$H(x,y)$ = "$x$ has $y$", $M(x)$ = $x$ is a man, and $I(x)$ = $x$ has an iPhone. Given these predicates, the English statement that has to be translated into its logical equivalent is "Every man has an iPhone." We can assume the domain is all living people.
Roughly, I interpreted the statement above to be that for every $x$ where $x$ is a man, it would imply that there exists some $y$ where $y$ is an iPhone and the man, or $x$, has the iPhone, $y$. The logical equivalent I came up with is:
$$
\forall x (M(x) \implies \exists y(I(y) \ \land \ H(x,y))
$$
Is my interpretation of the English statement correct? Also, just wondering, can the $\exists$ quantifier go with the $\forall$ quantifier in the beginning of the statement; why or why not?
Best Answer
The English sentence is an example of scope ambiguity. There are two readings1:
For every man it holds that he has at least one iPhone (and this might be different iPhones from man to man)
$\forall x (M(x) \implies \exists y(I(y) \ \land \ H(x,y))$
There is at least one iPhone which every man has (and this is one and the same iPhone, not excluding the possibility that some men might additionally have another iPhone)
$\exists y I(y)\ \land \ \forall x(M(x) \implies H(x,y))$
Intuitively, your interpreation (1.) is the more plausible one. This is called the linear reading of the sentence, because the surface structure in the sentence corresponds to the underlying logical form (where the univeral quantification has scope over the existential quantification).
If you understand the sentence as "Every man has an iPhone and it is not necessarily one iPhone that everyone has" (which is what you would intuitively understand), then your formalisation is correct.
The other reading is called the inverted reading: Here, the scope relation is the reverse of the one that is reflected in the surface sentence, i.e. the existential quantifier has scope over the unverisal quantifier although it appears later in the sentence.
If you understand the sentence as "There is (at least) one iPhone and this iPhone is owned by every single man", then (2.) is the correct formalisation. This is, in this case, not the most intuitive reading, but possible as well, because the natural language sentence is simply ambiguous. You could imagine other sentences like "Every child sings a song" where it is more understandable to assume that there is a song that is sung by everyone collectively instead of each child singing on its own.
So, yes, both your formalisation (my 1.) and a formalisation with the existential quantifier having scope over the rest (2.) are possible, but they have two different meanings, among which (1.) is more plausible in this context, but since natural language is often ambiguous (surface structure doesn't always reflect underlying logical structurein a 1:1 relationship), the formula shown in (2.) is also possible (but then with a different meaning).
Edit: Revisting your question whether "the ∃ quantifier [can] go with the ∀ quantifier in the beginning of the statement":
Changing the order of the quantifiers does change the logical meaning of the formula. If you move the existential quantifier in front of the univeral one, you get the alternative meaning I formulated in (2.).
Whether you then write $\exists y I(y) \land \forall x (...)$ or $\exists y \forall x I(y) \land (...)$, i.e. whether you move all quantifiers to the very front or place them just at the beginning of their respective scope area is just a matter of style - usually, it is treated the way that the quantifier should not be separated from its restriction, i.e. from where the variables it quantifies over appear in the formula; but the crucial feature that determines the scopal relations is the order of the quantifiers themselves, not the arrangement between the predicates (as long as the quantifier appears before the predicates containing the varible it quantifiers over, otherwise it doesn't make sense anyway), so both of the formulae mentioned in this last paragraph are equivalent in terms of the scopal relationship, as the order between univeral and existential quantifier is the same.
1 My formalisation was based on the assumption that you defined $I(x)$ as "$x$ is an iPhone" (instead of "$x$ has an iPhone") - I don't know whether this was a typo or intended; if you want to treat it as "has an iPhone", then the answer by @Lorenzo is correct and mine is wrong, but it seemed more useful to me to define $I(x)$ as "$x$ is an iPhone" (which I thought you did) so my whole post is based on that definition.