The Versatility of Universality in Principia Mathematica

January 2, 2011 11:18 History and Philosophy of Logic types

HISTORY AND PHILOSOPHY OF LOGIC, 00 (Month 200x), 1–21

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 predicative 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 predicative 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 arguments 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 variable,” “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

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 ϕ(t)) (xt) ϕ(t)(xt) ≡ A, where ϕ(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 ϕh2;oi) (xo) ϕh2,(o)i(xo) ≡ A (where h2; oi indicates that ϕ 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 (ϕt) A(ϕ) containing an apparent non-predicative variable of type t is interpreted as asserting that every formula A[F/ϕ] 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 equivalent, 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 := (ϕ ) χ(ϕ, zb). This is a non-predicative functional term of order 2. Accordingly, one can

use a non-predicative variable ϕh2,oi having F as a possible substituend. Now suppose A(ϕh2;oi) is, for

E instance, E ψh3;h2;oii (zo) ϕz ⊃ ψ(ϕ). Then A(F ) is ψh3;h2;oii (ϕh1;oi) (zo) χ(ϕ, z) ⊃ ψ(F ), where the apparent variable ψ 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 (ϕt) A(ϕ), and represent the counterpart in my epistemic universe of the use of bound predicative variables 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 propositional 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 xˆ (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

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. Thereby the application of the VCP is conditioned. Because truth and falsity are considered as propositional functions, and because the VCP states that (x) F x cannot be an admissible argument of F xˆ, the falsity peculiar to the proposition ‘(p) p is false’ has to be distinct from the function ‘pˆ is false’ (or simple falsity). Con- sequently a stratification is sketched, that seems to be easily applicable to the propositional functions in general, and so the idea of a hierarchy of propositional functions, modelled upon the case of truth and falsity, is launched. One might object that such a presentation is not self-evident. On the one hand, if we assume that F is a propositional function whose range of significance consists in a certain domain of propositions (for the sake of the argument only, we suppose that propositions exist), nothing requires that the proposition (p) F p be an admissible argument of a function F 0 homologous to F — homologous in the sense that two distinct truth predicates refer, despite their very difference of level, to the same intuitive and general notion of truth. On the other hand, any proposition allows a single truth predicate, so that two different truth predicates possess two disjoint ranges. But is this true of any propositional function? To put it another way, the collection of the propositional functions of the form ‘pˆ is true’ has two very particular characteristics: • every proposition falls under one truth predicate at the most; • every proposition falls under one truth predicate at the least. From the first characteristic, one deduces that two different types are necessarily disjoint. And from the second one, that if p designates any member of the range of significance of V , then there will be another function W to whose range (p) V p belongs. Furthermore, W will be the truth predicate germane to all the propositions of the form (p) f(p): there will not be a truth predicate W1 applying to (p) F p and another one, W2, applying to (p) Gp (where G is a propositional function with the same range as F ). Immediately above V stands only one truth predicate, not a multiplicity of different truth predicates. This merits further explanation. Let’s consider a proposition p1 and the falsity predicate, say F , germane to p1. Now, let p2 = (p) F p (where p1 is among the possible values of p). According to the VCP, (p) F p cannot be an admissible argument of F . That is all: nothing (except the intuitive universality of truth and falsity) further guarantees furthermore the existence of a predicate germane to p2. Let’s suppose nevertheless that such a predicate, say F 0, does exist. Why should F 0 be applicable to such an entity as 0 p3 = (p) Gp? Nothing guarantees the uniqueness of F anymore. In short, the particular case of truth substantiates the idea of a hierarchy, whereas the abstract VCP in itself does not. This echoes the following remark from Suzanne Langer: There is certainly an ambiguity in the expression Φ(Φˆx); Mr. Russell has sought to locate that ambiguity in the function. Φ, he claims, has more than one meaning, and the Φ outside the bracket is not really the same as the Φ within. The two have different ranges of significance; in the propositions “x is false”and “ ‘x is false’ is false,” the word “false” has two different meanings respectively. Now it is hard to convince ourselves that we really do not mean by “false” No. 1 what we mean by “false” No. 2; the systematic ambiguity of truth and falsehood is certainly not obvious to common sense, and it is only by reduction of the proposition to the standard form, and the application of the vicious-circle principle, that any statement may be known to contain similar confusions of meanings. A list of such “ambiguous” concepts is given in Principia Mathematica (p.

4I don’t claim that they are not equivalent (I won’t engage in that issue here), being content with remarking, after many commentators (in particular Gödel, in (Gödel 1944)), that they are different. 5(Russell and Whitehead 1910), Introduction, p. 41-42. 6Indeed, ‘(p) p is false’ cannot be a proposition with a truth value, because it involves the totality of all propositions, and, to begin with, because there is no ontology of propositions in Principia. January 2, 2011 11:18 History and Philosophy of Logic types

64); but there seems to be no method, except the chance discovery of paradoxes, for the recognition of such concepts7.

1.3. Functional orders and propositional orders Thus the consideration of truth levels8 does not owe anything to the VCP as such. Indeed, one could claim: V ((x) φx) = (x) V (φx) Df without violating the principle. So, in spite of the continuity between the analysis of propositional functions and the analysis of propositions, suggested in Russell’s 1908 “Mathematical Logic as Based on the Theory of Types,” the hierarchy of propositional orders and the hierarchy (to be defined) of propositional functions are not as such directly interderivable. As Whitehead puts it, in a letter to Russell, dated the 26th of May 1909: But there is one fundamental obscurity in the theory as ‘exposed’ in the American Journal article which I do not see through — viz — you nowhere explain the relation between the ‘values’ of functions of different types and the hierarchy of propositions. Whitehead’s remark refers to “Mathematical Logic,” where an ontology of propositions of different orders is embraced: the use of predicate variables is emulated substitutionally and the functional hierarchy is then derived from the propositional. In Principia there is no longer an ontology of propositions. Indeed, the hierarchy of propositions is a hierarchy of truth levels and relates to the multiple-relation theory of judgment, whereas the hierarchy of functional types of ramified type theory is connected with the combination of the VCP with an additional principle, namely the direct consideration according to which “a function is essentially an ambiguity9.” But the problem of the connection between the two hierarchies arises all the more. Actually, Russell does speak in Principia of a possible “derivation” of the propositional orders from the hierarchy of the propositional functions10. Still, this derivation is not warranted by definition, it has to come to some proof, however obvious it may seem to be: Propositions which are not elementary, which contain no functions, and no apparent variables except individuals, may be called first-order propositions. [. . . ] Thus elementary and first-order propositions will be values of first-order functions. [. . . ] Thus a function of an elementary or a first-order proposition may always be reduced to a function of a first-order function; and this obviously applies equally to higher orders. The propositional hierarchy can, therefore, be derived from the functional hierarchy [. . . ]. It is true that in the end propositional orders and functional orders amount to the same thing. This comes close to Landini’s thesis that the order part of the order\type distinctions among propositional functions relies upon the consideration of levels of propositional truth: With a no-propositions theory, “truth” and “falsehood” no longer need be regarded as primitive properties. Rather, there are now to be different senses of “truth” and “falsehood” as applied to statements differing in structures. The different senses, in turn, explain and philosophically justify the order part of the order\type indices on a predicate variable11. The propositional orders inherited from the functional hierarchy correspond to truth levels inherited from Russell’s theory of judgment, but this correspondence is not completely straightforward: the hierarchy of orders lands on its feet, so to speak, but we have really two prima facie distinct hierarchies to deal with. The reason for this is the two dualities related to the notion of type in Principia. As a matter of fact, a type, in the broad sense, constitutes both a range of significance12 and a legitimate totality, i.e., both an informal semantic parameter and a syntactic restriction matching the VCP. This coincidence is

7(Langer 1926), p. 223-224. 8(Russell and Whitehead 1910), p. 42 sq. 9(Russell and Whitehead 1910), p. 47-48. 10(Russell and Whitehead 1910), p. 54. 11(Landini 1998), p. 282-283. 12(Russell and Whitehead 1910), p. 161. See also (Russell 1908), p. 75. January 2, 2011 11:18 History and Philosophy of Logic types

not self-evident, and it is only on a functional level that it succeeds in being clariﬁed13. Besides (here comes the second duality), a type represents both a logical prototype and a truth level, and it is only on a propositional scale that this twofoldness succeeds in being explained. The two questions cannot be dealt with at one time, and this is why the order hierarchy of propositional functions and the order hierarchy of propositions go side by side, but without collapsing. Actually, there are cases where the two hierarchies do not exactly coincide. For example, let’s consider a proposition of the form (x) (y) Rxya. Through the recursive clauses of the correspondence theory set in the Introduction of Principia, the truth of that proposition can be traced back to elementary truth through a two-step genesis : (x) (y) Raxy is true** (second level truth) iff, for any x, (y) Raxy is true* (ﬁrst level truth) iff, for any x and any y, Raxy is true (elementary truth). Still, (x) (y) Rxyzˆ is a propositional function of order 1, not 2, since it involves only the totality of all individuals. Or, to put it another way, (x) (y) Rxyzb and (x) Rxbzb get the same order, and still their respective values do not belong to the same truth levels. Hence propositional order does not exactly coincide with the order of a corresponding propositional function14.

