A critique of The Vastness of Natural Languages by Langendoen and Postal
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Abstract
This paper looks at an argument in 'On the vastness of Natural Languages' by D. T. Langendoen and P. M. Postal.
The conclusion is that it does not pass mathematical muster.
The salient points are
saying "language rules impose no size limits" does not mean that one can say "there are arbitrarily large entities"; it simply means that one does not avail oneself of the former assumption, the latter assumption is just that: an assumption (or better axiom), not a consequence of not using the former
the existence proof for co-ordinate projections is mathematically unsound; it establishes that "Tom and Jerry" is a sentence built from the set {Laurel, Hardy}
the proof of the main result uses the assumption "there are projections of any given cardinality"; this assumption is equivalent to the conclusion of the theorem and this reduces the main theorem to a trivial tautology