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I think that the English sentence

Everyone admires someone who works hard.

...has two plausible, but non-equivalent, translations into predicate logic. Here they are:

$$ \forall x \, \forall y \, (Wy \rightarrow Axy )$$ $$ \forall x \, \exists y \, (Wy \; \wedge \; Axy )$$

...where $Wy$ says that "$y$ works hard" and $Axy$ says that "$x$ admires $y$".

The first translation interprets the original sentence as saying that every person $x$ admires any person $y$ who works hard. In other words: working hard guarantees universal admiration.

The second translation, on the other hand, interprets the original sentence as saying that, for every person $x$ there is at least one person $y$ such that (1) $y$ works hard and (2) $x$ admires $y$. IOW, it reads the original sentence as asserting the (rather curious) fact that not only each person admires a non-empty collection of people, but that among these objects of admiration there always happens to be at least one person who works hard.

I find the first interpretation far more natural than the second one, hence I was shocked to discover that my textbook mentions only the second one.

Neither I nor the author of my textbook is a native speaker of English, so I thought I'd ask for other opinions.

If my analysis is correct, is this particular source of ambiguity well known? IOW, does it have a name that one could Google for?


BTW, my preferred translation (i.e. the first one) derives from common colloquial expressions of the form

You have to admire someone who $Z$.

...where $Z$ stands for a conduct or deed that is so laudable that it simply renders admiration unavoidable, so much so, in fact, that one could safely assert the universal rule:

Everyone admires someone who $Z$.

Granted, this interpretation becomes less and less compelling in the measure that $Z$ becomes unimpressive. E.g.

Everyone admires someone who goes to work after highschool in order to put his younger siblings through college.

...sounds to me like it can admit only the first translation, whereas

Everyone admires someone who has ten toes.

...sounds to me absurd enough to admit either translation.

The case of $Z$ = "works hard", may be somewhere in-between. (After all, there people who rather look down on hard work as a sign of stupidity.)

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  • $\begingroup$ This is one of those sentences that may sound good when it comes out of the mouth, but loses a lot in being written down. I find this to be more of a lesson in appreciating the ambivalence of written language, which lesson you seem already to understand fully. $\endgroup$ – Lee Mosher Feb 19 '17 at 14:42
  • $\begingroup$ @LeeMosher: Point taken. The book I'm using is a book on logic for linguists. This book is very nice in many ways, but it has a perverse emphasis on the problem of translating English sentences to formal logic. If this were possible, one would not need formal logic to begin with. IOW, the reason for creating formal languages in the first place is precisely because human language is intractably ambivalent. Mind boggling... $\endgroup$ – kjo Feb 19 '17 at 14:49
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The first translation interpretation is linguistically much more frequent than the second one. It has to do with the aspect of the verb "admires" which can be understood as a punctual action with a general meaning or as the ongoing process "is admiring." The latter gives your second translation.

So I would say your first translation is right.

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