1.4. Ramif-types as propositional forms The above-mentioned example calls for a more fine-grained parameter than functional orders15. The “truth level” of a proposition depends on the number of quantifications of each functional order that occur in it. So functional orders have to be refined to catch up with propositional orders. Functional types discharge that task, since they keep track of everything possible in the structure of a propositional function. Thus, the situation is schematically as follows: • the truth of atomic propositions is a first-order truth, the truth of ‘(x) φx’ is a “second-order truth,” and so on. Orders of propositions relate directly to the consideration of levels within the meaning of truth. • Besides, the application of the VCP, combined with the addition of the “direct consideration,” results in the differentiation of orders of propositional functions. In some way, the order measures the depth of the presuppositions involved in any expression. Any expression presupposes the totality of the possible values of its variables, but also the totality of the entities presupposed by those values. This progression of orders carries out the subdivision of the previous simple type theory, and is sufficient to exclude propositional paradoxes similar to the paradox displayed by the §500 of the Principles. • Finally, the ramif-type16 of a propositional function, as opposed to its order, tends towards something else. It represents the tracing of the schematic expression of this function. The ramif-type of φ is determined by its order, but also by the number and respective orders of its possible arguments; it is also determined by the number and respective orders of the bound variables contained in it; and, in the case where its arguments are functions themselves, it is finally determined by the number and the respective orders of the arguments and the bound variables of its possible arguments, and so forth. To that extent, the ramif-type codes the most fine-grained logical structure possible.

13In (Goldfarb 1989) (p. 37), Goldfarb writes, about restrictions in range of variables to ranges of significance of propositional functions: “Once such variables are used, the question of the nature of the variable (as an entity) becomes far more urgent. Different variables can have different ranges; it then appears that our understanding of a proposition or a propositional function that contains quantified variables will depend quite heavily on an understanding of what those ranges are. The variable must carry with it some definite information; it must in some way represent its range of variation.” 14Russell acknowledges this fact in (Russell and Whitehead 1910), *9, p. 128. He writes (*12, p. 162): “First-order propositions are not all of the same type, since, as was explained in *9, two propositions which do not contain the same number of apparent variables cannot be of the same type. But [. . . ] their differences of type may usually be ignored in practice. No reflexive fallacies will result, since no first-order proposition involves any totality except that of individuals.” So indeed the question of the omission of the fine grained distinctions within orders is discussed in Principia. It emerges from Principia’s definition of “truth” and “falsehood.” Admittedly, if the fine grained feature of the recursion were fully represented it would be expressed by something like ‘1.2’, where ‘2’ represents the number of bound individual variables: precisely, it would not be expressed by a single number. In the case of higher order functional terms, it would be expressed by still more complex indices: ramified types. Orders are basically numbers, meant to show levels: they are devoid of the internal structure that makes it possible to represent the number and the respective types of all bound variables. 15As Landini himself acknowledges, see (Landini 2004), p. 277-278, and (Landini 1998), p. 283-284. 16Unless otherwise stated, “types” always refer to the ramified theory of types developed in Principia. I shall speak of “ramif-types” in order to avoid any confusion with simple types or Church’s r-types. January 2, 2011 11:18 History and Philosophy of Logic types

The first conclusion that follows is that ramif-types ramify orders, and not the other way round. In a way, the justification needed for the admission of ramif-types lies less in their technical necessity (in view of the resolution of the paradoxes) than in the continuity they express between ordinary language and mathematics. Indeed, they allow a systematic classification of all propositional schemata and, in return, the simple extensional hierarchy which prevails in mathematics is seen to stem from an intensional hierarchy with which it stays homogeneous. The fineness of the notion of ramif-types is maximal, in the sense that any further determination is impossible without mentioning a particular feature (a specific concrete propositional function). Ramif-types display all possible particular propositional forms that one can specify without overstepping the sphere of the schematic. Ramif-types are assigned to variables so it is usually thought that they basically consist in ranges17. But ramif-types are primarily types of propositional functions, and ramif-types of variables only in a derivative way. Ramif-types are not domains, but forms. This means at least that the ramif-type of any propositional function φxb can be conceived of as a diagram displaying not only the simple types of the apparent variables occurring in φ, but the number of such apparent variables, as well as their respective ramif-types; and not only that, but also the number and the respective ramif-types of all the real variables, so that the diagram of anything involved in the arguments, as well as (possibly) in the arguments of these arguments (if the arguments at issue are themselves functions), in the arguments of the arguments of the arguments, and so forth, in an a a r r inductive manner. So the general form of a functional ramif-type ought to be: ht1, . . . , tm; t1, . . . , tni, a r where the tj ’s are the respective ramif-types of apparent variables, and the ti ’s are the respective ramif- types of real variables, ordered according to their occurrences18. The ramif-type of individuals then becomes h−; −i =: o. The ramif-type of (x) ψ(x, yb) is ho ; oi, the ramif-type of (ψ) f!(φ!z,b ψ!z,b x) is hh−; oi; h−; oi, h−; oi, oi, that is h(o); (o), (o), oi, where (o) := h−, oi. The order is not mentioned, because it can always be determined directly from the ramif-type: the order of an individual is 0, and a a r r a r the order |ϕ| of a function ϕ of ramif-type ht1, . . . , tm; t1, . . . , tni is: max(|ti |, |tj |)1≤i≤m + 1.A 1≤j≤n predicative (as opposed to a non-predicative) propositional function can then be defined as usual, as a function whose order is the least possible with respect to their ramif-type (which means that its order is the order of its simple-type index). By extension, I shall call a ramif-type predicative when it is the ramif-type of predicative functions, and a variable predicative when all its values have a predicative ramif-type19.

2. Solution of the “no loss of generality” problem 2.1. Three questions and four different interpretations of the syntax of ramified type theory At this point, an account of the main existing interpretations of Principia’s ramified type theory is in order. To begin with, Hatcher sees the types of Principia’s theory of types as being those of simple type theory, only split into orders20. Hatcher follows the modern definition of simple-type symbols: ‘o’ is a type symbol and its order is 0; (ii) if t1, . . . , tm are type symbols, of respective order k1, . . . , km, then

17See (Chihara 1973) (p. 43) for an example. 18On this, see (Rouilhan 1996) (p. 234-240) and (Weiss 1994) (p. 183). 19It should be noted that Landini ((Landini 1998), p. 283-284) has a notion of very fine grained distinctions about the types of (predicative) variables, for which he introduces highly specified indices. Basically, Landini’s order\type symbols mention, for any given predicate variable, the order and the simple type of each of its admissible arguments. Since the order of an order\type symbol can be recovered from its type part, the order\type symbol of a predicate variable amounts to the nested sequence of all the simple-type symbols of its admissible arguments, of all the simple-type symbols of the respective admissible arguments of these admissible arguments, and so on recursively. Landini’s order\type symbols are, so to speak, ramified simple-type symbols, which falls short of fully ramified types. This is simply because Landini considers all order\type symbols to be types of predicative terms only. Landini’s very fine grained indices aim at representing more: the whole recursive process about the senses of “truth” and “falsehood” that are relevant in such and such case. Indeed, they keep track of the occurrences of negation and disjunction, as well as of the number and the order of quantifiers that occur in the wff under consideration. With respect to order\type symbols and very fine grained indices, ramif-types relate to predicative as well as to non-predicative terms, and follow a middle path: the ramif-type of each wff mentions the number and the respective ramif-types of all bound variables, as well as the number and the respective ramif-types of all admissible arguments, which is more than order\type symbols, but it does not represent the whole logical structure (connectives included) of the wff, which is less than maximally fine grained indices. 20See (Hatcher 1968), p. 106-113. January 2, 2011 11:18 History and Philosophy of Logic types

(t1, . . . , tm) is a type symbol, and has order n + 1, where n is the highest integer among the ki; (iii) these are the only type symbols. Variables of all types are assumed as terms. Then wffs and other terms are introduced. Basically, all type-matching expressions are wffs (which means that if A is a term of type (t1, t2, . . . , tn) and y1,..., yn are terms of types t1, . . . , tn, respectively, then A(y1, . . . , yn) is a wff), wffs are closed under quantification over (free) variables, and terms can be generated from wffs by circumflexion of some free variable. According to a cumulative conception of order hierarchy, the order of any term is defined as max(n1, n2 + 1), where n1 is the highest order of all its free variables, and n2 the highest order of all its bound variables. Following that interpretation, orders ramify types, and a functional term is predicative exactly when its order is the order of its type. The main fact is that variables are all predicative by construction, and so that non-predicative propositional functions, which (o) (o) o do exist (like (∀x1 )x1 (xc1), following Hatcher’s notations) are not ranged by any corresponding variable. Hatcher acknowledges afterwards that in Russell and Whitehead’s system “variables were thought ((o),(o,o))/5/(3,2) of as being differentiated not only by type, but further by order. [. . . ] For example, x1 would be a variable of type ((o), (o, o)), of order five, whose first argument was of type (o) and order 3 and whose second argument was of type (o, o) and order 221.” In particular, such a system allows for non-predicative bindable variables. But Hatcher considers it as a useless complication. This observation is certainly prompted by the dispensability argument (quoted above). I will come back to it in a moment. Contrary to Hatcher, Chihara considers that orders come first. The order of a propositional function “depends upon the structure of the formula of PM (which contains a circumflexed variable) used to de- note it22.” Thus propositional functions are sorted into types within orders, and each type, as a structure, determines a single order which is a positive integer, namely, for any propositional function of the type in question, the least integer greater than the order of all the bound variables (quantified or circumflexed) of that function (starting with individuals, of type T0 and order 0). So, according to Chihara, orders measure the presupposition depth of propositional functions, and their justification lies in the PCV. But the main feature of Chihara’s reading is that each variable ranges over one and only one type, whose order is the order of the variable. The question, then, is whether each type is ranged by corresponding specific variables. Chihara’s answer to it seems positive, but reserved. Since different types correspond to a same order, quantification over variables of same order but of different types can produce terms of the same order, and more and more so. That is why, whereas propositional functions of order 1 can take only individuals as arguments and are of the same type T1.0, there are two types of propositional functions of order 2 (for the sake of simplicity, Chihara only considers monadic propositional functions): T2.0, the type of propositional functions of order 2 taking individuals as arguments; and T2.1.0, the type of propositional functions of order 2 taking propositional n−1 functions of type T1.0 as arguments. Continuing in this way, one gets 2 types of order n. So each type exhibits not only the order of the arguments, but also the order of the arguments of the arguments, and so on. “In terms of this schema, a propositional function is predicative if its type is either T1 or Tk.k−1.σ (for some natural number k, with σ a sequence of the required sort).” So the hierarchy of types takes on finally the following form: T0,T1.0,T2.0,T2.1,T2.1.0,T3.0,T3.1,T3.2,T3.2.1,T3.2.1.0, and so forth. It is still a numerical (linearly ordered) hierarchy, like the simple theory of types. A function of type T3.0 must contain a bound variable of order 2, but the type does not reveal whether the type of the bound variable is T2.0 or T2.1. Chihara acknowledges that *9.131 refines the hierarchy of types, taking into account the types of bound variables as well as the number of occurrences of quantifiers, but quickly dismisses it: Roughly, the rationale for this enormous proliferation of types seems to be this: it is thought, for some reason, that one cannot get propositional functions of the same type from propositional functions of different types by merely changing argument variables into bound variables of quantification. [. . . ] But the logical justification for this principle is obscure23.

21Op. cit., p. 112. See also p. 124-127 for a formal exposition of the “system RT of ramiﬁed types” that Hatcher sees as the basis of Principia’s type theory. 22(Chihara 1973), p. 19. 23(Chihara 1973), p. 22-23. January 2, 2011 11:18 History and Philosophy of Logic types

But how could the two original functions be assigned different types in the first place, if the type and the number of free variables (or arguments) weren’t considered? And, since bound variables are as important as arguments to determine the order (and thus, the type) of a function, why oughtn’t the number and respective types of bound variables also matter in specifying the type of that function? Accordingly, Chihara (p. 29) deems the order distinctions and the type distinctions to be based alto- gether upon the “presupposition argument” to the effect that a propositional function always presupposes all the propositions that are its values. The “methodological” justification (namely, constructing propositional functions in stages) that he subsequently puts forth also puts the stress on orders (as presupposition indices) rather than on types. Types, as well as orders, derive from the VCP after all (p. 43). That is why Chihara is close to Hatcher’s hierarchy of types24, even though Chihara splits orders into types whereas Hatcher splits types into orders. The main point lies elsewhere: are there variables of any type, or only predicative ones? That issue is tackled head-on by both Church and Landini. (Church 1976) puts forward “r-types” as a way of mirroring within each type the respective types and orders of all arguments of the propositional function at stake and thus, owing to the recursive nature of the definition, the respective types and orders of the possible arguments of those arguments, and so forth (hence the properly ramified nature of r- types).

There is an r-type i to which the individual variables belong. If β1, β2, . . . , βm are any given r-types, m ≥ 0, there is an r-type (β1, β2, . . . , βm)/n to which there belong m-ary functional variables of level n, n ≥ 1. The r-type (α1, α2, . . . , αm)/k is said to be directly lower than the r-type (β1, β2, . . . , βm)/n if α1 = β1, α2 = β2, . . . , αm = βm, k < n. The intention is that the levels shall be cumulative in the sense that the range of a variable of given r-type shall include the range of every variable of directly lower r-type. The order of a variable is defined recursively as follows. The order of an individual variable is 0. The order of a variable of r-type (β1, β2, . . . , βm)/n is N + n, where N is the greatest of the orders that correspond to the types β1, β2, . . . , βm (and N = 0 if m = 0). — This is Russell’s notion of order as modified by the cumulative feature which was just described25. Types, in Church’s rendering of the ramified type theory, become much more fine-grained. The main loss of information has to do with order: only the highest order quantification within some propositional function is indicated. So, for example, (ψ) f!(φ!z,b ψ!z,b x), (ψ) (y) g!(φ!z,b ψ!z,b x, y) and (ψ) (χ) h!(φ!z,b ψ!z,b χ!z,b x) all get the same r-type, regardless of lower order quantifications. Once all those fine-grained r-types are granted, “there must be a separate alphabet of variables for each r-type,” predicative or not. This is the main feature of Church’s syntax. This is at odds with Landini’s reconstruction of formal ramified type theory26. Indeed, Landini introduces “order\type symbols” as simple types augmented with orders that turn them into predicative types in the context of ramified type theory. Order\type symbols are the straightforward ramified counterpart of simple types. Variables for each order\type symbol are introduced, and variables are the only terms. Since all variables are predicative by construction, all terms other than individuals are predicative — which eliminates the problem that would result from the existence of non-predicative terms that could not instantiate any variable. Wffs are then built up in the usual recursive way, starting with atomic wffs (t1,...,tn) t1 tn 27 of the form x (y1 , . . . , yn ). Matrices φ!(x), f!(φ!z,b x), . . . , are free variables in schematic letters’ clothing. (They are free variables, and typical ambiguity consists precisely in turning these into schematic letters, that is, in disregarding order.) Impredicative wffs can be formed by quantifying some variable but, since only predicative variables are allowed, such wffs are not terms, corresponding only to terms by virtue of the reducibility axioms. As a result, there is no need for types to type them, so that the ensuing ramified type theory ends up in very scarce a hierarchy (namely, a hierarchy quite akin to simple type theory).

24As noted by (Landini 1998) (p. 271). 25(Church 1976), p. 747-748 26(Landini 1998), p. 255-257. See also (Landini 1993), p. 378-384. 27The expression “matrix” is used here in an informal inductive characterization of the wffs of the language of the system, but in fact, as emphasized by Landini ((Landini 1998), p. 263), it has at *12 the technical sense of predicate variable only. January 2, 2011 11:18 History and Philosophy of Logic types

Let’s now pause to take stock of the four interpretations that we have just reviewed. By way of comparison, the following chart will indicate the similarities and discrepancies in the formalization of Principia’s system of types: Hatcher Chihara Church Landini Ramif-types as Forms

φ!x (o)/1/0 T1.0 (i)/1 (o\o)\1 (o) = h−; oi (x) φ!(x, y) (o)/1/0 T1.0 (i)/1 (o\o)\1 ho ; oi (φ) f!(φ!z,b x) (o)/2/0 T (i)/2 - h(o); oi (let t be its type) 2.0 (φ) f!(φ!z, ψ!z) ((o))/2/1 T2.1.0 ((i)/1)/1 ((o\o)\1)\2 h(o); (o)i t tb b (φ ) f!(φ z, φ!z) ((o))/3/1 T3.1.0 ((i)/1)/2 - hh(o); oi;(o)i t b b (ψ) f!(φ z,b ψ!zb) ((o))/2/1 T3.2.0 ((i)/2)/1 - h(o); h(o); oii Table 1: Different type symbols Along with that overview, one main issue stood out. It consists of a chain of three questions: 1) Are there bindable variables whose possible values are non-predicative terms? 2) Are there non-predicative propositional functions (even though there are no variables to range them)? 3) Are there at least predicative function expressions besides variables? Unsurprisingly, the chart of the respective answers of the above commentators is: Church Chihara Hatcher Landini Question 1 Yes No No No Question 2 Yes Yes No No Question 3 Yes Yes Yes No Table 2: Non-predicative and predicative terms As to the first question, the grammatical possibility to construct non-predicative terms (using variables of predicative variables, but this does not matter) calls for a positive answer. Indeed, how could there be regular terms without variables to range them, and how could there be genuine variables without the possibility to quantify them? On the other hand, non-predicative variables (i.e., variables of non-predicative functions) seem to be disposed of by the dispensability argument. By construction, as Russell recalls, any non-predicative propositional function of the n-th order is obtained from a predicative function of the n-th order by turning all the arguments of the (n−1)-th order (as well possibly as arguments of lower orders) into apparent variables. Oddly enough, Russell concludes that one can do without non-predicative variables. Each value of such a variable φt can surely be expressed using predicative variables only. But that does not mean that quantifying over φt can be captured by quantifying over predicative variables only. So the dispensability argument seems, in fact, to require quantification over non-predicative variables to be excluded. Moreover, the very description that Russell gives of the hierarchy as unfolded in stages clearly eschews any use of non-predicative variables. As to the second question, a positive answer is prompted by passages where Russell himself suggests that there are functional terms besides the predicative ones28. On the other hand, it is impossible to decide which non-predicative terms there are without embracing the particular point of view of some epistemic agent. This is actually what Russell’s own example points to when he considers the non- predicative having all the qualities that make a great general. Russell remarks that there is a genuine predicate equivalent to it: “For the number of great generals is finite, and each of them certainly possess some predicate not possessed by any other human being — for example, the exact instant of his birth. The disjunction of such predicates will constitute a predicate common and peculiar to great generals29.” Of course, we don’t have any access to all the exact birth instants of all the great generals, and that is why we are bound down to mentioning the non-predicative formulation rather than the equivalent predicate. But certainly God, were she existing, would use the predicate directly, because it would not outrun her cognitive abilities. Moreover, nothing requires being a predicate required in a great

28See (Russell and Whitehead 1910), p. 53 and 164. Landini ((Landini 1998), p. 263) argues that this is due only to Russell and Whitehead’s sloppiness when it comes to the distinction between variables and schematic letters, but the argument begs the question. 29(Russell and Whitehead 1910), p. 56. January 2, 2011 11:18 History and Philosophy of Logic types

general, or even being a great general, to exist. The formal system of Principia certainly cannot commit to anything of the sort. In other words, the existence of non-predicative functional terms is no more logically compelling than the axiom of reducibility itself. As to the third question, ﬁnally, Russell often lumps together φ!x and (y) φ(x, y) as examples of predicative propositional functions30; yet the second one clearly cannot be a variable. On the other hand, the previous argument against the existence of non-predicative terms also applies against that of terms other than variables.

2.2. Epistemic realizations of ramif-types as translation patterns This last discussion prompts a sharper distinction between two separate perspectives: the formal setting of Principia’s ramiﬁed type theory, and such and such particular epistemic realization of it, i.e., its application to some cognitive subject’s epistemic universe. The following passage is a typical example of the mixing that can occur between the formal system and the epistemic realizations of ramiﬁed type theory:

[. . . ] “φ!x” is a function which contains no apparent variables, but contains the two real variables φ!zb and x. (It should be observed that when φ is assigned, we may obtain a function whose values do involve individuals as apparent variables, for example if φ!x is (y) ψ(x, y). But so long as φ is variable, φ!x contains no apparent variables.)31 Formally considered, φ!x here is a variable, even though its possible value in some epistemic realization could be (y) ψ(x, y) — where ‘ψ(x, y)’ here is not another variable, but a schematic letter standing for some actual binary relation. Distinguishing these two contexts allows us to shed some light on a passage otherwise very obscure, and to better understand the status of non-predicative propositional functions. Looking on the matter that way, indeed, we are led to take one step further, and to interpret non- predicative terms as the equivalent, within one system, of what can be seen, within another system, as a predicative function. For example, what you mention as a predicate ψ!zb, can only be to me a non- predicative second order functional term, namely the property of having all the qualities that make a great general. The syntactic possibility of non-predicative propositional functions accounts of course not for non-predicative entities, but for the shadows that other epistemic universes leave upon mine. Truth can be brought into play only in the realm of an interpretation. But there isn’t any simple translation between the abstract hierarchy of logical ramif-types and its epistemic interpretations. This is the reason why I suggested at the beginning that there isn’t any automatic clutching between matters of vicious circle about propositional functions and matters of truth levels about propositions. The theory of ramif-types lends itself to a plurality of realizations, each of which is a differently structured epistemic universe. The understanding of the import of the theory of ramif-types in Principia requires the acknowlegment of this plurality. In the first place, this acknowledgement neutralizes the issue of the ontological nature of propositional functions and of the supposed platonism or constructivism to be ascribed to Principia on that score32. In table 2, the cascade of answers is linearly ordered (which gives table 2 its diagonal shape): generally, commentators don’t answer “yes” to some question without also answering “yes” to the next one. (Of course, it remains possible to answer “no” all the way, as Landini does.) In a reversal from that, the view I shall support is that you can answer “yes” to question 1, and “no” to question 2. Ramif-types of non-predicative functional terms, and corresponding variables, are indeed laid on by the formal system of Principia’s ramified type theory, in order to provide for non-predicative terms in such and such realization. Still, non-predicative terms, and also predicative terms besides variables, never live outside some epistemic universe and express the connections (and corresponding translation rules) of this universe with others. This goes hand in hand with a substitutional semantics for non-predicative complex functional terms made by circumflexion and also for non-predicative variables, but not for predicative

30See (Russell and Whitehead 1910), p. 51. 31(Russell and Whitehead 1910), p. 52. 32See, among others, (Linsky 1999) (chap. 2), (Goldfarb 1989), and (Chihara 1973) (p. 24-28 and p. 42-43). January 2, 2011 11:18 History and Philosophy of Logic types

variables. In (Landini 1998) and (Landini 2004), Landini has rightly pointed out that 1) Russell’s conception of the universality of logic does not preclude any object language/meta-language distinction; 2) the terms of the system of Principia are not ontologically loaded. This suggests that Principia’s ramified type theory is much more like a Tarski-type logical system than one would usually make out (following van Heijenoort). But the interpretations of ramif-types are left up in the air, as well as any model theory that would fit Principia’s framework and vindicate the idea of their being perfectly compatible with a language/metalanguage distinction. A model theory is still lacking to substantiate Landini’s claim that the logic of Principia Mathematica does not balk at meta-theoretical considerations. This is a suggestion I would like now to flesh out a bit. Principia’s ramified type theory allows us to distinguish among the statements from (1) to (4), even between (3) and (4) — hence the utility of fine-grained ramif-types. Indeed, these statements express propositions corresponding to propositional functions of different ramif-types, even if these propositions correspond, in some way, to a ‘same’ state of affairs. The objective referent of the situation will thus vary from subject to subject, and so will, more generally, the realization of the whole theory of ramif- types. The existence of a plurality of semantic applications is perfectly compatible with the purity, or the abstract universality, of logic. And it even turns out to be its condition. The epistemological perspective underlying the Introduction of the Principia enables us to leave aside the problem of the possibility of an integral analysis of reality, an analysis which otherwise would seem to be presupposed by the Principia without being likely to be given by them. There is no univocal analysis of reality, which would be available and which Russell would have put on hold. Russell’s account, on the contrary, is bound to a neutrality which explains that the only propositional functions to which the Principia are committed are the variables of propositional functions. Typical ambiguity is described by Russell himself as a form of context relativity33. There are actually non-predicative functions but they appear later, with reference to a particular epistemic perspective, and only then can the opposition between predicative propositional functions (and variables thereof) and non-predicative ones be drawn. All terms turn out to be predicative, because they all get interpreted by predicates (in the case of first order variables) or by higher order predicates (in the case of higher order variables) — which ones depends on the epistemic universe that happens to be considered to interpret the formal ramif- type theory. To take Russell’s example34: “Napoleon was a great general,” as uttered by some subject S1, is understood by some other subject S2 as: “(ψ) f!(ψ!zb) ⊃ ψ!(Napoleon).” (We suppose here that ‘being a quality that makes a great general” is a predicative second order propositional function f!(ψ!zb) 35 common to both S1 and S2.) Ramif-types keep track of properties that some subject cannot identify with any predicate that would be available to her. For instance, the translation rule [Gx] ≡ [(ψ) f!(ψ!z) ⊃ ψ!x] S1 b S2 adds no non-predicative terms to S2’s universe: ‘Gxb’ (rather than ‘G!xb’, which makes sense only in S1’s perspective) is not added, but only emulated from S2’s point of view, and introduced as a mere symbol in order to account for some of S1’s sentences. Still, S2 needs to be able to quantify over such symbols, which means using variables ranging over (in S1’s perspective) a domain of non-predicative terms, but only in a substitutional way. Here the complex ramif-type of S2’s counterpart of G!xb underlies such a substitutional quantification, in addition to providing the translation of terms such as ‘Gx’ with a pattern. Let’s look back now on the previous three questions. The table below summarizes the answers I advocate, according to whether they are given with respect to the formal system or to some realization of it. It is actually essential to underscore that duality. In connection with the third question, the above- mentioned passage of Principia (about ‘φ!x’, which may contain apparent variables, but, as a variable, does not contain any) is otherwise doomed to obscurity.

33(Russell and Whitehead 1910), p. 65. 34(Russell and Whitehead 1910), p. 56. 35Russell points out that since there have been finitely many great generals in history, and since each of them possessed some predicate not possessed by any other human being, the disjunction of such predicates will do. So in fact, in that particular case, the quantification over ψ is not essential, since it can be replaced with a finite disjunction. But, in other cases, the use of genuinely non-predicative terms won’t be avoided if S2 is to render all of S1’s statements. Appealing to a non-predicative term will then be only an epistemic matter, but I take epistemic matters to be the real source of non-predicative terms. January 2, 2011 11:18 History and Philosophy of Logic types

As far now as the second question goes, it must be said that no real ontological status is bestowed on non-predicative terms. In the realm of any epistemic realization, predicative terms (of whatever order) correspond to universals or attributes (hence, according to the Problems of Philosophy, to entities), but non-predicative ones are confined to the values of variables whose quantification is merely substitutional. Nonpredicative terms have but a schematic role in the formal system; they only provide for the impredicativity phenomenon that arises as soon as the match-up of two different epistemic perspectives is expressed. Still, in the realm of a formal system, they are perfectly legitimate terms, hence the positive answer. Finally, the first question also calls for a balanced answer. The need to speak of someone else’s predicates in a general way motivates the use of non-predicative variables in epistemic realizations of the theory of ramif-types. Now, within the formal system, how can we explain that one never finds bound predicate variables without the shriek? The dispensability argument is the main evidence here. But, as noted above (p. 9), by itself it does not go through. It makes sense only with the prospect of the reducibility axioms: then it is true that any apparent variable can be replaced, up to coextensionality, by a predicative one. Russell’s claim that one can do with predicative variables only, has given rise to an acute controversy. According to Hylton36, Russell has simply missed something and oversimplified the hierarchy. Landini, on the contrary, considers it as a textual proof in favor of his reading that all predicate variables are predicative. The distinction between the context of formal theory and that of realizations allows us to settle the issue in a more balanced way. On the one hand, the formal system set out in the Introduction of Principia does not allow for those variables syntactically. But, on the other hand, non-predicative variables occur in the realm of some realization only to internalize some other cognitive subject’s predicates and to quantify over these (even though none of them is available in that realization). The following table, which is at maximal variance with the previous one, summarizes what has been said so far: realizations formal theory Question 1 Yes No Question 2 No Yes Question 3 Yes No Table 3: Back to Table 2 Finally, table 4 summarizes what the commentators have said of the two issues at stake in this paper, namely the Types Within Orders vs Orders Within Types question and the No Loss of Generality problem: Types Within Orders? Solution of the No Loss Generality problem Hatcher No ∅ Chihara Yes ∅ Church Yes ? Landini Yes All predicate variables are predicative Table 4: The Types Within Orders question and the No Loss of Generality problem (Since Church allows for non-predicative variables, his interpretation should at least discuss the dispensability argument, this is why his answer can be considered as incomplete.) To sum up the solution of the “no loss of generality” problem advocated here, I suggest assigning various possible epistemic realizations to the ramified theory of logical types and understanding differences in order in Principia, not as differences within a single realization, but as differences between realizations. Each epistemic universe contains what the subject of this universe is acquainted with, that is, a specific stock of individuals and universals (predicative propositional functions being universals or higher order universals). Non-predicative terms are only the counterpart, within one epistemic universe, of an attribute F that is not available to the subject S2 of that universe, but that another epistemic sub-

36(Hylton 1990), p. 308-310. About the whole issue, see also (Linsky 1999), p. 79-83. January 2, 2011 11:18 History and Philosophy of Logic types

ject S1 is acquainted with. In such a case, S2 can still render F through a definition that may involve higher order quantification and thereby turn F into a non-predicative term [F ]S2/S1. On that basis, non- predicative variables can be introduced (in some epistemic universe) to match predicative quantification in some other epistemic universe, but their interpretation is only substitutional. Moreover, even though S2 is able to assert true sentences about F , the term [F ]S2/S1 is devoid of any ontological import: its reference does not exist in any realization, but only in some realization from the point of view of another one. In fact, ramif-types constitute fine-grained forms, and each non-predicative formal term is essentially a representant of its ramif-type. Viewed in that way, ramified type theory becomes a schematic metatheory of cross-checkings from one epistemic perspective to another: this corresponds to an epistemic contextualism (within a nominalistic semantics) when it comes to epistemic realizations of the order component of ramif-types. At this point, a caveat is in order. A contextualist reading of ramified type theory has nothing to do with any relativist interpretation of the formal system for logic itself. Two distinct epistemic hierarchies do not share the same basic domain of entities (individuals and universals), but, abstractly speaking, all the subjects share one single domain, corresponding to the formal entity variables. Moreover, different

realizations do not give rise to different deductions. For example, even though some subject has access

to some individual a only as to “the F ,” from ‘Ga’ as from ‘G(the F )’ (symbolically: ‘G(( ι x)(F x))’), one can equally derive ‘( E x) Gx’. Even though the ranges of the variables of the abstract hierarchy vary according to the interpretation of this hierarchy, the inferences remain the same, regardless of the cognitive context. Finally, I would like to add two short remarks. First, it seems reasonable to hold that ‘Kx’ meaning

“x is a subject” and the relation ‘Axy’ meaning “x is in acquaintance with y” are objects of a knowledge by acquaintance. Therefore, putting F x := ( E y) Ky & Axy, one may think that the class of all the F ’s will constitute a universal domain shared by all the subjects as basis of all epistemic hierarchies. It is nothing of the sort. Any outward meta-hierarchy collapses into a particular hierarchy. The argument presupposes that one could say (under this form) : ‘AmA’, where ‘m’ designates my mind. But this would violate the VCP, according to which a propositional function cannot itself be among its possible arguments. Actually, one can ﬁnd, in “The Nature of Acquaintance” (1914), a formalization of self- conscience as “experience of a present experience” (see (Russell 1912a), p. 166-167). But, precisely, Russell in that passage strives to defuse any circularity as well as any endless regression.

Second, the propositional contents and the propositional orders do correspond, but cannot correspond a priori to each other. As a matter of fact, ‘F a’ and ‘F (( ι x)(Sx))’ may be said to represent the same “propositional content” (as Frege says), since they share the same consequences, but have different orders. Thus, the setting of the correspondence between propositional contents and propositional orders implies the implicit adoption of a particular epistemic section of the universe, and therefore cannot be intrinsic, that is, inherent to the purely logical theory of ramif-types.

3. Epistemic model theory 3.1. Epistemic diagrams In Principia Mathematica, Russell decided to eliminate the propositions as incomplete symbols, through the multiple-relation theory of judgment. This is convincingly shown by Landini37. According to that new theory of judgement, a proposition becomes nothing but a disguised deﬁnite description, each time bound to a certain epistemic context (as will presently appear). To say that the proposition Socrates is human is true, is to say that there exists a fact p corresponding to the belief complex Bel{m,

Socrates, Humanity}. In other words:

E ‘F a’ is true =Df !( ι p)(Bel{m, a, F } corresponds to p).

Accordingly, ‘All men are mortal’ is true iff (x) ‘Humanity(x) ⊃ Mortality(x)’ is true iff

E E ι (x) !( ι p)(Bel{m, x, Humanity} corresponds to p) ⊃ !( q)(Bel{m, x, Mortality} corresponds to q). This truth condition is plainly of ﬁrst order, not elementary. And so is Russell’s diagnosis.

37See (Landini 1998) (p. 287-291) and (Landini 1991). January 2, 2011 11:18 History and Philosophy of Logic types

Landini38 concludes from these truth conditions that propositional functions cannot all stand for universals, or else there would be an entity U := Humanity-in-the-relation-(x) φˆ!x ⊃ ψˆ!x- to-Mortality such that: ‘All men are mortal’ is true iff Bel{m, Humanity, Mortality, (x) φˆ!x ⊃ ψˆ!x} corresponds to U, and the proposition ‘All men are mortal’ would then be an elementary truth. Still, that proposition, if not an elementary truth, could have been one. Indeed, Landini’s argument, although conclusive, suggests a multiplicity of epistemic analyses, each analysis depending on the stock of individuals and universals with which the subject under consideration is acquainted, as clearly noted by Nicholas Griffin about individuals39. This is in fact quite a general point, where the notion of ramif-type and the method of elimination of definite descriptions come into play together. Let’s consider the example of the proposition expressing that a certain individual a is green and that another individual b is a great general. If I am in acquaintance with both the individual a and the universal Green, I shall say, using the formal language of Principia: (1) Ga (elementary truth, first-order proposition).

On the other hand, having only access to a as to “the F ,” I shall say: (2) ( E x) (y) F y ≡ y = x & Gx (ﬁrst-level truth, ﬁrst-order proposition).

Having only access to green as to “the colour of grass” (‘CoG’), I shall say: (3) ( E φ) (ψ) CoG(ψ) ≡ ψ = φ & φa (second-level truth, second-order proposition).

Eventually, by combining the two descriptions, I shall say:

E (4) ( E x) ( φ) (ψ) CoG(ψ) ≡ ψ = φ & (y) φy ≡ y = x & φ!x (second-level truth, second- order proposition). In the third case, ‘green’ is nothing but an incomplete symbol. Nevertheless, its occurrence in any context will generate a second-order quantification. In view of this fact, green may be identified with a value of a second-order propositional function with one individual argument. Therefore my “epistemic diagram,” so to speak, will be the following: • Individuals : a, Being a great general; • Predicative propositional functions of individuals : x is a great general; • Second-order propositional functions of individuals : “x is green,” i.e., x has the color of grass. Whereas in the first case it will be: • Individuals : a, Green; • Predicative propositional functions of individuals : x is green; • Second-order propositional functions of individuals : “x is a great general,” i.e., x has all the predicates that make a great general. And so forth for the other cases. The quotation marks are meant to express the fact that the first epistemic subject does not speak for herself, but only uses a predicate that she borrows from another subject’s language. The contextual definition of ‘Green’ in the first case, or of ‘Being a great general’ in the second case, works as an interpretation rule from an epistemic universe to another.

3.2. A Russellian concept of model Up to now, we have used the term “realization” to designate the different epistemic interpretations which the theory of ramif-types keeps open. It may be useful to emphasize the originality of this Rus- sellian notion of interpretation. It is not just a matter of leeway in the choice of the starting level of the extensional hierarchy40, because the individuals, with respect to some diagram, are assigned to entities (in some other diagram) which do not have to stand in any single ramif-type. One may wish to compare the Russellian notion of realization with the model theoretic concept of structure interpreting a language. But there is a kind of duality between the variety of realizations of the language of any given logic, on

38(Landini 1991). 39(Grifﬁn 1980), p. 138: “terms like ‘individual’ or ‘ﬁrst-truth’ are not stable across contexts.” 40(Russell and Whitehead 1912), Prefatory Statement, p. xi-xii. January 2, 2011 11:18 History and Philosophy of Logic types

the one hand, and the variety of the logical analyses of reality, on the other. Unlike the model theoretic perspective, the Russellian version takes elementary equivalence (or the corresponding feature) for granted: as a matter of principle, two “corresponding” private statements have the same truth value (let us not clarify the “correspondence” which is at stake here). Thus, we are faced with a wider diversity of epistemic realizations than the mere plurality of the possible bases of the extensional hierarchy (or wider than typical ambiguity allowed in the Principia, which, again, is something else). But still, it is narrower than the wide open multiplicity of all possible model theoretic interpretations of a noncategorical theory. What is varying here is not a “domain of discourse,” since the respective domains of two given models are without comparison. They refer to the same whole of all entities, but the structuration, the epistemic diagram (and individualization pattern) of this whole is each time a different thing. Let me explain a little more. In a manuscript of May 1912 entitled ‘On Matter’, Russell writes: It is important to realize that, whenever mathematics is applied to the actual world, there must, if the analysis is pushed far enough, be some substitution of actual particular sense-data for the variables of pure mathemat- ics41. This is exactly what does not happen in the standard analysis of the deﬁnite description. Indeed,

‘F (the G)’ means: (5) ( E x) (y) Gy ≡ y = x & F x. Here the variable x is not likely to be replaced by any actual sense-datum: this is the very principle of a definite description as an expression of a knowledge by description. If a subject S1 knows a only as “the G,” she will assert (5) without being able to indicate the slightest possible instantiation for x. Now, if another subject S2 knows the G to exist and to be a, she shall say to S1 : “Well, take x := a!” In other circumstances, S1 will be able to pay S2 back the same. No subject is in acquaintance with all the possible instantiations of x, that is with everything. There is no way to take up the overhanging point of view of a mind capable of establishing a connection between all the epistemic diagrams. This does not detract “everything” from its meaningfulness. Every subject indeed resorts to an entity variable whose values do not boil down to her own actual objects of acquaintance, but include all the virtual objects of acquaintance that she is provided with by any other trustworthy subject, and in particular the lacking quasi value of x in (5). Thus, everything is the result of a kind of cooperation among all subjects. The admission of a, as a quasi value for an entity variable, into S1’s epistemic domain, hinges on S1’s believing that S2’s identification of (what S1 conceives of as) the G with some actual entity a is reliable. Granted that the G actually exists, “the G” becomes what Russell calls a construction, with the same status as other logical constructions, such as classes. This means in particular that this definite description then “has (speaking formally) all the logical properties of symbols which directly represent objects42”: it can, by extension, instantiate any individual variable. So, in the end, the variable x does not range a single collection of values (since its values depend on the subject considered), but at least a single all-inclusive range of values and quasi values: this justifies the real universality of the entity variables, as opposed to a mere formal one. There is no real comparison between the different model theoretic interpretations of a fixed logical language and the different realizations of ramif-types because, in the latter case, the resources of the language (especially the available proper names) are precisely the changing parameter. Well, one needs to say that there is another possibility left: in order to describe the ideal correspondence linking two statements made by two distinct subjects, one would like to speak of the interpretation of a given theory (i.e., of a set of statements), no longer in a certain structure, but in another theory. Now, this notion can be found in model theory43. In fact, the model theoretic concept of translation between theories does not describe, and is not aimed at describing, the relationship between Russellian realizations. Still, the former may help to bring out the specificity of the latter. Its definition is the following:

41(Russell 1912b), p. 83. 42(Russell and Whitehead 1910), *14.18, p. 180. See *20.71 about classes. The formal properties that Russell alludes to here are all the laws of the “theory of apparent variables,” such as *9.2 (rule of universal instantiation). 43Cf. for example (Hájek and Pudlák 1998), p. 148-150. Cf. also (Kreisel 1968) (p. 364-365) about the main relations between models. Among other things, Kreisel points out the notion of “commutation” between models, which differs from the notion of interpretation. January 2, 2011 11:18 History and Philosophy of Logic types

