Structured Variables - bhalimi.parisnanterre.fr · Structured Variables Brice Halimi∗ Drawing on...

Philosophia Mathematica (III) 21 (2013), 220–246.doi:10.1093/phimat/nkt013

Structured Variables

Brice Halimi∗

Drawing on Russell’s substitutional theory, this paper examines thenotion of ‘structured variable’, in order to compare Russell’s and Tarski’sconceptions of variables. The framework of syntactic fibrations, comingfrom categorical logic, is used as a common setting. The main objectiveof this paper is to make sense of the notion of structured variable beyondthe context of Russell’s theory, to question the Tarskian way of under-standing what it is to be a possible value for a variable, and to bring outsemantically structured variables as a renewed and fruitful way of con-ceiving of logical structure.

†Frege and Russell are the founding fathers of logicism. Still, contempo-rary neologicism is mainly Fregean. This paper endeavors to present oneway in which a specifically Russellian neologicism could be pursued. Onetenet of Russell’s conception of logic is that any suitable formal system forlogic should include variables of a single kind, insofar as logic rules anyentity whatsoever, without any distinction between kinds of objects. Rus-sellian logicism seems then doomed to failure because of the paradoxesthat plague unrestricted generality. However, Russell’s substitutional the-ory, as expounded in [Russell, 1906], constitutes a way of maintainingunrestricted untyped quantification while emulating simple type theorythrough substitutional constraints built into the syntax of the substitutionallanguage. Paradoxes other than Russell’s paradox have still to be faced,but I will leave that issue aside.

Indeed, the main upshot of Russell’s substitutional theory is that unre-stricted quantification is still compatible with restricted variation: any sin-gle quantification is unrestricted and unstructured, but a bunch of quan-tifications can be so articulated as to provide a proxy of a typed restrictedvariation (this will become clearer below). This has nothing to do withconditionalizing: ‘for any class x’ cannot be rendered with ‘for any entityx whatsoever, if x is a class, then . . .’, because being a class is not a gen-uine predicate. Structure alone is what explains how to produce a restrictedvariation using unrestricted quantification. Russell explains restricted vari-ation as unrestricted quantification over structured variables, namely over

∗ Universite Paris Ouest (ireph) & sphere, 92001 Nanterre cedex, [email protected]†I would particularly like to thank Sebastien Gandon. This paper, which was originally

part of a joint talk with him, greatly benefited from various discussions with him.

Philosophia Mathematica (III) Vol. 21 No. 2 C© The Author [2013]. Published by Oxford University Press.All rights reserved. For permissions, please e-mail: [email protected]

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a structured pattern of variables that Russell describes as a substitutional‘matrix’.

Structured variables cannot be reduced to a temporary tool in Russell’slogical work. Indeed, Russell’s logicism lies primarily in the idea that gen-erality has some kind of internal structure and hence can be analyzed assuch. That idea runs as a continuous thread from the theory of denotingconcepts in the Principles of Mathematics to Principia Mathematica’s the-ory of types.1 An accurate analysis of generality and a satisfying formal-ization of variation are fundamental targets of Russellian logicism. On thatscore, the use of structured variables constitutes a fundamental aspect, bothphilosophical and technical, of Russell’s logicism.

Russell showed us that unrestricted quantification is compatible withthe use of structured variables. But unrestricted quantification over a sin-gle unstructured variable is of course also possible. Besides, the quantifi-cation over domains formalized by Tarskian semantics can be described,as we will see, as both restricted and unstructured. Thus, the oppositionbetween unrestricted and restricted quantification does not coincide withthat between unstructured and structured variables. I am interested in thelatter, not in the former: the kind of reviving of Russellian logicism thatI would like to suggest does not consist in vindicating unrestricted quan-tification after all, but rather in thinking of variables anew, as structuredvariables, starting from the way in which Russell conceived of them whenbuilding up his substitutional logic.

My aim, in particular, is to compare Russell’s and Tarski’s conceptionsof variables. Such a comparison requires a framework where Russell andTarski can be put side by side. As we will see, the tool of fibrations, andof syntactic fibrations as developed by categorical logic, provides sucha framework. Indeed, fibrations are between the syntactic level at whichRussell’s substitutional theory stands, and the use of domains upon whichTarski’s semantics is based. Besides, the concept of cross section of a fibra-tion allows for formalizing both the structured variables that Russell hadin mind and the variable interpretation of the language that Tarski put for-ward. Accordingly, fibrations will be given a central role, even though acategorical presentation for logic in general is not embraced thereby.

The first part of this paper explains the Russellian notion of struc-tured variable, as opposed to the Tarskian interpretation of variables. Sec-ondly, I would like to make sense of that notion beyond the context ofRussell’s substitutional theory strictly speaking, towards a more Tarski-compatible setting; for that purpose, I will present the concept of fibration.I will then show, in the third place, how Russell’s structured variables and

1 It should be noted, however, that Principia’s variables are ‘internally limited’ by theirrespective conditions of significance, and as such are not structured variables in the senseof Russell’s earlier substitutional theory.



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Tarski’s interpretation of quantification can both be recovered in the uni-fying framework of fibrations. The fourth part is devoted to bringing outsemantically structured variables as a renewed and fruitful way of think-ing what a possible value for a variable is, and thus what a variable is.I will argue that semantically structured variables, while taking the formatof Tarskian semantics into account, go back to a fundamental inspiration ofRussellian logicism. Finally, I will tackle two possible objections againstthe adequacy of the fibrational account of structured variables that willhave been proposed.

1. Russell’s Variables vs Tarski’s Variables

While dealing around 1906 with the paradoxes that he had discovered ear-lier, Russell explored extensively several substitutional theories, the mainexpression of which is the ‘no-class theory’ advocated in [Russell, 1906].Russell’s framework is based on substitution, that is, on the replacementof some constituent of a given proposition by another entity. For a propo-sition p and a constituent a of p, the symbolic sequence ‘p/a; b!q’ istaken as a primitive, and means that q results from replacing a by b inp. Since Russell conceives of propositions as regular entities and doesnot distinguish between kinds of entities, substitution can actually be per-formed in any entity, and is not restricted to propositions: propositionsare singled out only as those entities p that imply themselves (i.e., aboutwhich the relation p ⊃ p holds). Anyway, supposing that p is a proposi-tion, ‘(x) p/a; x!q & q’ means that a replacement of a by x in p resultsin a true proposition, for any x whatsoever. Then ‘p/a; b’ is but a defi-nite description, namely ‘the result of replacing a by b in p’. Finally, theincomplete symbol ‘p/a’ constitutes a ‘matrix’ and can itself be thoughtof as representing the class of all x such that p/a; x is a true propo-sition. The definition of x ∈ p/a is: ( Eq)(p/a; x!q & q). In the sameway, u/x ∈ q/(p, a) is a shorthand for: ( Er)(q/p, a; u, x!r & r). But,owing to the very fine-grained identity conditions of propositions in Rus-sell, it should be borne in mind that a matrix such as p/a represents, reg-imented in a simple type theory, an attribute in intension, an intensionalpresentation of the corresponding class, and not that class directly. Substi-tutional theory emulates a simple type theory of attributes, hence a theoryof classes, but only in a derivative way.

The main point now is that universal quantification over both p anda when ‘p’ and ‘a’ occur only together in subcontexts ‘p/a’ within thescope of the quantification, amounts to a quantification over classes. Thisis the way in which a class variable is proxied by the specific combinationof two entity variables ‘p’ and ‘a’. The symbolic structure of the contextin which variables are quantified over specifies the interpretation of thosevariables, despite the type-free nature of substitutional logic. For instance,



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p∃a∀x(p/a; x ↔ φ(x)) expresses a comprehension schema: the jointquantification over p and a can be construed as a quantification overclasses p/a. Here, so to speak, the respective first occurrences of ‘p’ and‘a’ are absolutely unrestricted, but their second occurrences, in ‘p/a; x’,embody constraints upon the combined variation of p and a (because anyentity x has to be substitutable for a in p) that amount to type distinctions,even though no type whatsoever is mentioned.

