ and $\neg q(x)$, and we can simply write

$p(x) \to q(x)\\ \equiv \neg q(x) \to \neg p(x)$

and hence

$\forall x(p(x) \to q(x))\\ \equiv \forall x (\neg q(x) \to \neg p(x)).$

Note that $\neg$ is applied to the formula (the proposition) $p(x)$, not to the predicate $p$: The bracketing is $\neg(p(x))$. What we negate is the truth of the proposition "$x$ has the property $p$".