January 2, 2011 11:18 History and Philosophy of Logic types HISTORY AND PHILOSOPHY OF LOGIC, 00 (Month 200x), 1–21 The Versatility of Universality in Principia Mathematica Brice Halimi Paris Ouest University, France Received 00 Month 200x; final version received 00 Month 200x In this article, I examine the ramified type theory set out in the first edition of Russell and Whitehead’s Principia Mathematica. My starting point is the “no loss of generality” problem: Russell, in the Introduction ((Russell and Whitehead 1910), p. 53-54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell’s claim and to solve the “no loss of generality” problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (“ramif-types”). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other, makes it possible to give a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the “no loss of generality” problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed. Keywords: Principia Mathematica, ramified type theory, predicativity, epistemic diagram, reducibility, logical universalism. 1. The Principia’s theory of types and the “no loss of generality” problem 1.1. The “no loss of generality” problem The Introduction of the (first edition of) Principia Mathematica suggests that in practice only pred- icative predicate variables are adopted in the formal language and that there is “no loss of generality” in doing so: It is important to observe that all possible functions in the above hierarchy [the hierarchy of functions of increasing orders] can be obtained by means of predicative functions and apparent variables. [. ] Thus we need not introduce as variables any functions except predicative functions1. This is what I shall refer to as the “dispensability argument,” according to which one can do with pred- icative variables only. Russell and Whitehead continue: It should be observed that, in virtue of the manner in which our hierarchy of functions was generated, non- predicative functions always result from such as are predicative by means of generalization. Hence it is unnecessary to introduce a special notation for non-predicative functions of a given order and taking ar- guments of a given order [. ]; the only functions which will be so used [as apparent variables] will be predicative functions, because, as we have just seen, this restriction involves no loss of generality2. This is odd and requires explanation. How can there be no loss of generality? If non-predicative variables range over non-predicative attributes, then there is certainly a loss of generality. The solution of that problem is the main purpose of this paper. Before analyzing and explaining in greater detail the notions at stake here (“order,” “predicate vari- able,” “predicative,” “non-predicative”), let me set out the principle of the solution of the problem which I advocate. Were Principia to adopt non-predicative terms (variables and complex predicates obtained 1(Russell and Whitehead 1910), p. 53-54. 2(Russell and Whitehead 1910), p. 165. History and Philosophy of Logic ISSN: 0144-5340 print/ISSN 1464-5149 online © 200x Taylor & Francis http://www.informaworld.com DOI: 10.1080/0144534YYxxxxxxxx January 2, 2011 11:18 History and Philosophy of Logic types 2 by circumflexion), it would adopt a substitutional semantics for them. This substitutional semantics treats such terms as dummy schematic letters for wffs of a certain fixed structure matching the order component of the predicate terms. By the lights of a theory without such terms, these terms might be regarded as non-predicative. This is a non-standard use of “non-predicative” which normally means that comprehension axioms of the form ( E '(t)) (xt) '(t)(xt) ≡ A, where '(t) is not free in A, are adopted which make no further distinctions between the wffs allowed in A. Comprehension axioms involving non-predicative variables such as ( E 'h2;oi) (xo) 'h2;(o)i(xo) ≡ A (where h2; oi indicates that ' is an individual-level predicate variable of order 2)3 require that the wff A match in syntactic structure to the order 2 (i.e., no bound predicate variables of order 2 or higher may occur in A). I will argue that this substitutional semantics has to be understood epistemically. For example, some epistemic subject, other than me, mentions (what are to her) predicative attributes with which, however, I am not acquainted: I am nonetheless able to interpret these in my epistemic universe, but, in doing so, I cannot but understand them as involving some quantification that makes them non-predicative. I will then use a non-predicative functional term F (say, of type t) only as a proxy for what is, in some other epistemic perspective than mine, a predicative attribute. Accordingly, any formula ('t) A(') containing an apparent non-predicative variable of type t is interpreted as asserting that every formula A[F='] is true for any non-predicative such term F . Now given the substitutional semantics for non-predicate variables and circumflex predicate terms, a predicative and a non-predicative term of the same simple type which are provably extensionally equiva- lent, are not semantically distinct. In effect, under the substitutional semantics, the order amounts to just the order of the simple type (which is Principia’s syntactic definition of predicativity). Thus we have a way of explaining Principia’s otherwise perplexing statement that there is no loss of generality by omit- ting non-predicative terms from the language. The difference, in the context of a substitutional semantics, is epistemic (not ontological), in the sense that I have just suggested. So, for example, one can introduce h1;oi F zb := (' ) χ('; zb). This is a non-predicative functional term of order 2. Accordingly, one can use a non-predicative variable 'h2;oi having F as a possible substituend. Now suppose A('h2;oi) is, for E instance, E h3;h2;oii (zo) 'z ⊃ ('). Then A(F ) is h3;h2;oii ('h1;oi) (zo) χ('; z) ⊃ (F ), where the apparent variable is predicative (since it is of order 3 and has predicates of order 2 as arguments). Sentences such as A(F ) are substitutional instances of the non-predicatively quantified ('t) A('), and represent the counterpart in my epistemic universe of the use of bound predicative vari- ables in some other epistemic universe when the presumed values of those variables are not accessible to me as predicative attributes. In order to make that clear, we should first explain how orders and types are introduced in Principia and to what extent they constitute different features (section 1), then examine three preliminary questions to set out the main interpretations of orders and types in Principia’s syntax, before putting forward a solution to the “no loss of generality” problem that both supposes a substitutional interpretation of non-predicative variables (section 2) and leads to a new epistemic perspective on Principia (section 3). 1.2. The Vicious Circle Principle and the derivation of types The Introduction of the (first edition of) Principia Mathematica gives roughly three versions of the Vicious Circle Principle (VCP). By itself, this principle points at the vicious circle arising as soon as a collection of objects is supposed to include some member which cannot be defined otherwise than by referring to this very collection. The addition of the clause (C1), according to which a propositional function presupposes the totality of its values, gives rise to a new form (VCP 1) of the VCP: any propo- sitional function cannot have values which involve it. Inasmuch as any argument of any propositional function is a constituent of the corresponding value of this function (C2), no propositional function can be involved in its arguments (VCP 2). Finally, since ‘(x) F x’ involves F x^ (C3), the proposition (x) F x 3Here the index is the concatenation of the order and of the simple type. Later on, I will correct and refine the index so as to obtain genuine ramified-type indices. January 2, 2011 11:18 History and Philosophy of Logic types 3 cannot be an argument for F (VCP 3). To sum up: VCP 3 = VCP 2 + C3 = VCP 1 + C2 + C3 = VCP + C1 + C2 + C3. This is this final version (VCP 3) to which I will refer when speaking of the “vicious circle principle.” After formulating these successive versions of the VCP4, which he sets out rather abstractly, Russell illustrates it with the case of the truth value of the proposition ‘(p) p is false’5. The Principles of Math- ematics would claim that this proposition must be false, and therefore have a truth value, simply because its truth is self-refuting. Of course, this solution is no more practicable6. Nevertheless, Russell seems to hold on to the basic essential: the proposition cannot be devoid of any truth value.