Russell’s 1906 structured variables can be compared with two othernotions of ‘structured variables’. The first notion has to do with the useof several terms to emulate the terms of a domain with a given struc-ture. For instance, the construction of integers from natural numbers con-sists in identifying an integer with a class of couples of natural num-bers (−2= (0, 2)= (1, 3)= . . .) and in redefining arithmetic operationsaccordingly. In such a framework, quantifying over integers amounts toquantifying over both components of couples of natural numbers, and astructured variable is essentially a multivariable. Structure comes fromeach one of its values. The second notion can be found in Frege’s Grundge-setze (see §25): a first-level function is written ϕ(β), a second-level func-tion Mβ(ϕ(β)), a third-level function �ϕ(Mβ(ϕ(β))), and so on. Moreaccurately, ϕ(β), Mβ(ϕ(β)) and�ϕ(Mβ(ϕ(β))) are well-formed formulaswhere ‘ϕ’, ‘M’ and ‘�’ occur as bindable variables. Then (using modernnotations), extensionality of all second-level predicates, for example, isexpressed by: ∀M∀ϕ∀ψ(∀x(ϕx ≡ψx)→ Mxϕx ≡ Mxψx). In that case,a structured variable is a variable whose symbol is explicitly structured, itis an intrinsically structured variable: structure comes from the syntacticspecification of the variable symbol itself. Structured variables brought upby Russell’s substitutional theories are not of the Fregean kind, and arequite akin to multivariables, insofar, in particular, as they allow Russell toavoid any commitment to attributes and a fortiori to classes in quite thesame way as couples of natural numbers exempt the mathematician fromany commitment to negative integers. Still, Russellian structured variablesare not exactly multivariables: while the structure of a Russellian struc-tured variable cannot be read off its symbol, it cannot be found in eachvalue of the variable either. It rather consists in the mutual adjustment ofvalues of different variables, and thus has to do with the internal organiza-tion of a range of values.2

Russell himself mentions the notion of internal structure aboutvariables:

We may say that ‘φx is always true’ means ‘φx is true wheneverit is significant’ or ‘φx is never false’. We might then say that agiven function φx will always have a certain range of significance

2 Many thanks to Gregory Landini for suggesting this clarification.



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which will be either individuals, or classes, or classes of classes, ordual relations of individuals, or etc. The difficulty of this view lies inthe proposition (say) ‘φx is only significant when x is a class’. Thisproposition must not be restricted, as to its range, to the case whenx is a class; for we want it to imply ‘φx is not significant when x isnot a class’. We thus find that we are brought back after all to vari-ables with an unrestricted range. If this is to be avoided, the range ofsignificance must be somehow given with the variable; this can onlybe done by employing variables having some internal structure forsuch as are to be of some definite logical type other than individuals.[Russell, 1906, pp. 205–206]

This is precisely what a substitutional setting achieves. So, in specific con-texts, it makes sense to speak of ‘p/a’ as a class variable, even though,properly speaking, only entity variables (‘p’, ‘a’, . . . ) are available. In thesame way, variables of classes of classes are represented through (inten-sional) matrices q/(p/a), variables of classes of relations through (inten-sional) matrices q/(p/(a, b)), and so on. The symbolic structure of thesubstitutional environment is mirrored back to the variables that occur in it,giving rise to structured variables.

Think by contrast of the way in which variables are interpreted inTarskian semantics: they are assigned, from the outside, a given set-theoretic domain (namely, the domain of some structure for the language).The range of variation of a variable is not ‘given with the variable’: on thecontrary, it constitutes an external universe of discourse. So there is a deepopposition between the Russellian and the Tarskian conceptions of vari-ables. Russell’s substitutional theory manages to restrict variables throughsyntactic constraints that are built into the substitutional possibilities. Thispoint has been thoroughly emphasized by Gregory Landini.3 On the otherhand, Tarskian variables are externally restricted, which means that theirrestriction comes only from the semantic restriction of their range.

From this it could seem that a variable is either syntactically structuredor structureless. I will argue, on the contrary, that it is possible to steer amiddle course between these two extremes, namely to bring out interpretedvariables whose range is specified, but not from the outside. The basic ideaunderlying that claim is to take contexts seriously, which means workingwith semanticized contexts (in opposition to Russell’s syntactic ‘matrices’)without dealing with externally restricting ‘universes of discourse’ (as inTarskian semantics).

It is first necessary to set up a presentation that could embrace bothRussell’s and Tarski’s perspectives. On that score, the framework offibrations is a quite natural field to look at, since it is abstract and general

3 See in particular [Landini, 1987; 1998, p. 142; 2007, pp. 82–84].



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enough to gather both traditions. It supports in this paper the interactionbetween Russell and Tarski and leads, it will be argued, to a renewed con-ception of what a variable is. In particular, syntactic fibrations for typetheories are useful to remould the usual distinction between syntax andsemantics, insofar as syntactic contexts are used to make up a canonicalmodel, which puts one on the track of semanticized contexts. So this paperdraws on Russell’s analysis of variables in 1906 but also aims at imple-menting it so that it can be confronted with the modern Tarskian way ofdoing and presenting logic.

2. The Framework of Fibrations

Before expounding the concept of fibration, let us consider the well-knownexample of many-sorted first-order logic — that is, first-order logic whereindividuals of different types are considered in a language L0 of fixed sig-nature. The judgment to the effect that some constant a is of type σ iswritten a : σ . For a predicate symbol P of type (τ1, . . . , τk) and a1, . . . , akof respective types τ1, . . . , τk , P(a1, . . . , ak) is a well-formed proposition,which is written:

� a1 : τ1 . . . � ak : τk

� P(a1, . . . , ak) : Prop

Here as everywhere else, stands for a set of declared variables: = {x1 :σ, . . . , xn : σn}, where x1, . . . , xn are supposed to be all the free variablesin a1, . . . , ak .

The other rules for well-formed propositions are:

� ϕ : Prop �ψ : Prop � ϕ ∧ ψ : Prop

� ϕ : Prop �ψ : Prop � ϕ→ψ : Prop

, x : σ � ϕ : Prop � ∀x : σ.ϕ : Prop

In the last rule, ‘, x : σ ’ means that ‘x : σ ’ is added to all the declaredvariables of . The idea is that if, for any x : σ , ϕ is a well-formed propo-sition in context ∪ {x : σ }, then ∀x : σ.ϕ is a well-formed propositionin context .

Those formation rules then give way to the following usualtransformation rules in the form of a sequent calculus, where a sequent ‖ �ψ means that in the context of variable declarations x : σ the



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proposition ψ follows from the set of propositions . I only mention theintroduction rules:4

‖ � ϕ ‖ �ψ ‖ � ϕ ∧ ψ ‖ ,ϕ �ψ ‖ � ϕ→ψ

, x : σ ‖ � ϕx not free in .

‖ � ∀x : σ.ϕ

Actually, the ∀-rules (going both ways), namely:

, x : σ ‖ � ϕ ‖ � ∀x : σ.ϕ

can be seen as an adjunction �∗x � ∀x , where �∗

x is the ‘weakening func-tor’ that adds x : σ to any context, and ∀x is universal quantification overx : σ . This adjunction historically motivated the consideration of a ‘syntac-tic fibration’, that is, of a syntactic category of sequents above a syntacticcategory of contexts. Indeed, beyond the technicalities involved here, the∀-introduction rule can be understood as a shift from a logical relation incontext , x : σ to another logical relation in context . Thus it becomesnatural to consider a base category of logical contexts (variable declara-tions), above each of which lie all the possible logical relations that canbe derived in that context: those logical relations make up the ‘fiber’ thatgrows above the corresponding base context. A logical calculus then isnothing else but a set of connections between logical relations lying above(possibly) different logical contexts. All those logical relations put togethermake up a category of sequents, but each sequent lies above a single con-text. The syntactic fibration at stake here simply maps any sequent to itscontext, and keeps track of the correspondence that links the derivation ofa sequent from another and the transformation of the corresponding con-text. Let us expound that intuitive idea more formally.

