The brilliant mathematician John claims to be The Semantics

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					             The brilliant mathematician John claims to be:
         The Semantics of Modal Compatibility Relative Clauses

                         Alexander Grosu, Tel Aviv University
                Manfred Krifka, Humboldt University Berlin & ZAS Berlin

This talk deals with relative clauses of the following kind:

(1) a. The brilliant mathematician that John claims to be _ should have solved the problem.
    b. The happy couple that Charles and Diana were supposed to be _ never in fact existed.
    c. The abominable atrocity that the stoning of those women was _ should not go unpunished.
    d. The idealist he once was _ sincerely believed in the principles of Trotzkyism.

Such relative clauses have their internal “gap” is in post-copular position, as indicated in
(1). They must contain some modal (or temporal) operator, and must occur within a sen-
tence that contains a modal operator, like claim and should in (1.a). Without postcopular
gap, the modal operators are incidental and can be omitted, cf. (2); with it they are essential,
cf. (3).

(2) a. The brilliant mathematician that John claims to know _ should have solved the problem.
    b. The brilliant mathematician that John knows _ should have solved the problem.
    c. The brilliant mathematician that John claims to know _ solved the problem.

(3) a. #The brilliant mathematician that John is _ should have solved the problem.
    b. #The brilliant mathematician that John claims to be _ solved the problem.

Furthermore, the modal operators must be be compatible with each other, cf. (4), hence the
term Modal Compatibility Relatives, or MCR, for short.

(4) a. #The brilliant mathematician John claims to be _ should not go unpunished.
    b. #The idealist John once was _ sincerely believes in the principles of Trotzkyism.

We will present the following analysis of MCRs that explains these properties:

(i) The expression the brilliant mathematician that John claims to be refers to an individ-
ual concept that is only defined for the worlds of John’s claims; it picks out for each of
those worlds the individual that is John and that is a brilliant mathematician. Notice that
this concept is defined only if John claims to be a brilliant mathematician.

(ii) As the individual concept does exist only in certain modally accessible possible worlds
(the worlds of John’s claims), predications on this individual concept must be modalized
as well. In particular, the accessibility relation of the modal operator must access possible
worlds for which the individual concept is defined, that is, the two modalities must be
compatible with each other.

(iii) Examples without modal operators are odd because they make direct statements about
the world of evaluation, and for those the relative clause doesn’t contribute to the meaning:
The term the brilliant mathematician that John is, if defined, has the same meaning as
John.

We will show that (i) can be achieved within standard syntactic assumptions about relative
clauses and semantic assumptions about the definite article as an operator that identifies
the maximal unique object of a class. In the case at hand, it picks out the maximal individ-
ual concept with the property of being defined for the worlds of John’s claims, of being a
brilliant mathematician, and of being identical to John. An indefinite article could not pos-
sibly construct such an individual concept; this explains why MCRs must be construed
with a definite article.

(5) *A brilliant mathematician that John / a friend of mine supposedly is _
    should have solved the problem.

This is the informal sketch of our analysis. As for the formal implementation, we will con-
sider a standard model of intensional logic in they style of Montague (1973) and Gupta
(1980) in which definite descriptions are individual concepts and predicates apply to indi-
vidual concepts.

In this model a noun modified by a relative clause with modal operator can be interpreted
as follows, where i is a variable over worlds and x is a variable over individual concepts:

(6) [[ [N[ N brilliant mathematician] λ1[(that) supposedly [John is t1]]]]]
    = λi λx [∀i′∈DOM(x)[[[brilliant mathem.]](i′)(x)] ∧
          [[λ1[(that) supposedly [John is t1]]]](i)(x)]
    = λiλx[ ∀i′∈DOM(x)[BRILLIANT MATHEMATICIAN(i′)(x)]
         ∧ ∀i″∈SUPPOSED(i)[JOHN(i″) = x(i″)]]

The head noun brilliant mathematician applies to individual concepts x that are brilliant
mathematicians in all worlds i′ for which they are defined (which means that the individal
x(i′) is a brilliant mathematician at i′). The restrictive relative clause (that) supposedly John
is selects from this set those x that are identical to John in all worlds i″ that are compatible
with what is supposed to be true in i. Notice that in this analysis, the description brilliant
mathematician effectively gets under the scope of the operator supposedly even though it
is not c-commanded by it, as the individual concepts x are necessarily brilliant mathemati-
cians in all words in which they are defined. This makes it unnessary to assume head
movement of brilliant mathematician from the relative clause, as variously proposed (cf.
Bhatt 2002 and Hulsey & Sauerland 2004 for recent discussion).

The definite article presupposes that there is, for each world i, only one individual concept
x that satisfies the requirements. Uniqueness is only guraranteed for those worlds i″ that
are in SUPPOSED(i), hence the individual concept x is restricted to those worlds. The opera-
tor on individual concepts that achieves this is the following one:

(7) ιx[...x...] = y iff   (i) [...y...]
                          (ii) for all y′ with DOM(y′) = DOM(y) and [...y′...]: y′ = y
                          (iii)y is the individual concept with the largest domain
                               satisfying (i) and (ii).

(8) [[ [NP [Det the] [N brilliant mathematician that supposedly John is]]]]
    = λi ιx[∀i′∈DOM(x)[BRILLIANT MATHEMATICIAN*(i′)(x(i′))]
         ∧ ∀i″∈SUPPOSED(i)[JOHN(i″) = x(i″)]]

For each world i, this yields the indivdual concept x that is a brilliant mathematician
whereever it is defined, and identical to John for the worlds that are compatible with what
is supposed to be true at i. We can predicate a property, like should have solved the prob-
lem, on this individual concept:
(9) [[ [[NP the brilliant mathematician that supposedly John is] [VP should have solved the
    problem]] ]]
    = λi[[[should have solved the problem]](i)([[the brilliant mathem. that supposedly
         John is]](i))]
    = λi ∀i′∈ SHOULD(i)[SOLVE.THE.PROBLEM(i′)([[the brilliant mathem. that supposedly
         John is]](i))]
    = λi ∀i′∈ SHOULD(i)[SOLVE.THE.PROBLEM(i′)
         (ιx[∀i′∈DOM(x)[BRILLIANT MATHEMATICIAN*(i′)(x(i′))] ∧
              ∀i″∈SUPPOSED(i)[JOHN(i″) = x(i″)]])]

In prose, for all worlds i′ that should be the case in i, the individual concept x that is a bril-
liant mathematician and identical to John in all worlds that are compatible with what is
supposed to be true at i solved the problem in i′. For this to be the case, x must exist in i′.