The difference between $\forall x[R(x) \rightarrow P(x)]$ and $\forall x[R(x)\wedge P(x)]$.

Question

Apparently, it is a common mistake to write $\forall x[R(x) \rightarrow P(x)]$ instead of $\forall x[R(x)\wedge P(x)]$ in some cases, however I can't seem to find the difference between the two.

I did some research in ProofWiki and a Discrete Mathematics book, but I couldn't find this specific difference explained.

Could you please explain it to me and give some examples on when to use one and when to use the other?

Thank you for your time.

Answer 1

It's the difference between

1. "anyone who gets a perfect score on the quiz will receive a cookie," and

2. "everyone gets a perfect score on the quiz and will receive a cookie."

It's a bit tricky to translate number 1 from colloquial language into first-order logic, but when you do, it says $\forall x\,(Q(x) \to C(x))$ where $Q(x)$ means "$x$ gets a perfect score on the quiz" and $C(x)$ means "$x$ will receive a cookie."

Answer 2

Try with the following basic, intuitive definitions:

$$x\in\Bbb Z=\text{the set of integer numbers}\;,\;\;R(x):=x\;\text{is a prime number}\;,\;$$

$$P(x):=x\;\text{not divisible by}\;4$$

Then you try to explain the difference between "For any integer $\;x\;$ , if $\;x\;$ is a prime number then is is undivisible by four" and "For any integer $\;x\;$ , $\;x\;$ is both prime and undivisible by four".

Answer 3

Assume $[\forall x[Rx\to Px]]\leftrightarrow [\forall x[Rx\wedge Px]]$ and suppose $ \forall x[Rx\to Px] $. Then $ \forall x[Rx\wedge Px] $. Instantiate with $a$ for $x$. Now $Ra\to Pa$ and $Ra$, $Pa$. But then $\neg Ra$ is an option, a contradiction.

You can see these things clearly with tableaux.