The base category Ct has as objects the sets of variable declarations = {x1 : σ1, . . . , xn : σn}, and as arrows f : → ′ between = {x1 :σ1, . . . , xn : σn} and ′ = {x ′

1 : σ′1, . . . , x ′

m : σ′m} all sequences 〈 f1, . . . , fm〉

of terms such that � f j (x1, . . . , xn) : σ′j can be derived for any

1≤ j ≤ m. The category Ct = Ct (L0) refers implicitly to the signatureof L0, that is, to a list of primitive predicates of respective given types.

The fiber category above an object of Ct is the category ofpropositions proved to be well-formed in that context. More specifically,let us consider the logical category Lg:

4 See [Jacobs, 1999, pp. 171, 224] for further details.



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• whose objects are all couples (;χ) of a context and a propositionχ such that � χ : Prop is derivable;

• whose arrows f : (;χ)→ (′;χ ′) are all arrows f : → ′ inCt such that ‖ χ � χ ′[ f j (x1, . . . , xn)/x ′

j ] is derivable for each1≤ j ≤ m, for = {x1 : σ1, . . . , xn : σn} and ′ = {x ′

1 : σ′1, . . . , x ′

m :σ ′

m} as above.

A bunch Ax of specific axioms can be added to the purely logical sequentcalculus underlying Lg. In that case, Lg has to be written Lg(Ax).

So, for example, an object of Lg is a well-formed proposition ϕ(x) incontext x : σ , x : σ � ϕ(x) : Prop (which does not mean that ϕ(x) is true).And an arrow f in Lg from (x : σ ;ϕ(x)) to (y : τ ;ψ(y)) corresponds to aterm f (x) such that:

• x : σ � f (x) : τ ;• x : σ ‖ ϕ(x)�ψ[ f (x)/y] .

There is an obvious projection π0 : Lg → Ct which maps any proposi-tion in context (;χ) onto . All propositions in context make up asubcategory (Lg) = π−1

0 () of Lg, which is the so-called ‘fiber’ above. The projection π0 is called a fibration.

The exact technical sense of the concept of fibration does not mat-ter here:5 fibrations are simply the categorical generalization of surjectivemaps on objects. Basically, to each point b in the base space B correspondsa subcategory p−1(b)= Eb of the ‘total space’ E , the fiber above b, gener-alizing the set-theoretic notion of pre-image. The fundamental extra ingre-dient in comparison with surjective maps and pre-images is the fact thatthe base space can be endowed with any kind of structure, and that theexistence of a fibration requires then some systematic connection betweenthe relations between any two points in the base space, and the relationsbetween the two corresponding fibers in the total space (see Figure 1).

A simple example of a fibration is provided by set-indexed familiesof objects: the base category is the category Set of all sets, and the fiberabove any set I is the category FamI of all I -indexed families. This meansthat one considers the fibration π that maps any set-indexed family (Ai )i∈Ito I . The functor π is obviously surjective on objects, but it is surjective onarrows as well: indeed, for a given family (A j ) j∈J , any arrow u : I → Jinduces an inclusion arrow (Au(i))i∈I → (A j ) j∈J in E , where (Au(i))i∈Iappears to be the result of reindexing (A j ) j∈J .

5 See [Jacobs, 1999, pp. 19–27]. A fibration is a functor π : E → B such that for anyobject y ∈ E and any arrow u : i → π(y) in B, there exists an arrow u : x → y in E that is‘the Cartesian lift of u’, which means that : (i) π(u)= u; (ii) for any arrow g : z → y in E ,if there is w : π(z)→ i such that π(g)= u ◦ w, then there is a unique arrow h : z → x in Eabove w (i.e., π(h)=w) such that g = u ◦ h.



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Figure 1. A fibration in general.

Looking upwards instead of downwards, a fibration π : E → Bbecomes a family of categories Eb indexed by B, called an ‘indexed cate-gory’. Given a category B, an indexed category over B is a contravariantfunctor6 F : B → Cat . In other words, it is a functor that assigns to eachobject b of B a category F(b) and to each arrow u : b → b′ in B a functorF(u) : F(b′)→ F(b), written u∗, going in the opposite direction. Actu-ally, indexed category and fibration are almost two equivalent notions: allthe fibers Eb of any fibration π : E → B gives rise to the indexed categoryF : b �→ Eb over B, and conversely any indexed category F : B → Catgives rise to the fibration π : E → B whose fiber over b is exactly F(b).In what follows, anyway, I won’t distinguish between indexed categoriesand fibrations.

As a functor, an indexed category F sends any arrow u : a → b in thebase B to a functor F(u)= u∗ : F(b)→ F(a) between the correspond-ing fibers, called a ‘substitution functor’ (also, a ‘reindexing functor’).One can get a sense here that the base category of a fibration works asa control space, because, as has been said, the existence of a fibration (or,equivalently, of an indexed category) requires some systematic connectionbetween the relations between any two points in the base (arrows u), andthe relations between the corresponding fibers (functors u∗ between fibersin the case of an indexed category, Cartesian lifts u in the case of a fibra-tion). This is the case, in particular, of the base syntactic category Ct andthe total logical category Lg. As we will see, this is how structure canenter the scene and be bestowed on variables. The fibration π0 : Lg → Ctis a fibration between categories that have been lifted out of syntactical

6 To be exact, an indexed category is defined as a pseudo-functor, not as a functor prop-erly speaking (see [Jacobs, 1999, p. 50]), but the definition of a pseudo-functor is morecomplicated, whereas the distinction is not relevant to the points involved here.



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Figure 2. The syntactic fibration π0.

stuff (namely, the sequent calculus for many-sorted first-order logic); thatis why it is called a syntactic fibration (see Figure 2): in that fibration, ajudgment such as ‖ φ �ψ becomes a logical relation in the fiber over thetype context , hence each such type context becomes an index for a logicdescribing what holds in this context. Looked at from that point of view, asyntactic fibration becomes a ‘variable logic’, in the same way as a sheafof sets can be described as a variable set. Still, π0 gives rise to somethingsemantic — exactly as Henkin models in the proof of the completenesstheorem for first-order logic are syntactic in nature, but constitute mod-els nonetheless. Precisely, the working hypothesis of this paper is that π0affords the basis for semanticizing deductive contexts of the occurrencesof variables.

3. Tarski’s Fibration and Russell’s Fibration

Drawing on the categorical semantics for logic set out above, the idea isto consider structured values whose structure comes from that of contextvariations as captured by a fibration. The notion of variable context is par-ticularly relevant in the case of a syntactic fibration, but actually extendsto fibrations in general, because any fibration can be seen as a kind of syn-tactic fibration (see below, Section 5) and because the change of fiber inany fibration works as a change of context. Now, since we want to getRussell’s and Tarski’s conceptions of logic to interact, we first need toshow how Tarski’s semantics as well as Russell’s substitutional theory areamenable to the framework of syntactic fibrations. In what follows, the



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fibration associated with Russell’s substitutional theory will be a variantof the syntactic fibration π0, whereas Tarski’s semantics will be morenaturally cast in terms of a fibration built from structures for a language.

3.1. Tarski’s Fibration

Let us consider Tarski first. The typical clause for the interpretation ofquantification in Tarskian logical semantics is, for any first-order structureM and any assignment σ of values to the variables of the language:

M � ∀xφ(x)[σ ]iff, for any assignment σ ′ that differs from

σ at most at x,M � φ(x)[σ ′].

In other words, variables of the language are replaced with variables ofassignments. Hence everything hinges on assignments σ :Var → |M | ofvalues to all variable symbols of the language.

So the natural move to present Tarskian semantics in terms of a fibra-tion is the following. The base space consists of all possible ‘universes ofdiscourse’, and the fiber above any domain D is nothing but the set of allassignments in D.

More formally, let S be the category of all structures for a fixed first-order language L, whose arrows are given by the following rule: giventwo objects N and M in S, there is an arrow f : N → M in S each timethere is an L-homomorphism from M to N .7 For any M ∈ S, let AM be thecategory reduced to the set |M |Var of all assignments in the domain |M |of M . The total category A is the union of all AM , M ∈ S, and any arrowf : N → M in S gives rise to the functor f

∗: σ ∈AM �→ f ◦ σ ∈AN . One

can check that the mapping M �→AM , f �→ f∗

induces an indexed cate-gory, hence a fibration πT :A→ S, which sends any assignment σ in M ,to M . I shall call πT ‘Tarski’s fibration’ for logic.8

7 Given two structures M and N for L, f : M → N is a homomorphism if, for any itemo in the signature of L, f (oM )= oN .

8 Dynamic logic conceives of assignments as being possible worlds, so that the identityof two assignments on all variables but maybe x becomes an accessibility relation Rx .Existential quantification over x then becomes a possibility operator:

M, σ � �xϕ iff there exists θ such that σ Rxθ and M, θ � ϕ.

In particular, it is possible to place constraints on assignments, and to consider strongeraccessibility relations than Rx — meaning that all the assignments which are genuinelyaccessible from a given assignment make up only a subset of |M |Var. See for instance[van Benthem and Alechina, 1997] for details. Looking at dynamic logic as a way of struc-turing Tarski’s fibration is an approach worth pursuing. For lack of space, I must foregosuch a study in this paper.



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So the conception of variables exhibited by Tarski’s semantics is actu-ally amenable to a fibrational semantics. The collection of all structuresfor a language L make up a base category that is structured by the homo-morphisms existing between those structures in the set-theoretic universe.Hence the base category works as a control space because any travel withinthe total space has to follow specific paths: the paths that relate an assign-ment σ in M to another σ ′ in M ′ to the extent that σ ′ is the image of σ bysome homomorphism f : M → M ′.

3.2. Russell’s Fibration

Is it possible as well to frame Russell’s substitutional theory of classesin a fibrational setting? There are at least two reasons in favour of sucha project. The first one is that a regimentation of Russell’s and Tarski’stheories into the same framework would allow one to establish a bettercomparison between their respective conceptions of variables. The secondreason is the naturalness of introducing a variant of the syntactic fibrationπ0 to express Russell’s substitutional theory. Indeed, owing to the verydefinition of arrows in Lg, substitution within a proposition correspondsto the existence of an arrow in the total space of π0. In terms of the indexedcategory associated to π0, any arrow f = 〈 f1, . . . , fm〉 in the base categoryCt induces a substitution functor f ∗ between fibers that corresponds to themultiple substitution operation χ ′ �→ χ ′[ f j/x ′

j ] (1≤ j ≤ m). So the basicidea is to translate substitution within a proposition, in Russell’s substi-tutional logic, as the existence of an arrow in π0. In order to formalizeRussell’s theory through a fibration, a variant of the syntactic fibration π0,called ‘Russell’s fibration’ for substitutional logic, will be introduced. Thecorrespondence between Russell’s substitutional logic and Russell’s fibra-tion will thereby be explained.

Language of Russell’s substitutional logic The first requirement is lay-ing out a formal language R for Russell’s substitutional logic.9 The basicterms of R are just individual variables x , y, z, . . . . The rules for well-formed propositions of R can be given inductively as follows: (i) givenbasic terms α, β, π and δ, α ⊃ β and π/α;β!δ are (atomic) well-formedpropositions; (ii) A ⊃ B, A/B; C!D and (μ)A(μ) are well-formed propo-sitions whenever A, B, C and D are basic terms or well-formed proposi-tions and μ is an individual variable free in A.10

9 For a complete and detailed presentation of the syntax of Russell’s substitutional logic,see [Landini, 1998, pp. 102–105].

10 Negation is defined by ¬A := (μ)(A ⊃μ). The other propositional connectives ‘∨’,‘&’ and ‘≡’ are then defined from ‘⊃’ and ‘¬’ in the standard way. Identity is also definedby x = y := (p)(q)(r)(a)((p/a; x!q&p/a; y!r)⊃ (q ⊃ r)).



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Language of Russell’s fibration The language R of Russell’s substitu-tional logic can be translated into a many-sorted first-order language L Rthat is a variant of L0. Let us define L R first. In keeping with Russell’s con-ception of logic, the type theory over which L R is defined consists only ofthe type ε of all entities whatsoever and of the two subtypes At of basicterms (atoms) and Prop of well-formed propositions11 — where At andProp’s being subtypes of ε means that

� x :At � x : ε

and � x : Prop � x : ε

are taken as axioms of the type theory over which L R is defined.The terms of L R are individual variables x , y, z, . . ., propositional vari-

ables u, v, . . . , as well as individual constants a, b, . . . and propositionalconstants p, q, r , . . ..12 Besides, the signature of L R comprises two rela-tion symbols I (2) and S(4) corresponding to implication and substitutionas relations between basic terms in Russell’s substitutional logic. Thus L Rdiffers from L0 insofar as the only available types in L R are ε, At andProp, constant terms are permitted, and there are only two relation sym-bols allowed as primitive.

The rules for well-formed propositions of L R can then be given induc-tively as follows: (i) I (α, β) and S(π, α, β, δ) are (atomic) well-formedpropositions in contexts {α :At, β :At} and {π :At, α :At, β :At, δ :At},respectively; (ii) A → A′, A[B/C] and ∀μA(μ) are well-formed proposi-tions in a context whenever A′ : Prop, B : ε and C : ε are derivable from and μ is an individual variable declared in and free in A. In any con-text , ‘�’ will refer to some fixed logical truth in that context, such as

In Russell’s original substitutional logic, any well-formed proposition A can be nom-inalized into a term {A} which is the proposition A viewed as a single entity. Here, thefull strength of nominalization is not retained: nominalized propositions are not introducedamong terms, and for instance A/{B}; C!D and A/B; C!{D} (B and C being two well-formed propositions) are both identified with A/B; C!D. Nevertheless, the relations ofimplication and substitution are defined so as to have both basic terms and propositions aspossible arguments.

11 Actually, Russell’s universalism does not balk at its being blended with the exis-tence of several subtypes of individuals. The subtype Prop could itself be divided intomore refined subtypes, namely the subtypes Propn of all propositions having n constituents,for all n ≥ 1. For simplicity’s sake, that possibility will be disregarded.

12 Constant terms are needed because the occurrence of x : σ in a context ∪ {x : σ },by virtue of the ∀-introduction rule, amounts to quantification over x : σ in context . Yetone wants to be able to mention a particular class p′/a′ and say that it belongs to q/(p/a),without being committed to saying that q/(p/a) contains all classes. That is the reason whyconstant terms are admissible, for each α of which α : ε or α : Prop is the correspondingdeclaration in the current context, as in the case of a variable, although constant terms arenot bindable.



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p → p for some well-formed proposition p in context . Note that con-texts can include declared constant terms on top of declared variables.

A sequent w.r.t. L R is finally defined as in the case of L0: it is a state-ment of the form ‖ φ �ψ , where φ and ψ are well-formed propositionsin the context .

Translation of the first language into the second one The translationof the language R of Russell’s substitutional logic into L R should not beconfused with the partial translatability of substitutional logic into simpletype theory, where arbitrary high orders are considered.13

That translation needs first the definition of an ‘erasing’ operation.Erasing, within a proposition ϕ of L R , a strict sub-proposition ψ of ϕ,gives a proposition e(ψ, ϕ) which is defined inductively as follows:

• e(A, A → B)= B;• e(B, A → B)= A;• e(B, A[B/C])= e(C, A);• e(B,∀μA(μ))= ∀μ e(B, A).

