If anyone can do it, then Jones can do it, too

Question

1. If anyone can do it, then Jones can

2. If Jones can do it, then anyone can

I am confused whether to use forall or there exists.

Answer 1

"If anyone can do it, then Jones can (do it)," says, "For anyone: if they can do it, then Jones can do it."

$$\forall x~\big(D(x)\to D(j)\big)$$

• $\small\text{ Which is equivalent to }(\exists x~D(x))\to D(j)~$

"If Jones can do it, then anyone can (do it)," reads: $D(j)\to\forall x~D(x)$ or $$\forall x~\big(D(j)\to D(x)\big)$$

Answer 2

hint

If $ X$ can do it, we will write : $P(X)$.

The conditionnal : if p then q is represented by $ p \implies q$.

So, the proposition : if any can do it then Jones can do it becomes

$$\forall X \;P(X) \implies \; P(Jones)$$

The second one is $$P(Jones)\implies \forall X P(X)$$ Here, it clearly means that if Jones can, everyone $\forall$ can.