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I'm considering the sentence

Some man and some woman visited a garden

Obviously it's not 100% unambiguous how many gardens there are, but I think most people would agree there is just one common garden.

The sentence has the following formal grammar:

[NP [NP Some man] and [NP some woman]] [VP visited [NP a garden]]

My problem is that since a VP has the type entity to truth value (E -> T) the only way I can think of interpreting it is λv.∃x.garden(x)∧visit(v,x), however now how the only path I see forward leads me to

∃a.man(a)∧(∃x.garden(x)∧visit(v,x)) ∧ ∃a.woman(b)∧(∃x.garden(x)∧visit(b,x))

Which is exactly what we didn't want: that that the garden might be different persons.

To summarize, given the way the VP is interpreted, is there any way we can ever end up with the intended formula (below)? Or is my interpretation of the VP simply incorrect?

∃a.man(a)∧(∃a.woman(b)∧(∃x.garden(x)∧visit(a,x)∧visit(b,x)))

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    Note that friend is itself a relational predicate, and the ambiguity stems from the fact that a friend may mean either ∃x.friend(x,a) or ∃y.friend(y,b). If you don't represent that relationship, it's not clear that lambdafication will do any good. – jlawler Apr 26 '13 at 17:42
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    You could change "a friend" to "a garden" to remove the ambiguity that jlawler covered. – prash Apr 26 '13 at 17:48
  • I balk at the thought of writing a proper answer to this question, because I don't know if I'll be able to do it justice with a short answer. But I will mention the terms quantifier scope and Cooper storage. You can get a nice overview in this set of slides. – prash Apr 26 '13 at 18:08
  • OK, now the problem is whether they visited the [identical] garden together, in company, whether they visited separately but as a joint venture, or whether their visits were completely unconnected. Ready, aim, parenthesize! – jlawler Apr 26 '13 at 18:51
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I'll attempt an answer.

The key issue is the contrast between lay intuition, and mathematical logic. Lay intuition tells us that nouns and verbs are the most 'contentful' parts of a sentence. Mathematical logic brings its rules for operators and their scopes.

If we have the expression ∃x(F(x)), we can't have the x scoping wider than the outermost parentheses here. If an x had some sort of a universal scope, for example, over G too in G(x)∧∃x(F(x)), we'd soon have a formalism that only talks about constant terms in the universe. Such xs would be useless as variables.

Now, note that the logical terms that we introduce for describing sentences are based on the principle of compositionality. If we had used "man" => λx.man(x), and "some" => λF∃x.F(x) to derive "some man" => ∃x.man(x), this would not correspond to what we understand of the phrase "some man". Instead, it would correspond to "there is a man".

The proper expression for "some" would be λPλQ∃x.P(x)∧Q(x). So, "some man" => λQ∃x.man(x)∧Q(x), and (after a bit of α-conversion to avoid name clashes) "some woman" => λR∃y.woman(y)∧R(y).

Here are two more pieces of the puzzle: "visit" => λSλu.S(λv(visit(v,u))), and "and" => λHλJ.H∧J.

We now have NPs that have become functions that accept Vs and VPs as arguments. Determiners, that play a subordinate role in Phrase Structures, play a key role here in controlling the scope of variables, and the order in which functions apply over arguments.

Given this, if you were to apply the object over the verb, and then apply the subject over the resultant, you'd have the man and the woman each visiting a garden, that may or may not be the same garden. However, if you were to apply the subject over the verb, and then apply the object over the resultant, you'd have them both visiting the same garden.

In all this, we haven't brought in (Neo-)Davidsonian Event semantics, so we have made no commitments at all about whether or not the visit(s) to the garden(s) happened at the same time.

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  • This doesn't look right: ∃y.woman(y)→R(y) will always be true, independent of R, as long as our domain includes persons that aren't women. – Thomas Ahle Apr 27 '13 at 15:31
  • The . means that you should take the widest possible interpretation. So, interpret it as λR∃y(woman(y)→R(y)). People use this notation because some of us can't read expressions as easily if there are too many brackets. – prash Apr 27 '13 at 15:41
  • But say there is a non-woman in my domain, then there will exist y so the anticedent of woman(y)→R(y) is false, meaning the entire thing ∃y(woman(y)→R(y)) is true without even considering R. – Thomas Ahle Apr 27 '13 at 15:42
  • You wrote ∃y(woman(y)→R(y)) twice. Note that I wrote λR∃y(woman(y)→R(y)). R(y) is not a free-standing expression. – prash Apr 27 '13 at 15:46
  • I'm not sure if I got your point. But if you're asking what prevents λR∃y(woman(y)→R(y)) from taking λx.garden(x) as an argument, the answer is that (1) you'd have to jump across many other words to do that. I have not seen an need for doing that in English so far, (2) you'll have all the other terms from all the other words waiting to be resolved. You don't have a successful parse until you have dealt with all the words, and (3) discourse representation theory, which we have not spoken about, comes with its own rules about what entities are available in the universe beyond thesentence – prash Apr 27 '13 at 16:10
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I ended up finding a nice way to do it. Inspired by the way "All men visited a garden" which doesn't have any problems with the high number of entities, I make my NP have the meaning

λvp.∃x(man(x) ∧ ∃y(woman(y) ∧ ∀z(z=x ∨ z=y → vp z)))

That is, I let

np1 `and` np2 = λvp.np1(λx.np2(λy.∀z.z=x ∨ z=y → vp z))

It looks slightly less nice than what's been suggested, and you'll need different ands for different typesl, but the nice thing is that it works even if our original sentence was something complicated like:

Every man and every woman and some child ...

Which simply becomes

λvp.∀x(man(x) → ∀y(woman(y) → ∃c(child(c) ∧ ∀z((z=x ∨ z=y v z=c) → vp(z)))))
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  • So, what does one do with such a nicely represented sentence? Is it machine-washable? – jlawler Apr 27 '13 at 18:35
  • Actually I'm not quite sure. I'm using dynamic semantics, and the whole advantage that you can write "If a man visit a garden, he cuts it", without scoping problems, disappear with the above construction, because "a garden" gets trapped in so many scoping layers. – Thomas Ahle Apr 28 '13 at 17:22
  • That's generally been my experience with the proper treatment of quantifiers. I don't require consistency in a semantics (because certainly natural language semantics doesn't), so 2nd order QPC is fine in principle, though I've never needed to rely on it. – jlawler Apr 28 '13 at 17:38

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