Each well-formed proposition A of R can then be translated into asequent A∗ = A ‖ ϕA �ψA w.r.t. L R , according to the following rules:

• (α ⊃ β)∗ (for basic terms α and β) is α :At, β :At ‖ � � I (α, β).• (π/α;β!δ)∗ (for basic terms π , α, β and δ) is π :At, α :At, β :At, δ :

At ‖ � � S(π, α, β, δ).• (A ⊃ α)∗ is A ∪ {α : ε} ‖ ϕA →ψA � I (�, α).• (α ⊃ A)∗ is A ∪ {α : ε} ‖ I (�, α)� ϕA →ψA.• (A ⊃ B)∗ is A ∪ B ‖ ϕA →ψA � ϕB →ψB .• (A/B; C!D)∗ is A[C/B] ‖ A[C/B]� A[C/B], where A[C/B] isA with every occurrence of ‘B : ε’ replaced with ‘C : ε’, or everyoccurrence of ‘B : Prop’ replaced with ‘C : Prop’.

• If ‘A/B; C!D’ occurs in a proposition � as a strict sub-propositionand ‘D’ also occurs in � outside ‘A/B; C!D’, then �∗ is(e(A/B; C!D,�)[A/D])∗[C/B].

• If ‘A/B; C!D’ occurs in a proposition � as a strict sub-proposition but ‘D’ does not occur otherwise in �, then �∗ is(e(A/B; C!D,�))∗.

• ((μ)A(μ))∗ is A ∪ {μ : ε} ‖ � � ϕA →ψA.

13 A translation of simple type theory into Russell’s substitutional logic is set out in[Landini, 1998, pp. 140–144]). But it is not true that every formula of the type-free lan-guage of Russell’s substitutional logic can be translated into the language of simple typetheory: well-formed formulas such as ‘(x) x ⊃ y ⊃ x’ or ‘( Eq) p/a; x!q & q’ cannotbe translated. This is due to Russell’s use of ‘⊃’ and ‘&’ as relations between terms, whichoriginates in his refusal to distinguish between propositions and entities.



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The assertion of a proposition obtained by substitution is a bit awk-wardly rendered by Russell through ‘ & q’, as in ‘p/a; b!q & q’. Asone can see, the basic idea here is to express the assertion of q, not bythe explicit addition of ‘ & q’ any more, but simply by the derivabil-ity of q in the sequent calculus upon which π0 is built. Actually, thecase of ‘( Eq) p/a; x!q & q’ (see above, note 13) confronts one withthe assertion ‘ & q’ of some bound variable q which cannot be identi-fied with some particular well-formed proposition. Still, this case can beaddressed. In fact, since existential quantification is defined through nega-tion and universal quantification in the usual way, and negation is definedthrough universal quantification by ¬A := (μ)(A ⊃μ), it is sufficient toexplain how to translate a well-formed proposition of the form (μ)(A ⊃μ). But, owing to the clauses that define A∗, one has that ((μ)(A ⊃μ))∗is A ∪ {x : ε} ‖ ϕA →ψA � I (�, x).

Interpretation of Russell’s substitutional logic The interpretation ofRussell’s substitutional logic into L R is the formal theory TR defined asa set of sequents by the following rule: a sequent s w.r.t. L R belongsby definition to TR iff s = θ∗ for some theorem θ of Russell’s substi-tutional logic as expressed in R. Owing to that definition, the corre-spondence θ �→ θ∗ constitutes an interpretation of Russell’s substitutionallogic into TR (in the sense of the interpretation of a formal system intoanother one).

Then, the theory TR can itself be interpreted (now in the sense of theinterpretation of a theory by a model), namely by a syntactic fibration πR ,exactly as many-sorted first-order logic can be interpreted by π0. The basecategory Ct R and the total space LgR of πR are defined by analogy withthose of π0.14

So we have in the end a two-step interpretation: the interpretation ofRussell’s substitutional system into TR , and then the interpretation of TRby πR . According to that process, any theorem of Russell’s substitutionallogic corresponds to a derivable sequent in TR , and that sequent itself cor-responds to an arrow in the total space of πR . That principled transfor-mation of any theorem of Russell’s substitutional logic into an arrow inthe total space of Russell’s fibration will henceforth be what is meant bythe ‘correspondence’ between the former and the latter. Let us now lookat examples of that correspondence.

14 Introducing propositional terms (constants and variables) in L R causes the followingsubtlety: an arrow (x : σ � ϕ(x) : Prop)→ (y : τ �ψ(y) : Prop) in the total space of πR isa term f of L R such that x : σ � f (x) : τ and x : σ ‖ ϕ(x)�ψ[ f (x)/y]. In case τ = Prop,f (x) is a proposition in the context (x : σ), and thus an object of LgR , whereas f (i.e.,f (x) as a function) is an arrow in LgR . But that subtlety will not have to be dealt with inall that follows about Russell’s fibration.



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3.3. Instances of Arrows in Russell’s Fibration

Let us first consider a proposition such as p ⊃ q, supposed to be atheorem of Russell’s substitutional logic. For simplicity’s sake, one canassume that the respective translations p∗ of p and q∗ of q in L R have thesame context and are of the form ‖ � �ψp and ‖ � �ψp. One can fur-ther assume that p can readily be read off ψp, so that ‘ψp’ can be replacedwith ‘p’ without risk of confusion, and ‘ψq ’ replaced with ‘q’ in the sameway: this is but a notational convention that makes the correspondenceeasier to handle. Then, p ⊃ q’s theoremhood in Russell’s substitutionallogic corresponds to the derivability of ‖ � → p � � → q or, equiva-lently, to the derivability of ‖ p � q and thus, owing to the definition ofπR , to15 id : (; p)→ (; q) being an arrow in LgR .

In the same way, let us consider a theorem of Russell’s substitu-tional logic of the form p/a; b!q & q (which is Russell’s definition forb ∈ p/a). By definition, (p/a; b!q & q)∗ is:

(e(p/a; b!q, p/a; b!q & q)[p/q])∗[b/a]= p∗[b/a]= p[b/a] ‖ � � p[b/a]

(with the same convention that φp can be taken to be � and that ψpcan be identified with p). So the theoremhood of p/a; b!q & q corre-sponds to the derivability of both p[b/a]� b : ε (which is an axiom)and p[b/a] ‖ � � p[b/a], and thus to idp[b/a] : (p[b/a]; �)→ (p; p)being an arrow in LgR .

Finally, since the occurrence of x in a context is a way of express-ing the quantification over x in a proposition written on the right sideof a sequent in context , a proposition such as (x) p/a; x!q & q cor-responds to the derivability of both p, x : ε � x : ε and p, x : ε ‖ � �p[x/a], and thus to idp∪{x :ε} : (p ∪ {x : ε}; �)→ (p; p) being an arrowin LgR .

Those first examples should suffice to show how theorems of Rus-sell’s substitutional logic are coded by arrows in Russell’s fibration, stillwith the notational convention that the well-formed propositions of L Rare written so as to be identified with their counterparts in Russell’s sub-stitutional language. Let us try now to catch up more specifically withRussell’s substitutional treatment of classes and classes of classes. In thesubstitutional language, p1/a1 proxies a class and q2/(p2, a2) a dyadicrelation. Owing to what has just been shown, the assertability of p1/a1 ∈q2/(p2, a2) in Russell’s substitutional logic amounts to the derivability of

15 If = {x1 : σ1, . . . , xn : σn}, id is simply the sequence 〈 f1, . . . , fm〉 of projectionsgiven by fi (x1, . . . , xn)= xi for each 1≤ i ≤ n.



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p1 : ε, a1 : ε ‖ � � q2[p1/p2, a1/a2] (for simplicity’s sake, the other possi-ble declared terms of q2 are disregarded), and so to 〈idε, idε〉 : (p1 : ε, a1 :ε; �)→ (p2 : ε, a2 : ε; q2) being an arrow in LgR , where idε is the func-tional term given by idε(t)= t for any t : ε.

