Option C.
"No female likes a male who does not like all vegetarians" is the negation of "Some female likes a male who does not like all vegetarians". To get the answer, we negate the WFF of the latter.
The phrase "there exists a female" suggests that we need to use the $\exists$ quantifier. And, since the domain is unrestricted, we use the $\wedge$ connective with the $\exists$ quantifier. So, the WFF is -
$\sim \exists x\left \{ female\left ( x \right ) \wedge \mathbb{P} \right \}$
Here, $\mathbb{P}$ = "$x$ likes a male who does not like all vegetarians". Basically, $\mathbb{P}$ is where we quantify the males and show that $x$ (representing female) likes them.
Two types of ambiguity occur while interpreting $\mathbb{P}$ -
1. Quantity of "a"
$\mathbb{P}$ = "$x$ likes a male who does not . . ." has 2 interpretations -
a. "$x$ likes some male who does not . . ."
b. "$x$ likes every male who does not . . ."
It is very likely that in first reading, one would make the 1st interpretation where the "a" suggests the $\exists$ quantifier. But, after a second reading the 2nd interpretation can be made, where "a" suggests the $\forall$ quantifier.
Examples -
1. Some illiterate people can’t read a word.
a. "There is an illiterate person who can not read some word".
$\exists x\left \{ illiterate\left ( x \right ) \wedge \exists y\left ( word\left ( y \right ) \wedge \sim read\left ( x,y \right ) \right ) \right \}$
b. "There is an illiterate person who can not read any word at all".
$\exists x\left \{ illiterate\left ( x \right ) \wedge \forall y\left ( word\left ( y \right ) \rightarrow \sim read\left ( x,y \right ) \right ) \right \}$
2. Every man loves a woman.
a. "Every man loves some woman".
$\forall x\left \{ man\left ( x \right ) \rightarrow \exists y\left ( woman\left ( y \right ) \wedge love\left ( x,y \right ) \right ) \right \}$
b. "Every man loves every woman".
$\forall x\left \{ man\left ( x \right ) \rightarrow \forall y\left ( woman\left ( y \right ) \rightarrow love\left ( x,y \right ) \right ) \right \}$
In such cases, we must use intuition. For e.g., in the 1st statement, the adjective "illiterate" makes it clear that an illiterate person can't read at all, which is exactly what the second interpretation means. Also, because the first interpretation allows the possibility that an illiterate man can read some words, it is intuitively weak. So, "a" can be safely interpreted as the $\forall$ quantifier. In the 2nd statement, the second interpretation is wrong because every man can't love every woman. Intuitively, it is only possible that every man can love some woman. So, the $\exists$ quantifier is suitable here.
Now, if no additional context is available and neither of the two interpretations makes more sense than the other, then it is always safe to make the standard interpretation of "a", which is the $\exists$ quantifier. For our question, the statement for $\mathbb{P}$ is arbitrary and there is no way to know whether a female dislikes some or every male. So, we make the standard interpretation -
"$x$ likes SOME male who does not like all vegetarians".
2. Scope of negation
Examples -
1. All that glitters is not gold.
a. "Not everything that glitters is gold", OR "Some things that glitter are not gold".
$\sim \forall x\left ( glitter\left ( x \right ) \rightarrow gold\left ( x \right ) \right )$ OR $\exists x\left ( glitter\left ( x \right ) \wedge \sim gold\left ( x \right ) \right )$
b. "All that glitters is non-gold", OR "None of the things that glitter is gold".
$\forall x\left ( glitter\left ( x \right ) \rightarrow \sim gold\left ( x \right ) \right )$ OR $\sim \exists x\left ( glitter\left ( x \right ) \wedge gold\left ( x \right ) \right )$
2. All cats are not dogs.
a. "Not all cats are dogs", OR "Some cats are not dogs".
$\sim \forall x\left ( cat\left ( x \right ) \rightarrow dog\left ( x \right ) \right )$ OR $\exists x\left ( cat\left ( x \right ) \wedge \sim dog\left ( x \right ) \right )$
b. "All cats are cats, and not dogs", OR "No cat is a dog".
$\forall x\left ( cat\left ( x \right ) \rightarrow \sim dog\left ( x \right ) \right )$ OR $\sim \exists x\left ( cat\left ( x \right ) \wedge dog\left ( x \right ) \right )$
Here, ambiguity is due to the usage of "not". In the 1st interpretation of both examples, "not" is applied to the whole statement, i.e., its has a broad scope, which is equal to the scope of the $\forall$ quantifier. Whereas in the 2nd interpretation, it is applied only to the word following it, i.e., it has a narrow scope which is internal to the scope of the $\forall$ quantifier. All such statements of the form "All M not N" are ambiguous.
If such statements are asked in exam, then the best choice is the option "Ambiguous statement". But, if there is no such option and both the interpretations are present in the options, then we must rely on intuition. For e.g., for the 1st statement above, the second interpretation is wrong because we know that gold glitters. And, because some glittering things (diamond) are not gold, the first interpretation is correct. Hence, "not" should have a broad scope. However, for the 2nd statement, the second interpretation is intuitively correct because it is a known fact that no cat is a dog. Also, the first interpretation is intuitively weaker because it allows the possibility that some cats can be dogs. Hence, a narrow scope is appropriate for "not" here.
