Proof that $\exists x(R(x) \to\forall yR(y))$ is logically valid

Question

Let $R$ be a predicate symbol of arity $1$.

Show that $\exists x(R(x) \to\forall yR(y))$ is logically valid.

Answer 1

How are you supposed to show this is valid?

There are several options:

• Just informally talking about the logical operators involved
• Using formal semantics
• Using equivalence principles
• Using inferences (formal proof)
• A truth tree
• Resolution
• Etc.

And if any of these, what specific rules or logic principles do you have to work with?

Just informally: Classcial logic assumes that the domain we talk about is non-empty (indeed, this claim would not be true in an empty domain). And given any (non-empty) world, there are two options: either everything has property $R$, or not everything has property $R$. If the former, then clearly $\forall y R(y) $ is true, and hence you can pick something from the domain (doesn't matter which) such that the conditional is true. If the latter, then there is something that does not have property $R$. For that something, the conditional will be true, since the antecedent is false. So, either way, the claim is true.

Here is a proof using equivalence principles:

$\top \Leftrightarrow$ (Complement)

$\forall x R(x) \lor \neg \forall x R(x) \Leftrightarrow$ (Commutation)

$\neg \forall x R(x) \lor \forall x R(x) \Leftrightarrow$ (Implication)

$\forall x R(x) \rightarrow \forall x R(x) \Leftrightarrow$ (Replacing bound variables)

$\forall x R(x) \rightarrow \forall y R(y) \Leftrightarrow$ (Prenex Law)

$\exists x (R(x) \rightarrow \forall y R(y)) $

And here is a formal proof using one out of many different proof systems: