Discourse Representation Theory

Fourteenth-century logicians such as Burley and Buridan note that quantifiers not only express relations between terms but also introduce items in discourse that can be referred to anaphorically. The problem this raises is not one of truth conditions, as we have seen—it is not difficult to give first-order representations of sentences such as ‘an animal is running and it is a man’ or ‘Every farmer who owns a donkey beats it’— but rather of compositionality. The problem, in other words, is typically not a lack of appropriate truth conditions but a rule-governed way of generating them. In Aristotelian logic, we can represent ‘an animal is running’ as ‘Some A is R,’ but we then have no way of indicating that the running animal is a man. Appending another categorical proposition will not help. In first-order logic, similarly, we can represent ‘an animal is running’ as ∃x(Ax ∧ Rx). Having done that, however, we have no way of adding another formula representing that the x in question is a man. What we have to do in both cases is extend the original representation, writing ‘Some A is R and M’ or ∃x(Ax ∧ Rx ∧ Mx). Since the anaphora occurs within the same sentence, we could perhaps write rules requiring this. But similar anaphora can occur across sentential boundaries. Buridan’s example could just as easily have been ‘an animal is running. It is a man.’ These could moreover be separated by intervening discourse: ‘an animal is running. People look on in surprise. It is uncommon to see such a sight in the square, amidst the crowds, in the midday heat. It is a man.’ This makes a strategy of waiting to close off the representation of the first sentence until all later anaphors have been collected implausible. Sometimes, the problem is a lack of appropriate truth conditions. Consider this discourse: