Sunday, October 6, 2019

Syntactic Puzzle

(partially taken from Facebook post on August 16, 2018)
I came up with this puzzle while visiting Erich Groat at his cabin in upstate NY.

Find a word X, such that no matter how many times X is repeated, the result is a complete grammatical sentence. So X is a complete grammatical sentence, and so is XX, XXX, XXXX, etc.

Proposed Solution:
Consider the verb help. It can be used as an intransitive (with an implicit object), as in the following sentence:

(1)       I promise to help.

If the intransitive is used in the imperative, there is a complete sentence with n=1:

(2)       Help!

But help can also be used as a control verb without the infinitival marker to (once again with an implicit object):

(3)       a.         I will help wash the dishes.
            b.         Help wash the dishes!

Combining the intransitive help in (1-2) and the control help in (3) in the imperative, we have n=2:

(4)       Help help!

This sentence is a bit strange, but there does not seem to be anything wrong with it grammatically. But once the pattern in (4) is established, it is easy to generate:

(5)       a.         Help!                                       (n=1)
            b.         Help help!                               (n=2)
            c.         Help help help!                       (n=3)
            d.         Help help help help!               (n=4)

The crucial assumption here is that the intransitive and control verb help count as the same word, which seems reasonable. Pauline Jacobson also came up with this solution in response to my original Facebook post.

Are there any other words like this? fish comes close. The crucial property is that fish can be used as an intransitive verb, a transitive verb or a noun (similar comments hold for buffalo). One can have:

(6)       a.         Fish!                            (V)
b.         Fish fish.                     (NV)
c.         Fish fish fish.             (NVN)
            d.         Fish fish fish fish        (NNVV)
                        (NNV is an object relative clause construction: “Fish that fish fish”)
            e.         Fish fish fish fish fish (NNVVN)

But since fish does not embed a verb (unlike help) it is trickier to work out in the general case. If you are willing to allow the phrase fish fish (meaning something like genuine or real fish), then I think it works (a possibility brought up by Connor Quinn). I leave it to the reader to show this. Of course, this solution assumes that the verbal and nominal fish count as the same word.

Another possible solution involves the verb say (if one ignores quotation marks):

(7)       a.         Say!                 (a bit odd, but possible in context meaning “Say it!”)
            b.         Say “Say”!
            c.         Say “Say “Say””!
            d.         Say “Say “Say “Say”””!

Another possibility (proposed by Mark De Vos) is:

(8)       a.         No!
            b.         No! No!
            c.         No! No! No!

I think that this solution needs to be ruled out, since it seems that each instance of No! is a separate sentence as in:

(9)       Leave! Leave! Leave!             (repeated for emphasis)

So (9) does not count as a complete grammatical sentence, rather it counts as three complete grammatical sentences.

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