Let T and T 0 be two theories of respective languages L, L0. To interpret L0 in T means to deﬁne the following in T : • for the variables of L0, a predicate Ω of L such that T ` ∃y Ωy, which represents the range of these variables in T 0 • for each constant c of L , a formula ϕc(y) of L such that T ` (∃!y)(Ω(y) ∧ ϕc(y)) 0 • for each n-ary predicate symbol P of L , a formula ϕP (x1, x2, . . . , xn) of L 0 • for each k-ary function symbol f of L , a formula ϕf (x1, x2, . . . , xk, y) of L such that

k ^ T ` Ω(xi) → (∃!y)(Ω(y) ∧ ϕf (x1, . . . , xk, y)). i=1

One may thus replace every predicate symbol, every function symbol and every constant of L0 by a corresponding predicate symbol, function symbol or constant of L. Consequently, to each formula Φ of L0 corresponds a formula Φ? of L. 0 0 Then ? is said to be an interpretation of T in T iff, for each axiom A(x1, . . . , xn) of T ,

n ^ ? T ` Ω(xi) → A (x1, . . . , xn). i=1

This model theoretic notion is used for giving a precise meaning to the idea that some theory ‘contains’ another, as one says of a theory that it is a conservative extension of another, or as one speaks of a theory which contains the Peano Arithmetic (for example, in the case of the interpretation of Peano Arithmetic in ZF, Ω(x) will be the set-theoretic formula expressing that x is an ordinal). It could seem to be the concept we are looking for. Still, its counterpart in the Russellian context would be placed under a further constraint, namely a kind of reciprocation of all interpretations: the theory peculiar to any subject has to be interpreted in any other, and such a universal interpretability is unlikely to give something practicable in a model theoretic frame. Finally, there is at hand a notion of synonymy, or translational equivalence, in the perspective of “universal logic.” The main principle guiding this perspective consists in delimiting families of logics (for example the family of temporal and modal propositional logics) and in establishing a one-to-one application between sets of theorems of the two theories, rather than a dictionary between formulae44. This further notion may succeed in grasping some aspects of the Russellian idea of a plurality of epistemic realizations. But there is a difﬁculty here: let m be acquainted with a and m0 know a only as “the G.” Then m will know that what she calls “a” is what m0 calls “the G,” but m0 will not, and it seems hard to render this asymmetry in terms of a synonymy between two logics.

3.3. Epistemological necessity Let’s now summarize what has been seen so far. The relationship between the abstract theory of ramif-types and the different epistemic diagrams (seen as models) is reducible neither to the relationship between a formal theory and interpreting structures, nor to the notion of arbitrary level in the extensional hierarchy. The relation between two epistemic diagrams (seen as sets of statements) is reducible neither to the interpretation of a formal theory in another, nor to the synonymy between two theories. Hence, the variability of the realization of the theory of ramif-types is unlikely to be recovered through any model theoretic feature (in the usual sense of “model theory”). It is neither purely syntactic (because two realizations are always deductively equivalent), nor purely semantic (because the language to be interpreted, especially its genuine constant symbols, varies from one realization to another).

44Cf. (Pelletier and Urquhart 2003). January 2, 2011 11:18 History and Philosophy of Logic types

Furthermore, it hints at a new concept of necessity. As regards the doctrine of the Principles of Math- ematics, a propositional function is said to be necessary when all its values are true propositions45. This definition accounts for the special validity of logical laws. It equates logical validity with truth in virtue of logical form. According to a diagnosis shared by several commentators46, accounting for logical necessity constitutes a real problem to Russell. I won’t delve into the details of that deep issue, and only point out that, besides logical necessity, an epistemic notion of necessity may be devised in the Russellian framework. As soon as 1905, Russell thought of logical necessity in terms of fully general truth. Landini47 argues that this is maintained in Introduction to Mathematical Philosophy (chap. XV), but that logical necessity of a first order wff then consists in the truth of its full second order closure. He adds: “Russell’s analysis of necessity parallels Tarski’s semantic definition of logical truth. Russell gets at Tarski’s idea of different interpretations of the predicates of a first-order formula (over domains of differing cardinality) by turning the predicate letters into predicate variables and universally closing the formula.” I think that the Tarskian idea of giving different interpretations of predicates after turning them into variables can be paralleled in another way in Russell’s perspective, with epistemic realizations instead of Tarskian interpretations. Of course, it is not about giving different extensions to the same predicate symbols, but rather to carving up the intensional hierarchy of propositional functions in different ways. The truths keep the same (the epistemic realizations are all elementarily equivalent, so to speak), so the spread of realizations does not filter logically true sentences, but rather “epistemically necessary” sentences, that is, sentences of which all epistemic variants are true, in the following sense. Let Cxyz express the fact that x designates y by means of z (here ‘x’ designates some mind, and ‘z’ some expression). One may suppose that this notion is available to every subject (i.e., that every subject is in acquaintance with it). Let ‘F a’ be an elementary statement, asserted by some mind m in acquaintance with the universal F and the individual a. Let m0 be some other mind, which is acquainted with the universal F , but knows a only through the description ‘the S’; m knows m0 as “the Φ0,” and m0 knows m as “the Φ.” If m ` F a & C(the Φ0, a, ‘the S’) and m0 ` F (the S) & C(the Φ, the S, ‘a’), let’s write: m[F a]m0, i.e., m and m0 are congruent regarding ‘F a’ (they both know the “same” state of

affairs and agree to say so). (More exactly, we should write: m[σ]m0, with σ(m) = ‘F a’ and σ(m0) = ‘F (( ι x)(Sx))’.) Eventually, let’s set: ‘F a’ is epistemically necessary iff, for all minds m, m0, m[F a]m0 (Df). Quantifying over minds here refers to a remark that can be found in Russell’s Theory of Knowledge48. The main point is the following. In the case of logical necessity, one makes the arguments vary for a given implicit mind m as subject of the belief : φam, φbm, φcm. . . are true (where am, bm, cm, etc.,

are objects of acquaintance of a given mind m). In the case of epistemic necessity, one makes the

subjects vary for a given argument a as object of discourse : φa = φam1 , φ(( ι x)(F x)) = φam2 , ι m φ(( x)(Gx)) = φa 3 ... are true (where m1 is acquainted with a, m2 knows a as “the F ,” m3 knows a as “the G,” and so forth). There is thus a kind of complementarity between these two necessities. In both cases, necessity is not a kind of super-truth, only a degree along one of the two axes onto which any truth may be projected. The intricacy of the notion of logical necessity in Russell’s early doctrine has been pointed out by Grifﬁn49. According to (Russell 1905), necessity characterizes any propositional function of which all values are true propositions, and, in a secondary sense, any proposition which is some value

45Cf. (Russell 1905). 46See (Griffin 1980) (p. 121-126), (Landini 1998) (p. 294) and (McKeon 1999). In particular, McKeon underlines the ambivalence of the way Russell handles logical necessity: in some respect, he throws it back onto factual universality (in (Russell 1905)), but in some other respect (in Introduction to mathematical Philosophy) he seems to define it by reference to possible worlds. 47(Landini 2007), Appendix B, p. 266-278. 48“The strongest objection which can be urged against the above analysis of experience into a dual relation of subject and object is derived from the elusiveness of the subject in introspection. [. . . ] We are thus forced [. . . ] to ask ourselves whether our theory of acquaintance in any way implies a direct consciousness of the bare subject. If it does, it would seem that it must be false; but I think we can show that it does not. Our theory maintains that the datum when we are aware of experiencing an object O is the fact ‘something is acquainted with O’. The subject appears here, not in its individual capacity, but as an ‘apparent variable’ [. . . ].” ((Russell 1913), p. 36-37) 49(Griffin 1980), p. 121-126. January 2, 2011 11:18 History and Philosophy of Logic types

of a necessary propositional function. But, since a propositional function doesn’t assert anything, it can’t be true. Hence Russell’s definition requires, oddly enough, to admit that necessity doesn’t involve truth. This problem at least does not occur in the case of epistemic necessity. Epistemic necessity is nowhere explicitly mentioned as such in Russell, admittedly, but it is still quite a natural concept to bring up as soon as epistemic realizations are introduced and understood as models in a non-Tarskian sense — and epistemic realizations themselves are a natural component to bring up as soon as Principia’s ramified type theory is understood as an abstract scheme to be applied in a natural language environment. Mathematics is a science of forms, but those forms remain the forms of ordinary language sentences. Besides, describing epistemic realizations like so many perspectives upon ramified type theory is a way to give Russell’s view of logic and knowledge a Leibnizian flavor that is not irrelevant after all.

