16.1.3 Formulas as Trees

Thinking of the FOL formula as a tree, we can use dominance constraints to underspecify the structural relations holding between these three components and thereby get one compact representation for the two readings. To do this, we use a meta-language which is underspecified with respect to dominance and therefore can describe several lambda terms. The intuition is this. Suppose we represent the meaning of the above sentence by the following two trees:

apply apply ' ` ' ` ' ` ' ` apply lambda(2) apply lambda(1) ' ` ! ' ` ! ' ` ! ' ` ! a guru apply every yogi apply ' ` ' ` ' ` ' ` apply lambda(1) apply lambda(2) ' ` ! ' ` ! ' ` ! ' ` ! every yogi apply a guru apply ' ` ' ` ' ` ' ` apply var(1) apply var(1) ' ` ' ` ' ` ' ` has var(2) has var(2)

The indices (1) and (2) on the lambda and var nodes represent binding links between quantifier variables (lambda nodes) and predicate variables (var nodes). For instance, the (2) co-indexing indicates that the quantifier variable in $\lambda P (\exists x (guru(x) \wedge P(x))$ binds the lambda variable z in the predicate $\lambda z.\lambda u.has(z,u)$ .