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My understanding (mistaken, see below) is that in basic lambek/categorial grammar, the basic objects are strings. Does anyone know of variants where one can have multiple trees for the same string?

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I'm not entirely sure what you mean by "basic" Lambek/categorial grammar, or by "basic objects". But you can certainly have multiple trees for the same string, at least in a bidirectional categorial grammar, i.e. a categorial grammar with two concatenation operations (\ and /).

A bidirectional CG:

  1. The basic categories are n (for noun) and s (for sentence).
  2. If A and B are categories, then (A\B) and (A/B) are categories.
  3. If a is an expression of category A, and b is an expression of category A\B, then ab is an expression of category B.
  4. If a is an expression of category A/B, and b is an expression of category B, then ab is an expression of category A.

Lexicon:

  • men and women both have category n.
  • old has category n/n.
  • and has category (n\n)/n.

The expression (string) old men and women now has two different parses: (i) [old men] and women, and (ii) old [men and women].

[old men] and women:

    old       men           and          women
    n/n        n          (n\n)/n          n
    -------------
          |
       old men              and          women
          n               (n\n)/n          n
                          --------------------
                                    |
       old men                  and women
          n                        n\n
       ----------------------------------
                       |
               old men and women
                       n


old [men and women]:

    old       men           and          women
    n/n        n          (n\n)/n          n
                          --------------------
                                    |
    old       men               and women
    n/n        n                   n\n
              ---------------------------
                          |
    old             men and women
    n/n                   n
    -----------------------------
                  |
          old men and women
                  n
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  • Thank you for your answer. To clarify, my question concerns whether in the deductive system presented by lambek, the 2 terms of type n/s that you have constructed above are equal.
    – Colin
    Apr 1 '15 at 20:11
  • It seems they are not, though.
    – Colin
    Apr 1 '15 at 20:31
  • In a natural deduction system (or sequent calculus) for a Lambek grammar, you would have two proofs of "old men and women", with both proofs resulting in the same syntactic category but differing in semantics. What I gave was a categorial grammar derivation, but the idea is the same in a Lambek grammar proof. See Carpenter 1998 Type-Logical Semantics (ch. 5 in particular) for more details. Apr 2 '15 at 2:48
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Yes, in the Combinatory Categorial Grammar, type-raising X->T/(T\X) yields trivially ambiguous derivations such as:

  X        Y\X
---------------<
       Y

vs

  X        Y\X
------->T
Y/(Y\X)
-------------->
     Y

While this seems counterproductive, type-raising is invoked to allow a category to become a functor over a functor (see 'Argument Cluster Coordination' in https://www.cl.cam.ac.uk/teaching/1011/L107/clark-lecture3.pdf)

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