3.4. Epistemic and meta-systemic considerations But there is more than that. There is a follow-up to introducing epistemic necessity (as defined above) as the natural counterpart, in a Russellian epistemic setting, of Tarski’s concept of logical validity. This bears on reducibility, which lends itself to an epistemic interpretation: Although our access to a function ϕ is mediated by quantification over other functions, this in no way precludes the existence, within the hierarchy, of extensionally equivalent predicative functions or functions of order 1 [. . . ]. The axioms of acquaintance and reducibility postulate (respectively) the possibility of knowing individuals and classes in terms of functions that possess a certain epistemic transparency, a transparency embodied by acquaintance in the one case and the absence of complex forms of quantification in the other. Classes occur in the hierarchy only under the guise of predicative functions, which are the means by which they are known. Reducibility thus postulates a concordance between mathematical reality and our knowledge of it that the ramified theory is otherwise unable to demonstrate50. Once the epistemic background of Principia’s ramif-types is given some attention, the reducibility schemata can be understood as meaning that, for any propositional function, there is always an epistemic realization in which that function, as a symbol, is interpreted by a genuine term. In the case of individuals, any object introduced by a definite description and assumed to exist (as opposed to “the actual king of France”), must correspond, up to a shift in epistemic context, to an actual individual. This is a kind of quasi-acquaintance principle. In the case of propositional functions, the axiom schema says that any nonempty functional term corresponds, in some epistemic universe, to an actual predicate, going, in fact, with a whole list of conditions. Indeed, let’s consider a translation pattern such as [(φ) f!(φ!z, x)] ≡ [F x] (?). (What b S1 S2 S1 mentions as the complex propositional function ‘(φ) f!(φ!z,b xb)’ corresponds to what S2 knows as the attribute F .) This equivalence holds even though, among all the predicates φ at stake in (?) that S1 has access to, there may be some for which S2 does not have any predicative counterpart. Suppose that f!(φ,b zb) belongs itself to S2’s epistemic universe. Then, for any particular G also belonging to that universe, F x ⊃x f(G!z,b x) has to be valid in S2’s realization. If, on the contrary, f is not part of S2’s own stock of binary relations, then [Θ(φ, z)] ≡ [f!(φ, z)] (??), where Θ is the higher order complex b b S2 b b S1 propositional function that constitutes S2’s equivalent of f, and this second translation rule (concerning f) adds to the first one (concerning F ), in spite of going in the reverse direction. Then, for any particular G, F x ⊃x Θ(G!z,b x) has to be part of all the truths that S2’s realization validates. The additional pattern (??) can be said to be a package of the original pattern (?). This package, as well as all the truths involving a particular G, are part of the unraveling of (?) in S2’s universe. As a set of sentences, this unraveling has to be consistent and, at some point, it has to come to an end: the process through which S1 and S2 come to an agreement about a given translation cannot involve any infinite chain. The translation of any predicate gives rise to a complex term that is generally non-predicative. Now, just as any consistent first order theory has a Tarskian model, the reducibility schema tells us that any non-predicative propositional function, presumably associated to a consistent well-founded unraveling,

50(Clark and Demopoulos 2005), p. 158. January 2, 2011 11:18 History and Philosophy of Logic types

corresponds to an actual predicate in some epistemic realization. This is a kind of completeness principle (in the sense of a completeness theorem). Thus, theory of knowledge appears as a way to graft the analogue of a model theory onto Russell’s formal system for logic. It should be added that epistemic subjects mean nothing else here but complete epistemic diagrams, that is, ways reality could be carved up. Hence, they are not metaphysically loaded and, besides, remain quite remote from actual concrete cognitive subjects. So, in fact, a formal semantics based on epistemic diagrams has only loose connections with epistemology, properly speaking51. Now, speaking in terms of epistemic models shows an important sense in which Russell allows for “meta-systemic” considerations. Following on that idea, I would like to point out, in conclusion, that universality in Russellian logic is not monolithic at all. Indeed, logical universalism is a label that is commonly pinned on to Russell, but it is important to distinguish two different features traditionally attributed to logic, but which that single label covers. First, “logical universality” consists in logic being about absolutely everything, or in the existence of a single unrestricted range of values for entity variables. “Logical fundamentality,” on the other hand, corresponds to there being the one and the same logic that any reasoning must obey, if it is to be accepted as a reasoning. It may sometimes be synonymous with the acknowledgment of classical logical statements as being laws. The association of logical universality with logical fundamentality is anchored in the tradition of philosophical logic. It goes back to the Kantian Logic, which combines and, after all, assimilates both elements. From a Kantian point of view, indeed, logic is to be described as the science of the “universal and necessary rules of thought in general,” and just as well as a discipline “which applies to all objects in general.” The collapse of logical universality and logical fundamentality occurs not only in Kant but also even in Frege52, and, albeit in a negative way, in van Heijenoort53 and in Hintikka54. Hintikka basically merges the two ideas of universality and fundamentality as special cases of the conception of language as a universal medium, whose semantics is ineffable. I think that the admissibility of a meta-theoretical sphere is essentially what matters to Hintikka, which is perfectly legitimate. But precisely the universality of logic as well as its fundamentality do not, per se, if considered separately, preclude meta-theoretical considerations: they do so only when they are both identified to the inescapability of a language or a context we may be doomed to speak within. The formal calculus for logic given in Principia uses one single ramif-type of entity variables, still it is fully compatible with meta-theoretical considerations and, to be specific, with a model theory where epistemic realizations stand for models. To that extent, universality does not imply “fundamentality,” as Hintikka describes it. Besides, as far as realizations themselves go, two distinct epistemic perspectives do not share the same basic domain of entities, and yet abide by the same logical deductions. To that extent, fundamentality does not imply “universality,” as Hintikka describes it. This suffices, I think, to urge to more care when speaking of logical universalism. In particular, the system of Principia, which is often claimed as Russell’s final renouncement of the universality of logic, should rather be seen as inviting a more flexible combination of two components of logical universalism that turn out to be independent.

Finally, let me summarize the solution of the “no loss of generality” problem that I have advocated in this article. I first remarked that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (“ramif-types”). In particular, types ramify orders,

51On that score, there is an issue about whether it is possible or impossible, in Russell’s view, that objects exist without belonging to any epistemic universe. I leave that question aside. 52“The truths of arithmetic govern all that is numerable. This is the widest domain of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought?” ((Frege 1884), §14, p. 21) 53“Another important consequence of the universality of logic is that nothing can be, or has to be, said outside of the system. And, in fact, Frege never raises any metasystematic question [. . . ].” ((van Heijenoort 1967), p. 326) Frege and Principia Mathematica are lumped together as precluding any semantics, owing to their conception of the universality of logic: “Questions about the system are as absent from Principia Mathematica as they are from Frege’s work. Semantic notions are unknown.” (ibid) 54On the one hand, the fundamentality of logic is “the universality (in the sense of inescapability) of logic. We cannot, as it were, get outside our logic and its intended interpretation” ((Hintikka 1997), p. 162). On the other hand, logical universality is the universality of language: “Semantics is ineffable. Interpretation cannot be varied. Model theory [is] impossible (or irrelevant). Only one world can be talked about. [There is only] one domain of quantiﬁcation in the last analysis. Ontology is the central problem. Logical truths are about this world” (op. cit., p. 227-228). January 2, 2011 11:18 History and Philosophy of Logic types

20 REFERENCES

and not the other way round. Then, comparing different important interpretations of Principia’s theory of types, I considered the three questions as to whether Principia allows for non-predicative variables, at least for non-predicative propositional functions, or at least for functional terms other than predicative variables. The usual answers to those questions follow a natural order: each question is more likely to be answered positively than the previous one. I put forward that an important distinction has to be made between the formal system of ramified type theory, on the one hand, and its realizations in such and such epistemic context, on the other. Owing to that distinction, each of the three questions gets a different answer depending on whether it is considered formally or epistemically. As a consequence, the “no loss of generality” problem gets the following solution. Non-predicative functional terms are only the counterpart, within one epistemic universe, of an attribute F that is not available to the subject S2 of that universe, but that another epistemic subject S1 is acquainted with. They are not ontologically loaded, but to be understood as symbols allowing S2 to render a propositional function that, if it were to be taken as part of S2’s universe, would result in a violation of the VCP. Accordingly, non-predicative variables are to be interpreted substitutionally. So 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 ramif-types. Non-predicative terms and non-predicative variables are relative in nature: they occur only as a way to refer to some epistemic realization from the point of view of another one. In fact, a whole epistemic model theory can be built on that intuition and added to the formal system of types, giving rise to an epistemic concept of necessity that parallels logical validity. In that perspective, the reducibility schema takes on the form of a completeness principle. At any rate, the universality of logic (when couched within a ramified type theory) does not preclude semantics, and thus metatheory. This shows that the contrast that is usually recognized between the Russellian and the Tarskian conceptions of logic should be qualified, if not reconsidered.

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