Translate given sentence to predicate logic

Question

Problem given:

Assume the following predicate and constant symbols:
$W(x,y):$ $x$ wrote $y$
$L(x,y):$ $x$ is longer than $y$
$N(x): x$ is a novel $a$: Amit $h$: Harshal
Which of the following predicate logic formula represents the sentence:
"Harshal wrote a novel which is longer than any of the Amit's novels"

Answer given:

$\exists x \left(N(x) \wedge W\left(h,x\right) \wedge \forall y \left( N\left(y\right) \wedge W\left(a,y\right) \implies L\left(x,y\right)\right)\right)$

Doubt

I think using implication is incorrect in this problem. Instead of implication, there should be AND $(\wedge)$, since existence of novels by both Harshal and Amit dont imply Harshal's novel is longer than Amit's. These are three indepndent facts:

1. Amit has written some novels
2. Harshal has written some novels
3. Some Harshal's novel is longer than all of Amits

Being independent, shouldnt we AND?

I understand that, without those novel exists, Harshal's novel cannot be longer than all of Amit's. But mere existence of them does not imply anything. The fact of one of Harshal's novel is longer than Amit's is given independently / explicitly and is not an implication of existence of novels.

Just to rephrase, is there really a sense of implication in this statement?: "Harshal wrote a novel which is longer than any of the Amit's novel"

Also, implication lets consequent $L(x,y)$ be true if antecedent ($\exists x \left(N(x) \wedge W\left(h,x\right) \wedge \forall y \left( N\left(y\right) \wedge W\left(a,y\right)\right)\right)$) is false, which I guess, wont be correct here. Right?

Answer 1

The given answer is correct, but you've been misreading the symbol "$\implies$". By the way, $\implies$ should be used for modus ponens and not within formulas, where you should use $\to$ for clarity : $\phi \to \psi$ reads "if $\phi$ then $\psi$".

Let's inspect the formula $\varphi(x) := \forall y ( N(y) \wedge W(a,y) \to L(x,y))$ :
Take $x_0$ such that $\models \varphi(x_0)$, that is to say :
For any $y$, if $y$ is a novel written by Amit, then $y$ is shorter than $x_0$.

In natural language, one would say :

Any novel written by Amit is shorter than $x_0$.

Likewise, the formula $\forall x, \, 4|x \to 2|x$ reads "any number divisible by 4 is divisible by 2".

On the other hand, if you write $\forall y ( N(y) \wedge W(a,y) \wedge L(x,y))$ like you suggested, this reads : "Any $y$ is a novel written by Amit that is shorter than $x$."

Answer 2

"Harshal wrote a novel which is longer than any of the Amit's novels" $$\exists x \left(N(x) \wedge W\left(h,x\right) \wedge \forall y \left( N\left(y\right) \wedge W\left(a,y\right) \implies L\left(x,y\right)\right)\right)$$

These are three independent facts:

1. Amit has written some novels
2. Harshal has written some novels
3. Some Harshal's novel is longer than all of Amits

Being independent, shouldnt we AND?

The given formalisation is correct (though, echoing Oliver, I'd replace ${\implies}$ with $\to$): the correct symbol preceding $L(x,y)$ is $\;\to\;$ instead of $\;\land\;$ because $(1)$ is not a given fact.