Propositional Ontology and Logical Atomism Francisco Rodr´ıguez-Consuegra Abstract. 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, I 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” no- tions) 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. ‘Logical atomism’ is perhaps an appropriate expression for referring to the most important philosophy developed by Bertrand Russell, which is associated with his most important contributions to logic, foundations and philosophy of mathematics. These contributions were mainly presented in Principia Mathe- matica [23]. I will regard logical atomism as extending, broadly, from 1910 to 1918. The ontology held by Russell in former periods of his philosophy (mainly in The Principles of Mathematics [13]) consists of terms (logical subjects), concepts (including properties and relations) and propositions, these last be- ing genuine entities, possibly to be regarded as a special sort of terms. Here, classes are terms, and are used to define numbers and the rest of mathemat- ical entities. The status of propositional functions should, on this scheme, be equivalent to that of concepts, and they, despite their being genuine primitive notions, could easily be introduced in terms of (albeit not reduced to) propo- sitions, given that a propositional function, when a value is assigned to its variable, is transformed into a proposition (as its final value).1 The problem in logical atomism, beginning in Principa or even earlier, is that propositions 1For the details of the whole construction in Principles see my [6]. 2 (in the sense that they are expressed by sentences) are no longer regarded as genuine entities, but merely as sets of entities, so the symbols used to denote them are no longer true symbols, but incomplete symbols. So, propositional functions can hardly be explained through propositions, unless we first clarify the ontological status of propositions. In Principia propositional functions, together with the usual quantifiers, are used to define-eliminate: descriptions; classes, through the axiom of re- ducibility (which is also named “axiom of classes”); relations, through the corresponding axiom of reducibility (or “axiom of relations”), and also the fun- damental notions of identity and membership. From the viewpoint of math- ematical ontology, that means that although numbers, the basic notion on which mathematics is founded, are defined as classes of classes. And, since classes are nothing but their members and everything that you may want to say about classes can be said through propositional functions, numbers, as well as the rest of the mathematical entities, can legitimately be reduced to propo- sitional functions. Yet, propositional functions are nothing but properties, so numbers are ultimately properties. Let us first examine how Russell describes the relationship between propo- sitional functions and propositions: “By a ‘propositional function’ we mean something which contains a variable x, and expresses a proposition as soon as a value is assigned to x” ([23]: vol. I, 38). So, propositional functions cannot even be understood without appealing to the notion of proposition. When you assign a value to the variable in a propositonal function (φxˆ) you obtain, as the value of the function, a proposition (φx). Since propositonal functions are genuine entities, and since the values you assign to their variables are entities too (values which belong to the classx ˆ(φx)), you accordingly have to obtain entities as final values. That is, you can have sentences as values of propo- sitional functions if these are just expressions; you can have propositions as values if propositional functions are treated as entities, but you cannot have propositional functions as entities and sentences as values. Therefore, propo- sitions should be true entities, and according to Russell, they are denoted by propositional functions. Although denotation was interpreted merely as a re- lation between words and entities (not as a logical relation between concepts, as it was in Principles), propositions should be seen as “objects”, as Russell writes: The function itself, φxˆ, is the single thing which ambiguously de- notes its many values; while φx, where x is not specified, is one of the denoted objects, with the ambiguity belonging to the manner of denoting. ([23]: vol. I, 40.) Yet the problem starts when Russell writes in Principia that propositions are not “single” entities, but sets of them which are in need of judgment to reach the necessary unity, judgment being a “multiple” relation. This theory 3 of judgment, which was held by Russell in unpublished manuscripts as early as 1906 or 1907 (see my [5]). However, it was not admitted in public until the official proclamation in 1910 ([23]: vol. I, 41–44) ([14]: 155–82), where judgment is openly explained as a multiple relation between the mind and several terms which, taken together, constitute a complex as a consequence of the judgment itself: Owing to the plurality of the objects of a single judgment, it fol- lows that what we call a “proposition” (in the sense in which is distinguished from the phrase expressing it) is not a single entity at all. That is to say, the phrase which expresses a proposition is what we call an “incomplete” symbol; it does not have meaning in itself, but requires some supplementation in order to acquire a complete meaning. This fact is somewhat concealed by the cir- cumstance that judgment in itself supplies a sufficient supplement, and that judgment in itself makes no verbal addition to the propo- sition. Thus “the proposition “Socrates is human”’ uses “Socrates is human” in a way which requires a supplement of some kind be- fore it acquires a complete meaning; but when I judge “Socrates is human,” the meaning is completed by the act of judging, and we no longer have an incomplete symbol. ([23]: vol. I, 44.) Thus, propositions are not single entities, but rather complexes of entities, which are joined through a special relation. However, propositions (sentences) are now regarded as incomplete symbols, for they need the complement of the judgment itself to reach full meaning, and it is unclear whether this suggests that propositions as complexes are, in fact, nothing at all. In principle, the complex, being a mere creation of the mind, would be something subjective, and this would be a sign that Russell was also trying to avoid one of the forms of Bradley’s paradox concerning the statement of any form of correspondence.3 The problem lies in explaining how it is possible that the (judging) mind is not a part of the complex which is supposed to be the object of the judg- ment, while it is, at the same time, to be taken as a part of the multiple relation of judging itself. For in this way the mind can impose a certain order among the terms with which it is acquainted, an order which does not exist on its own. Thus, the status of relations as terms, i.e. as genuine realities, is doubtful, given that, although we accede (through acquaintance) objectively to the constituents of the complex, yet the relation joining them is a mental product (the judging relation). And this, again, turns into the problem about the pre-eminence of relations over terms or vice versa. 2Curiously enough, these pages are part of the new essay that Russell wrote to replace the third section of [8] (which was here eliminated with no explanation), where he still presented the multiple relation theory only as one more possibility. 3”I.e., the relation of correspondence between two things is in need of another relation which in turn relates it to the two things, and so on. 4 On the other hand, it is truly surprising that, for Russell, an incomplete symbol (the proposition before being judged by the mind) can be transformed into a complete one (the proposition once judged by the mind); for an incom- plete symbol designates nothing, while a complete one designates something extra-linguistical. So the mind seems to have the incredible capacity to trans- form nothing into something. Russell’s continued defense in 1907 of the objective status of external rela- tions as the only correct alternative to monism, while he was really espousing, in other manuscripts, the infeasibility of treating propositions as complexes, as well as the subjectivity of the judging relation, is, in my view, incoherent. As usual, he had to find a compromise between two strong forces pulling in opposite directions. On the one hand, he needed to point out relations as terms4 to escape from the image he constructed about monism. On the other, he also needed to deny the reality of these relations as terms in one of the most important instances (that of propositions and judgments) in order to avoid Bradley’s paradox (now under the form of the impossibility of regarding complexes as terms), while, at the same time, he should avoid any correspon- dence theory of truth as falling down into another form of the same paradox.