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I have in the lectures a sentence in English, which I have to translate to First Order Logic.

Someone who loves all animals loves all humans.

Textbook solution:

∀x.(is_human(x) ⇒ is_animal (x))

⇒

∀y.(is_human(y) ∧ ∀z.(is_animal(z) ⇒ loves(y, z)) ⇒ ∀z.(is_human(z) ⇒ loves(y, z))))

My solution: I would have interpreted the someone as an existential quantifier.

∀x.(is_human(x) ⇒ is_animal (x))

⇒

∃y.(is_human(y) ∧ ∀z.(is_animal(z) ⇒ loves(y, z)) ⇒ ∀z.(is_human(z) ⇒ loves(y, z))))

Why would this be wrong and why assuming that keywords like someone just comes with Existential quantifier is not always correct. Since I have learned that some word just introduce either existential or universal quantifier. But here in this example this is not the case anymore.

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    $\begingroup$ This is more a question on the interpretation of an English sentence than on logic… $\endgroup$
    – Nathaniel
    Dec 2, 2022 at 16:07

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The main issue here is that someone is ambiguous. You are correct in that usually, it translates to an existential quantifier. But it doesn't always.

To find an example, I Googled for "someone who * will also", which produced this page, with the sentence:

Someone who really respects you will also respect your boundaries about sex [...]

We can read this in the existential way, as meaning

There exists a person who really respects you, and who will also respect your boundaries about sex.

But clearly, that is not what is being meant here. The intended meaning is:

Anyone who really respects you will also respect your boundaries about sex.

or in other words:

Everyone who really respects you will also respect your boundaries about sex.

So in this case, someone translates to a universal quantifier.

Natural language is full of subtleties like this, which is the reason first order logic and other forms of logic were developed in the first place.

Now let's get back to the example you were given:

Someone who loves all animals loves all humans.

This is ambiguous in the same way: it may be used to mean

Someone exists who loves all animals and who also loves all humans.

but it is more likely to be used to mean

Everyone who loves all animals also loves all humans.

Incidentally:

  • both of your first order logic examples are missing an opening bracket, which makes them ambiguous, too;
  • regardless of where you put those brackets, neither of them will mean either of these two things.

So there is some more work left for you to do.

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  • $\begingroup$ It's worth noting that natural language isn't being completely irrational here: "someone" represents a free variable, say x, with LovesAllAnimals(x) -> LovesHumans(x), and no other constraints on x. In this case it is allowed to universally quantify over x. So it's very much the same vein of "take a generic person I know nothing (additional) about" which we are using in both natural language and formal logic. $\endgroup$
    – cody
    Dec 6, 2022 at 18:29
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    $\begingroup$ But why is it allowed? That is determined to whether the clause who loves all animals is interpreted as constraining (defining) or not, and the rules for that in natural language are subtle. $\endgroup$
    – reinierpost
    Dec 6, 2022 at 20:50
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Consider the following sentence:

The green men live on Mars.

How would you translate this into first order logic?

Be sure to note that the sentence does not claim that any green men actually exist.

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