Actually, p1/a1 is not a class as such, but rather, as pointed outabove (beginning of Section 1), an intensional presentation of a class.Suppose that p′

1/a′1 = p1/a1, i.e., that p′

1(x) := p′1[x/a

′1] expresses a

condition which is extensionally equivalent to p1(x) := p1[x/a1]. Thenx : ε ‖ p1(x)� p′

1(x) and x : ε ‖ p′1(x)� p1(x) are both derivable,

which means that g := idε : (x : ε; p1(x))→ (x : ε; p′1(x)) and g′ := idε :

(x : ε; p′1(x))→ (x : ε; p1(x)) are two inverse arrows in LgR , making

(x : ε; p1(x)) and (x : ε; p′1(x)) isomorphic as objects of LgR . The rela-

tion existing between those two objects is obviously an equivalence rela-tion, and the class p1/a1 = p′

1/a′1 is nothing but their common equivalence

class w.r.t. that equivalence relation. That point shows that there is a wayto account in πR for Russell’s following remark:

Although a matrix [such as p/a] is more akin to a class or relationin extension than in intension, it is not quite extensional; for even ifp/a and p′/a′ define the same class, they can still be distinguishedif p is different from p′ or a from a′. Thus the theory advocatedis intermediate between that of intension and extension.16 [Russell,1973, p. 175]

Let us consider further examples of the correspondence between state-ments in Russell’s substitutional logic and arrows in πR . Given that a1 isan actual constituent of p1, there is a natural way of looking at p1, whichconsists in focusing on the class presented by p1/a1. The two followingsequents are obviously derivable:

a1 : ε ‖ p1 � p1;x : ε ‖ p1(x)� p1[x/a1].

Keeping the hypothesis that p′1/a

′1 = p1/a1, one gets (again, simply by

definition of arrows in LgR):

(a1 : ε; p1)f =idε

�� (a1 : ε; p1) (x : ε; p′1(x))

k=idε��

(x : ε; p1(x))g=idε

��(a′1 : ε; p′

1) .

h=idε

��

16 On this, see [Landini, 1998, p. 148], which defends the view of substitutional logicas essentially an intensional calculus.



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The arrow f presents the class p1/a1, the arrow g presents the same classas p1(x)/x , h as p′

1/a′1, and k as p′

1(x)/x . However simple, that principledmultiplicity of presentations is the fibrational equivalent of what Russellconsiders to be the very core of the ideas of substitution and of exten-sion. Indeed, to mention Russell’s own notations, it is essential that a class(φa)/a is referred to independently of the index a taken as the initial value((φa)/a = (φx)/x = (φx ′)/x ′) and independently of the proposition takenas the intensional presentation of the class ((φa)/a = p/a = p′/a′).17

Now, let us consider a class of classes q2/(p2/a2), and suppose thatp1/a1 ∈ q2/(p2/a2). Russell writes:

A class of classes [. . .] is a particular kind of dual relation, namelythe kind defined by a function of p and a, when p and a appear onlyin the form p/a.18 [Russell, 1973, p. 176]

In other words, q/(p, a) is a class of classes if a is a constituent of qonly because it is a constituent of the proposition p (being granted thatp and a are constituents of q). In that perspective, p1/a1 ∈ q2/(p2/a2)can be rendered by a1 : ε � p1(a1) : Prop and a1 : ε ‖ � � q2[p1(a1)/p2]being derivable, in other words by the existence of the arrow p1(a1) : (a1 :ε; �)→ (p2 : Prop; q2) in πR , where p1(a1) is the functional term x �→p1[x/a1]= p1(x).

Furthermore, one has p2 : Prop � p2 : ε as an axiom, hence anarrow (p2 : Prop, a2 : ε; q2)→ (p2 : ε, a2 : ε; q2), and another arrow(p2 : Prop; q2)→ (p2 : Prop, a2 : ε; q2) comes from weakening, hence acomposite arrow ι : (p2 : Prop; q2)→ (p2 : ε, a2 : ε; q2).

Besides, the derivability of u : Prop, x : ε ‖ u ↔ p1(x)� q2[u/p2] cor-responds in πR to the arrow idProp : (u : Prop, x : ε; u ↔ p1(x))→ (p2 :Prop; q2), and the derivability of p1 : ε, a1 : ε ‖ � � p1 ↔ p1(x)[a1/x] tothe arrow 〈idε, idε〉 : (p1 : ε, a1 : ε; �)→ (u : Prop, x : ε; u ↔ p1(x)).

Finally, p1/a1 ∈ q2/(p2, a2) guarantees, as above, that (p1 : ε, a1 :ε; �)→ (p2 : ε, a2 : ε; q2) is an arrow in LgR .

One ends up with the diagram in Figure 3. In Figure 3, the arrow fpictures the fact that p1/a1 belongs to the class of classes q2/(p2/a2),whereas the arrow g pictures the fact that p1/a1 belongs to the dyadicrelation-in-extension q2/(p2, a2). So the distinction made by Russell

17 See [Russell, 1973, pp. 172–173].18 The translation into L R of the formal definition given by Russell is: q/(p, a) is a

class of classes, written q/(p/a), iff

u : Prop, v : Prop, x : ε, y : ε, z : ε ‖ u[z/x]↔ v[z/y]� q[u/p, x/a]↔ q[v/p, y/a]

is derivable. Here I gloss over the complications studied by Landini in his [1998,pp. 149–152].



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Figure 3. Bundle of arrows attached to a matrix.

between the general case of a dyadic (or ‘dual’) relation and the morespecific case of a class of classes can be accounted for in the setting ofRussell’s fibration.

Moreover, exactly as (a1 : ε; p1) was the common target in the previ-ous diagram, the matrix q2/(p2, a2) gives rise to a bundle of arrows ofthe form (u : ε, x : ε; q2(u, x))→ (p2 : ε, a2 : ε; q2), that start from differ-ent domains, but have the same target (p2 : ε, a2 : ε; q2) (see Figure 3).(Here, as above, ‘q2(u, x)’ is just a shorthand for ‘q2[u/p2, x/a2]’.) In thesame way, a matrix r3/(q3, p3, a3) would correspond to arrows startingfrom different domains, but with the same target (q3 : ε, p3 : ε, a3 : ε; r3).All those arrows whose target is (p2 : ε, a2 : ε; q2) make up a subcategoryLgR(q2) of the category of all arrows in LgR and correspond to differentways of looking at q2(p2, a2).

And again, as in the case of classes, the extensional equivalence oftwo relations q2/(p2, a2) and q ′

2/(p′2, a′

2) can be emulated by the deriv-ability of both sequents u : ε, x : ε ‖ q2[u/p2, x/a2]� q ′

2[u/p′2, x/a′

2] andv : ε, y : ε ‖ q ′

2[v/p′2, y/a′

2]� q2[v/p2, y/a2], which amounts to the exis-tence in LgR of the diagram:

(u : ε, x : ε; q2(u, x))〈idε ,idε〉 ��

(v : ε, y : ε; q ′2(v, y))

〈idε ,idε〉�� .

Schematically, one ends up with arrows

α : (; q)→ (p2 : ε, a2 : ε; q2), α′ : (′; q ′)→ (p2 : ε, a2 : ε; q2),α′′ : (′′; q ′′)→ (p2 : ε, a2 : ε; q2), . . .,

with α′ = α ◦ f for some mediating arrow f : (′; q ′)→ (; q), α′′ =α′ ◦ g for some mediating arrow g : (′′; q ′′)→ (′; q ′), and so on. Thearrows α, α′, α′′, . . . , thus draw gradually a path whose endpoint is(p2 : ε, a2 : ε; q2) and whose longer and longer portions α, α′, α′′, . . . , cor-respond to various presentations of the relation q2/(p2, a2) which, whilebeing extensionally equivalent, lie above different contexts.

