Crossing corefernece in discourse representation theory

Sentences a, ith crossing coreference (Bach-Peters-sentences) are notoriously difficult to eq/lain ill a natural nmnner. An intriguing parallel with certain properties of t'mlog suggests a modificatiml to Discour~ Represenlation Thexn'y which allows a simple and coherent explanation of these, and related, sentences. The Probteln In English there is due type of sentence that has caused major problems for practically all linguistic theories that have tried to explain it, and none of Ihe explanations put forward is very convincing. The sentences in question are those with cross'lag corcferenee, the so-called Bach-Peters-sentences. The standard exam-pies arc: l) "The htutter who shot al:, it hJl; the ]ion that chased him and, with cxpiicit quarldfier expressions: 2) Every man who wants it will get the prize he deserves What is the difficulty wilh this type of sentence? 'Illey contaiu two uoun phrases each of which contains a pronoun that refers to the other notnl phrase, and tile first pronoun is filrthermore a case of "backwards anaphora", or "cataphonl". These sentences are admittedly tare, bet sentences with simple (non-crossing) camphora are quite frequent ill real world English (Carden 1982). And Bach-Peters-sentences are nevertheless perlectly reguhu', and so they should find a natural exph/nation. Moreover, they are key exanlplcs of sentences where Gala. pllora cannot, in principle, be mphlced by anaphora (cf. also Mittwoch 1983). This is imporlant because one of tile standard approaches to cataphora has been to define it away as stylistic variant of anaphm'a, which can be "rectified" by a simple transposition. Ill other words: Since we have to find a way to explain cata-phora for Bach-F'eters-sentences anyway, we carl save us the lfouble to devise such tricks for the siutpler cases. Are Bach-Peters-Pronouns Descriptional l'ronouns? The non-reAucibility of cataphofic to anaphoric pronouns in Bach-Peters-sentences becomes clear if we try to explain them in tile traditional manaer. It seems that both of/tie two traditional interpretations of prononns, tile "de~:rip-tional" as well as the "denotafionar' one, fail to explain the intuitive truth condi-lions of Baeh-l'ete~.-sentences. In Transformational Grammar the deseriptional approach is taken, trod pronouns are always expanded to the sl,rface syntax form of the norm pbaase they anaphoric~dly refer to (in other words, pronominalization is an obligalxlry cyclic rule). But then we get, for tile example above, a double infinite embedding of relative clauses: The hunter who shot at (the lion that chased (the hunter who shot at (tile lion that chased. . …