One might wonder "Is there any standard interpretation here like in the previous type of ambiguity? If not, then can we always resolve this ambiguity by intuition ?". No, there isn't any standard interpretation here. And, intuition can't always be used to resolve this ambiguity. There are some statements for which both options are intuitively wrong. For e.g., consider "All men are not mortal".
a. "Not all men are mortal", OR "Some men are immortal".
b. "All men are immortal", OR "No man is mortal".
Clearly, both the interpretations are wrong because we know that all men are mortal. So, the ambiguity can't be resolved.
There are also some statements for which intuition can't even be applied. For e.g., in our question, $\mathbb{P}$ = "$x$ likes some male who does not like all vegetarians" can be interpreted in two ways -
a. "$x$ likes some male who does not (like all vegetarians)", OR "There exists a male liked by $x$ such that not all vegetarians are liked by him", OR "$x$ likes some male who dislikes some vegetarians". Here, the scope of "not" is broad, and completely takes over the scope of the universal quantifier "all" that follows it.
b. "$x$ likes some male who does not (like) all vegetarians" OR "$x$ likes some male who dislikes all vegetarians". Here, "not" is applied only to the word "like", meaning that the scope of "not" is narrow, and is restricted inside the scope of the universal quantifier "all".
Since, the statement for $\mathbb{P}$ is arbitrary and has nothing to do with the reality, we can't use intuition. So, here too, ambiguity can't be resolved.
If such statements are asked in exam, then assuming that the "ambiguous statement" option is not present, we can't choose an answer unless only one of the two interpretations is present in the options. We must solve both interpretations and select the one present in the options. If both are present, then the question is challengeable and grace marks can be obtained.
The WFFs for both the above interpretations of $\mathbb{P}$ -
1. "$x$ likes some male who dislikes some vegetarians".
Here, the phrase "some male" indicates that we need to use the $\exists$ quantifier, and since the domain is unrestricted, we use the $\wedge$ connective with the $\exists$ quantifier. So, the WFF for $\mathbb{P}$ looks something like -
$\exists y\left ( male\left ( y \right ) \wedge \mathbb{Q}\wedge like\left ( x,y \right ) \right )$
Here, $\mathbb{Q}$ indicates that "$y$ dislikes some vegetarians". So, again due to the phrase "some vegetarians", we need to use the $\exists$ quantifier along with the $\wedge$ connective. So, $\mathbb{Q}$ becomes -
$\exists z\left ( veg\left ( z \right ) \sim like\left ( y,z \right ) \right )$
So, $\mathbb{P}$ becomes -
$\exists y\left \{ male\left ( y \right ) \wedge \exists z\left \{ veg\left ( z \right ) \wedge \sim like\left ( y,z \right ) \right \} \wedge like\left ( x,y \right ) \right \}$
So, the final answer becomes -
$\sim \exists x\left [ female\left ( x \right ) \wedge \exists y\left \{ male\left ( y \right ) \wedge \exists z\left \{ veg\left ( z \right ) \wedge \sim like\left ( y,z \right ) \right \}\wedge like\left ( x,y \right ) \right \} \right ]$
We can convert this formula to Prenex form by pulling out the quantifiers and placing them together as -
$\sim \exists x\exists y\exists z\left [ female\left ( x \right ) \wedge male\left ( y \right ) \wedge veg\left ( z \right ) \wedge \sim like\left ( y,z \right ) \wedge like\left ( x,y \right ) \right ]$
2. "$x$ likes some male who dislikes all vegetarians".
The WFF for the this interpretation of $\mathbb{P}$ is exactly the same as the one for the previous interpretation. The only difference is in $\mathbb{Q}$. Here, $\mathbb{Q}$ indicates "$y$ dislikes ALL vegetarians". So, we need to use the $\forall$ quantifier along with the $\rightarrow$ connective. So, $\mathbb{Q}$ becomes -
$\forall z\left ( veg\left ( z \right ) \rightarrow \sim like\left ( y,z \right ) \right )$
So, $\mathbb{P}$ becomes -
$\exists y\left \{ male\left ( y \right ) \wedge \forall z\left \{ veg\left ( z \right ) \rightarrow \sim like\left ( y,z \right ) \right \} \wedge like\left ( x,y \right ) \right \}$
And, the final formula becomes -
$\sim \exists x\left [ female\left ( x \right ) \wedge \exists y\left \{ male\left ( y \right ) \wedge \forall z\left \{ veg\left ( z \right ) \rightarrow \sim like\left ( y,z \right ) \right \} \wedge like\left ( x,y \right ) \right \} \right ]$
By converting this formula to Prenex form, we get -
$\sim \exists x\exists y\forall z\left [ female\left ( x \right ) \wedge male\left ( y \right ) \wedge \sim \left ( veg\left ( z \right ) \wedge like\left ( y,z \right ) \right ) \wedge like\left ( x,y \right ) \right ]$
Option C matches the WFF based the first interpretation above, but there is no option for the second interpretation. Hence, Option C.