So a dyadic relation q2/(p2, a2) does not coincide with any of itspresentations, but only with the collection of all of them strung together



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Figure 4. Russell’s fibration πR : equivalent presentations of q2/(p2, a2).

along all the mediating arrows that are part of the structure of LgR . Itreally consists in the web of arrows α, α′, α′′, . . ., and mediating arrowsf , g, . . ., between them. But such a web corresponds exactly to what canbe called a (partial) section19 of the fibration πR . The section that is thefibrational translation of q2/(p2, a2) can be visualized as the path drawnin LgR by α, α′, α′′, and so on, which picks out one object in each fiberof πR that it crosses (see Figure 4). The structure carried by the fibrationπR rules that section and allows one to understand the latter as a structuredcomplex. Since a relation expressed in the substitutional language is inter-preted by a section of Russell’s fibration, a relation variable can be definedas a variable each value of which is a section of that kind. The same holdsabout classes of classes and more generally about matrices of any kind.

4. Semantically Structured Variables

Let us get back to Tarski’s fibration πT to see the difference between Tarskiand Russell. In Tarskian semantics, given a fixed interpretation structureM ∈ S for a language L, each assignment gives a specific value in M toeach variable symbol of L. Let us turn things around: let us consider afixed variable symbol ‘x’ and consider the value that it gets in each pos-sible structure for the language. One thus gets an ‘x-generalized assign-ment’, which is nothing but a choice function α ∈ ∏

M∈S |M |, that selectsa member α(M) ∈ |M | of the domain of each structure M as the value

19 The concept of section comes primarily from the study of fibrations between topo-logical spaces (‘fiber bundles’). A (partial) section of a fibration p : E → B is a couple〈B′, s〉, where B′ is a subcategory of B and s : B′ → E a functor such that p ◦ s = idB ′ .So a section can be represented as the particular choice of one object in the fiber aboveeach object of B′, and of an arrow above each arrow in B′.



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Figure 5. Tarski’s fibration πT restricted to ‘x’: all the paths are x-assignments.

for ‘x’ in M . If one does not focus on a single variable symbol ‘x’ andconsiders all the variable symbols of L, the result is a generalized assign-ment, namely a map β :Var → ∏

M∈S |M | that assigns, to each variablesymbol v ∈ Var and each structure M in S, a given element β(v)(M) of thedomain of M . One can check that a generalized assignment is nothing buta section of Tarski’s fibration πT . Figure 5 is a picture of a simplified ver-sion of Tarski’s fibration, where only the variable ‘x’ is taken into account,so that the fiber above each L-structure M is not |M |Var, but |M |{x} � |M |.

The fact that a generalized assignment coincides with a section of πTelicits two points. The first one is that the conception of variables exhibitedby Tarski’s semantics lends itself to a fibrational treatment, where fibers arenot deductive contexts any more, but universes of discourse.20 It is nowpossible to take stock. Russell’s substitutional logic can be interpreted inRussell’s fibration πR and, in that framework, the semantic values that areclasses, relations, classes of classes, and so forth, correspond to sectionsof πR . On the other hand, Tarki’s semantics can be put into perspectivethrough Tarski’s fibration, which collects all structures for a given lan-guage instead of considering them one by one, and, in that framework,the genuine semantic values are not point values but sections of πT . The

20 The novelty of Tarski’s semantics relied primarily on the introduction of multipleuniverses of discourse and on the machinery of assignments. That is precisely what iscaptured by Tarski’s fibration. Actually, the part of Tarski’s semantics that can thus becaptured is a bit larger. Indeed, for any (partial) section α of πT , the generalized satisfactionof a formula ϕ by α is readily definable:

α �g ϕ(�x) iff, for any M ∈ dom(α), M � ϕ(�x) [α(M)].Then, if one considers sections of πT along elementary chains M0 ≺ M1 ≺ . . .Mk ≺ . . .only (instead of chains of homomorphisms), one has that, for any formula ϕ, either α �g ϕor α �g ¬ϕ.



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distribution of the possible values for a variable, in Tarski’s semantics,and in Russell’s substitutional logic as well, is shaped according to theleeway that is left for a section in Tarski’s fibration and in Russell’s fibra-tion, respectively. The structure of a variable, in both cases, lies in the wayin which the shift from a possible value to another operates, and this cor-responds to the way in which the structure of the fibration is reflected byits sections. So the framework that has been built gives one the means of aclear comparison between the two authors.

But – and this is the second point – an important distinction has to bemade between Tarski’s semantics and Tarski’s fibration. Indeed, all thedifferent values that an x-generalized assignment α gives to ‘x’ can beconceived of as a sequence of counterparts of an original value. So if αgives to ‘x’ the value a in the domain |M |, a′ in |M ′|, a′′ in |M ′′|, andso on, the sequence a′, a′′, . . . , could be viewed as what a becomes whenone shifts from context |M | to context |M ′|, then to context |M ′′|, andso on. According to Tarski’s fibration, such a sequence is allowed on thecondition that a′ = f (a) for some L-homomorphism f : M → M ′ and thata′′ = g(a′) for some L-homomorphism g : M ′ → M ′′. Stronger conditionscould be set if L-homomorphisms were replaced, for instance, with ele-mentary embeddings. Yet, according to Tarski’s semantics, there need notbe any connection between a, a′ and a′′, because the choice carried outby α is entirely arbitrary. Otherwise put, Tarski’s semantics corresponds tothe weak case where any map f : M → N whatsoever (even if it is not anL-homomorphism) is taken to induce an arrow f : N → M in S. In thatcase, the base category S, thus redefined, does not work as a real controlspace. Indeed, in Tarskian semantics, the choice of the value of ‘x’ in anyinterpretation structure M is completely free, apart from the external con-straint placed by the domain of M : the choice is free, it has only to bemade within |M |. So Tarski’s semantics deletes any constraint that couldbe placed on sections of Tarski’s fibration: in Tarski semantics, all sec-tions are allowed, regardless of which arrows exist in the base category.On that account, Tarski’s semantics constitutes a kind of limit case, whereno constraint, and thus no structure, is set up.

This stands in sharp contrast to the syntactic fibration πR as used toexpress Russell’s logical construction of classes and relations in exten-sion. As a matter of fact, arrows between contexts in the total spaceLgR express sequent derivations, and are thus constrained by the logi-cal rules of the substitutional calculus. Accordingly, arrows in the totalspace, in our examples, express ways of looking at some class, at someclass of classes or at some dyadic relation, and are constrained insofar asthere is, to get back to the example of q2/(p2, a2), only one equivalenceclass of objects above a given context that is linked to (p2 : ε, a2 : ε; q2).At each point, the choice is bound down to picking out one object amonga prescribed equivalence class at most (if the relation is taken up to



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extensional equivalence), whereas, in Tarski’s semantics, it is only bounddown to picking something out.

In this respect, Russell’s substitutional theory points to somethingimportant: what is at stake is getting hold of a structurally restricted rangeof values for a variable. Indeed, Russell’s substitutional theory does notbuild variables that would be structured in themselves, but on the con-trary uses unstructured entity variables which are supposed to be availableat the outset. Still, Russell manages to delineate specific ranges of possi-ble values without externally restricting those ranges, through conditionsembedded into the syntax. Hence structured variables follow from built-informal constraints upon their possible values. The environment of a fibra-tion follows that idea, with the further advantage of bringing to the forevariables that are both structured as in Russell and interpreted as in Tarski.

