How to Invalidate this first order logic statement formally ?

Question

$(\forall (x)[\alpha] \to \forall (x)[\beta] ) \to \forall (x)[\alpha \to \beta]$

I have already disproof this statement using an intuitive example but i want to know if this can be done using formal proof method quickly.

Answer 1

Formally, you must find a model where the statement fails. Hint: find $\alpha,\beta$ s.t.:

• $\forall (x)[\alpha]$ is false, i.e. exists $x_0$ s.t. $\alpha(x_0)$ is false. But... $\alpha$ isn't always false (why the second condition?).
• $\forall (x)[\alpha\implies\beta]$ is false, i.e., for some $x_1$, $\alpha(x_1)$ is true and $\beta(x_1)$ is false.

Answer 2

If by formal proof method you mean a formal ederivation, then no: you can't use a formal proof to demonstrate a statement to be invalid, except for when the statement is a contradiction, in which case a formal derivation of its negation would show it to be a contradiction, and thus invalid.

There are 'formal methods' to demonstrate invalidity though, e.g. Truth trees, though for FOL even those methods will be incomplete.

In general, then, you'll want to find a counterexample to demonstrate invalidity. It sounds like you did just that ... and that's a fine demonstration; nothing more is needed.