Propositional ontology and logical atomism

In the following I will briefly indicate the role of propositional functions in Principia, then point out the way in which they continue to be introduced through propositions, in spite of the doubtful status of these pseudo-entities. I will then try to see if the notion of judgment, which is used by Russell to explain propositions, can meet the requirements which are needed in a serious, coherent ontology. After a survey of the different attempts to build up a convincing notion of proposition and judgment carried out by Russell between 1910 and 1918, 1 will conclude that the ontology of logical atomism was finally a failure, and so that the mathematical ontology usually associated with Russell's logicism was also a failure. In doing so, I will often show that almost all the problems in which Russell was involved in that period are somehow dependent upon what I will call Bradley's paradox of relations, according to which we cannot consider relations (or other similar "incomplete" notions) as genuine terms, or logical subjects. If we do that, then we should give an account of the way in which these relational terms are in turn related to other terms. But this leads to an unavoidable infinite regress.