This paper presents a sound and complete proof system for the first order fragment of Discourse Representation Theory. Since the inferences that human language users draw from the verbal input they receive for the most transcend the capacities of such a system, it can be no more than a basis on which more powerful systems, which are capable of producing those inferences, may then be built. Nevertheless, even within the general setting of first order logic the structure of the formulas (...) of DRS-languages, i.e. of the Discourse Representation Structures suggest for the components of such a system inference rules that differ somewhat from those usually found in proof systems for the first order predicate calculus and which are, we believe, more in keeping with inference patterns that are actually employed in common sense reasoning.This is why we have decided to publish the present exercise, in spite of the fact that it is not one for which a great deal of originality could be claimed. In fact, it could be argued that the problem addressed in this paper was solved when Gödel first established the completeness of the system of Principia Mathematica for first order logic. For the DRS-languages we consider here are straightforwardly intertranslatable with standard formulations of the predicate calculus; in fact the translations are so straightforward that any sound and complete proof system for first order logic can be used as a sound and complete proof system for DRSs: simply translate the DRSs into formulas of predicate logic and then proceed as usual. As a matter of fact, this is how one has chosen to proceed in some implementations of DRT, which involve inferencing as well as semantic representation; an example is the Lex system developed jointly by IBM and the University of Tübingen ). (shrink)

Russell alleged that the version of the cosmological argument he debated with Copleston involved type confusions, but the definitions of plural descriptive functions and the ancestral in Principia Mathematica can be used to reformulate the argument in a type-safe way via a notion of causally self-sufficient classes. Although the argument depends on the assumption that the class of contingent things is not causally self-sufficient, if that assumption is weakened to say only that it may not be so, then a new (...) modalized version emerges which has similarities with recent ontological arguments but weaker and more intuitive assumptions. (Familiarity with Principia notation and conventions is presupposed throughout.). (shrink)