Predicate Abstraction Rule: Let α be a branching node with daughters β and γ, where β dominates only a lambda binder λx. Then, for any variable assignment g, [[α]]^g = λz. [[α]]^g[z/ x].
Why not instead have the semantic value of `Fx' be a function from assignments to extensions -- and the semantic value of `λx' be a function from functions from assignments to extensions to functions from individuals to extensions? Then the semantic value of `λx.Fx' can be calculated from the semantic values of `λx' and `Fx' as follows.
[[λx.Fx]]^g = [[λx]]^g (λg.[[Fx]]^g) = λi. [[Fx]]^g[i/x]
To see the similarity to the discussion of Kaplan's LD below notice that we could add a non-compositional rule to patch up the semantics of Kaplan's LD.
Quantifier Construction Rule: Let α be a branching node with daughters β and γ, where β dominates only a quantifier ∀x. Then, for any variable assignment g, [[α]]^g = ∀z. [[γ]]^g[z/ x)].
This would allow semantic calculations of the relevant constructions but would be explicitly non-compositional.
Why would one opt for having these non-compositional rules when the system can easily be transformed into a compositional system with no such non-compositional rules? Is it that much of a cost to have a few non-compositional rules like the predicate abstraction rule?
In the discussions of temporalism/eternalism in relation to the operator argument why don't the eternalists just respond by claiming that temporal constructions need special non-compositional rules?
Past Tense Construction Rule: Let α be a branching node with daughters β and γ, where β dominates only the temporal operator PAST. Then, for any time t, [[α]]^c,t = ∀t' before t. [[γ]]^t'.
In this case it seems that the eternalist theory with a few non-compositional tense rules is a bad theory. But if we are happy with the Predicate Abstraction Rule why should a few more such rules really bother us?
I guess what I have ended up doing was pushing some kind of slippery slope worry -- predicate abstraction is a gateway rule that will lead to nasty ones like the past tense construction rule. That is probably not a very threatening argument. But I just don't understand why we would opt for predicate abstraction in the first place when a truly compositional semantics is close at hand.