Categorial Grammar and Discourse

∀v((u1 6= v ∧ . . . ∧ (un 6= v) → V (v)(i) = V (v)(j)) ,

a term which expresses that states i and j differ at most in u₁ ,. . . ,u_n; i[ ]j
will stand for the formula ∀v(V (v)(i) = V (v)(j)). We impose the following
axioms.

AX1 ∀i∀v∀x ∃j (i [v]j ∧ V(v)(j) = x)

AX2 ∀i∀j (i [ ]j → i = j)

AX3 u 6= u⁰, for each two different discourse referents (con-
stants of type π) u and u⁰ .

AX1 requires that for each state, each pigeon-hole and each object, there
must be a second state that is just like the first one, except that the given
object is an occupant of the given pigeon-hole. AX2 says that two states
cannot be different if they agree in all pigeon-holes. AX3 makes sure that
different discourse referents refer to different pigeon-holes, so that an update
on one discourse referent will not result in a change in some other discourse
referent’s value.

Type logic enriched with these three first-order non-logical axioms has
the very useful property that it allows us to have a form of the ‘unselec-
tive binding’ that seems to be omnipresent in natural language (see Lewis
[1975]). Since states correspond to lists of items, quantifying over states cor-
responds to quantifying over such lists. The following lemma gives a precise
formulation of this phenomenon; it has an elementary proof.

Unselective Binding Lemma. Let u₁ ,. . . ,u_n be constants of type π, let
x 1,... ,x_n be distinct variables of type e, let φ be a formula that does not
contain j and let = be the result of the simultaneous substitution of V(u 1 )(j)
for x 1 and ... and V(u_n)(j) for x_n in φ, then:

=AX ∀i (∃j (i [u 1,... ,u n ]j ∧ φ⁰) → ∃x 1... ∃x n φ)

=AX ∀i (∀j (i [u 1,... ,U n ]j → ψ⁰) ^ ∀x 1... ∀x n φ)

We now come to the emulation of the DRT language in type logic. Let us fix
some type s variable i and define (u = = V(u)(i) for each discourse referent

(constant of type π) u and (t = = t for each type e term t, and let us agree

to write

P τ for λiP (τ=,

τ 1 Rτ 2    for λi (R(τ 1 = (τ 2= ),

τ 1 is τ2 for λi((τ 1 = = (τ2= ),

<<   4   5   6   7   8   9   10   >>

More intriguing information