Fwd: linear and non-linear terms

[Ash Asudeh]

Dear Carl and Howard (especially Howard)

I realized belatedly that the message below comes across as more in your
face than I intended it to. I've since looked at Howard's FG paper from
this year and see the linguistic consequences.

The basic intuition dovetails nicely with Carl's recent discussion of
logics and the Curry-Howard isomorphism. This excerpt from Howard's FG
paper is helpful:

"The relevant implication connective is based on the principle that the
premises in a deduction must combine in a non-trivial way to produce the
conclusion. In terms of natural deduction, the rule of implication
introduction requires the non-trivial discharge of an assumption. This
corresponds to a consequence relation in which the combination of premises
is not subject to the structural rule of Weakening." (Gregory 2002, p. 54)

In particular, this reveals that the issue is not vacuous abstraction in
the *meaning language* or linguistic semantics, just in the proof terms.
As Howard points out in his paper, Weakening is excluded in linear logic
(unless the fragment includes the "of course" connective, !), as is
Contraction. Relevance logic retains Contraction however. (I should point
out, by the way, that the reason Howard gives for retaining Contraction,
which is argument re-use in, e.g., Equi (see his footnote 9) has been
dealt with for Glue Semantics, which also dispenses with Contraction in
the !-free fragment. In fact, it has been argued that the lack of
Contraction leads to a situation in which the scope facts for Equi simply
fall out of the logic. This is also discussed in the coordination paper I
mentioned recently).

The reason I got alarmed and came across like a jerk in the message below
is that I actually need vacuous abstraction in the meaning langauge in the
theory of resumption that I'm working on for my thesis (at least I'm
pretty sure I need it, but the thesis is still in progress). We also use
vacuous abstraction in our coordination paper, but there its use could be
eliminated; it just makes it more straightforward to deal with the
recursion in n-ary coordination.

Ash

 On Sun, 20 Oct 2002, Ash Asudeh wrote:

>
> Dear Carl and Howard,
>
> I was just wondering what kind of linguistic motivation there is for
> banning vacuous abstraction, if there is in fact any. Isn't a lambda term
> with vacuous abstraction just another kind of constant function?
>
> Ash
>