Thanks for your contributions here, Ray! The question whether "that" can be a relative pronoun is indeed a very interesting topic. BillJ, I did not mean to minimize the importance of the distinction you wish to draw in formal syntax.
I only meant to minimize its importance insofar as deepcosmos's question in this thread is concerned. He thought the NP/DP "a paradox that rational models of decision-making fail to represent" might not contain a relative clause at all!
It has occurred to me that the relative pronoun "which" doesn't have a posessive form, either. (Well, we could say its possessive form is also "whose," but is it etymologically justified?) Do you also deny "which" is a relative pronoun?
I had exactly the same thought, Ray, when reflecting on BillJ's point that "that" lacks a "that's" possessive ("that's" only works as a contraction); the same may be said, as you say, of "which":
- This is the car which's headlights need to be replaced.
- This is the car whose headlights need to be replaced.
I learned two beautiful empirical arguments on this topic from the world-class syntacticians at U.C. Santa Cruz, the institution whence the almighty Geoffrey Pullum hails. The first argument is historical; the second relates to Pied Piping.
First, in Middle English, some relative clauses began with "which that" and "who that," indicating that the relative clause subordinator was perceived to exist separately from the relative pronoun and to have a separate function.
Second, Pied Piping is impossible with "that." We can't say the "the horse on that he rode" instead of "the horse on which he rode." Interestingly, it has occurred to me that this also applies to the "of"-variant of the possessive:
- This is the car the headlights of that need to be replaced.
- This is the car the headlights of which need to be replaced.
I shall add a third argument of my own. In recent years, it has occurred to me that the compatibility of relative-adverb relative clauses with "that"-relatives and their incompatibility with "which"-relatives shows "that" to be neutral:
- I remember the time when we saw a rattlesnake.
- I remember the time that we saw a rattlesnake.
- I remember the time which we saw a rattlesnake.
Finally, during my earliest days on the Grammar Exchange, a member named Chuncan Feng asked what I still believe to be the hardest question I have ever encountered on the Grammar Exchange. He asked about these two sentences:
- I phoned the two patients that every doctor will examine tomorrow.
- I phoned the two patients who(m) every doctor will examine tomorrow.
Initially, I thought that all that needed to be cleared up was the "who"-"whom" distinction. It seemed to me that the versions with "whom" and "that" were equivalent in meaning. But Seiichi Myoga convinced me otherwise.
That thread single-handedly led me to study generative grammar! I had to learn why, as Seichii convinced me, it was possible for the sentence with "that" to mean, not that 2 patients were called, but 2 x (the number of doctors)!
Roughly a decade later, with a lot of generative grammar and Chomskyan theory under my belt, I believe that I am a bit closer to understanding why that meaning is (may be?) possible, but I am still not crystal clear on it!
In any case, this conundrum suggests, too, that "that" is importantly different from a relative pronoun, even in "that"-relatives. But the importance of this is relative to the case! I do not believe deepcosmos's example here to be one.