The lesson to be learned from Russell is thus that structure does not liein values or in variables, but comes from the way in which a total rangeof values is divided and thus organized. This amounts, in the fibrationalsetting adopted here, to considering the total space of a fibration as a rangeof ranges and to structuring it along prescribed orbits or sections. So letus define a structured value as a section of a fibration such as π0, πR ,or πT , and a structured variable as a generalized variable whose possiblevalues are structured values. An example of a structured value is providedin Tarski’s semantics by an x-generalized assignment, in Russell’s substi-tutional theory by a particular bundle of arrows with the same codomain(p2 : ε, a2 : ε; q2): in both cases, a structured value is a section of the fibra-tion at work. The smaller the leeway in the choice of single values to intro-duce a section, the more constrained are the structured values, and the morespecific is the range of the corresponding structured variable.

In π0 or in πR , structured values are directly defined by a syntactic con-straint: the fact that specific sequents have to be derivable in certain typecontexts. This is a difference between these structured values and the sec-tions of Tarski’s fibration πR . But a yet more significant difference existsbetween the sections of Tarski’s fibration and the unconstrained choicefunctions through which Tarski’s semantics can be rephrased within πT .It explains why Tarski’s interpretation of quantification, even though itobviously involves structures in the set-theoretic sense, does not rely onstructure in the sense of a structured fibration, and thus corresponds onlyto a limit case. At the same time, the introduction of Tarski’s fibration is aninvitation to reinterpret Tarski’s semantics and to incorporate some struc-ture in it by insisting on the different possible choices of the arrows of thebase category S.

So, on the one hand, the framework of semantically structured variablesachieves some kind of conciliation between Russell and Tarski: it makes itappear that Russell’s syntactic structured variables and Tarski’s semanticsfor quantification can be combined. But, on the other hand, it suggests that



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we should think of variables as primarily structured, and of unstructuredvariables as being only a special case. And, in this respect, it is a way ofstanding back from Tarski’s semantics and from the usual practice, in orderto reactivate a fundamental intuition underlying Russell’s logicism.

5. Two Objections

But two objections seem to stand in our way of speaking rightfully of‘semantically structured’ values and variables.21 Indeed, one could argueeither that they are not internally structured, or that they do not reallybelong to a semantic setting. Let us consider the two objections in turnto conclude.

The first objection is that syntactic fibrations do not really provide anyinternal structure, because the underlying type theory is explicit instead ofbeing built-in. Since simple type theory is enough to get ‘structure’, what isthe point of working with such an awkward framework, which comes withsimple types anyway? Moreover, since types and contexts are supposed tobe given in the same way as domains, in Tarskian semantics, are supposedto be given with the background set-theoretic universe, a type assignmentappears to be finally as external as a domain restriction. A sharper way ofputting things might be the following: structured values in the context ofπ0 or πR seem to blend both types and structure (namely, types comingfrom an explicit simple type theory and structure coming from a fibrationabove contexts), whereas the whole point of Russell’s substitutional theorywas precisely to emulate types in a type-free way, so as to have typesthrough structure, not types and structure.

The short answer is that the structure of structured values (in the senseof sections of π0 or of πR) does not pertain to types themselves, but to sys-tematic connections between type contexts. Besides, the type contexts andthe arrows between propositions in context are based on a logical calculus,which is very different from having an external restriction. Furthermore,nothing precludes the construction of a fibration built onto an untypedlambda-calculus (where variables are declared without type).

But the better answer is more fundamental: a new notion of value hasto be recognized as the true notion of value. Structured values are not iso-lated values, but families of isolated values which organize a total range.From a logical point of view, structured values are the genuine values.Indeed, the whole comparison of Russell and Tarski calls for the dis-tinction between single values (values in the usual sense, represented asobjects in different fibers) and structured values (sections of a fibration),and for the recognition of the latter as the only place where the whole

21 I wish to thank Marco Panza for raising both objections.



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semantical scheme becomes visible. Now, to answer the specific objec-tion above: types and structure do not relate to the same kind of items. Thestructure of a structured value cannot consist in a simple type, but only in asequence of connected choices of single values of different types. Assign-ments of simple types are only about single values, whereas structure isonly about structured values. So there is no blend of types and structure inthe end.

The second objection that could be raised is that the framework of alogical fibration does not really provide a semantic interpretation, since itconsists in syntactic derivations above syntactic contexts. But in fact, asalready said, syntactic fibrations are no less semantic than Henkin mod-els. Moreover, any fibration can be seen in fact as a syntactic fibration.Indeed, any fibration p : E → B admits of an ‘internal logic’, namely thetype theory:

• whose types are all the objects of B,• whose propositions in context are all x : b � P(x) : Prop where P is

an object above b (i.e., such that p(P)= b),• and whose sequents are all derivations x : b ‖ P(x)� Q(x) associ-

ated to the arrows between P and Q in Eb.

Now, as already explained, any type theory (in the form of a sequent calcu-lus) gives rise to a syntactic fibration. Thus, any fibration becomes the syn-tactic fibration of its own internal logic, where any fiber Eb of p appearsas a set of logical relations relative to a logical context. So finally π0 andπR are no more ‘syntactic’ than any other fibration: the semantically struc-tured variables that have been introduced above are as genuinely semanticas they are genuinely structured.

6. Conclusion

Structured variables put forward by Russell’s substitutional theoryexpresses a fundamental and too often neglected understanding of vari-ables, and a cornerstone of Russell’s mature logicism. But they seem to betoo distant from the modern Tarskian view on variables. The main objec-tive of this paper was to question, from a Russellian point of view, theTarskian way of understanding what it is to be a possible value for a vari-able, and, as a consequence, what it is to be a variable. But it was first tosuggest a common framework where Russellian substitutional logic andTarskian logical semantics could be plunged and interact fruitfully.

Fibrations, and in particular syntactic fibrations coming from categor-ical logic, stood out despite their technical nature, because they consti-tute a genuine synthesis of syntactic environments (built-in type theorya la Russell) and semantic universes (set-theoretic domains a la Tarski).



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Moreover, sections of a fibration constitute ruled progressions that fit wellthe fundamental idea of structured values and variables. As a matter offact, a fibration is an environment varying over a base space in keepingwith the relations that organize the base space, and a structured value cor-responds to a particular way of picking out an item in each fiber. Structure,and this is quite a general point, consists exactly in the constraint set uponthe possible sections of a total space.

The categorical apparatus brought up in this paper is not intended to bemore than a tool for the specific purpose of comparing Russell and Tarski.It shows that there is actually some leeway in the conception of contex-tuality (contexts being represented as fibers), that structuration can occurin a semantic setting, and thus that neither structure nor semantics has tobe sacrificed. It both extends the syntactic basis of Russell’s substitutionaltheory and allows one to implement structured quantification in the seman-tics inherited from Tarski, by extending Tarski’s semantics on the basis ofTarski’s fibration or some enriched version of it.

The upshot intends primarily to revive some of Russell’s logicalinsights. By the same token, it puts a new spin on Russellian substitutionaltheory, through a new connection between substitutional type theory andcategorical semantics for modern type theory. It may also lead to a con-sistency proof for a variant of Russell’s substitutional theory.22 But this ismatter for further work.

References

Jacobs, Bart [1999]: Categorical Logic and Type Theory. Studies in Logic andthe Foundations of Mathematics; 141. Amsterdam: Elsevier.

Lackey, Douglas, ed. [1973]: Essays in Analysis. London: Allen and Unwin.Landini, Gregory [1987]: ‘Russell’s substitutional theory of classes and rela-

tions’, History and Philosophy of Logic 8, 171–200.——— [1998]: Russell’s Hidden Substitutional Theory. Oxford: Oxford Univer-

sity Press.——— [2007]: Wittgenstein’s Apprenticeship with Russell. Cambridge: Cam-

bridge University Press.Russell, Bertrand [1906]: ‘On “Insolubilia” and their solution by symbolic

logic’, in [Lackey, 1973], pp. 190–214. English translation of his ‘Les

22 Russell’s original substitutional theory lays itself open to an inconsistency that Lan-dini has called the ‘p0/a0 paradox’ (see [Landini, 1998, pp. 204–205]). This could be aproblem, because any contradiction in Russell’s substitutional logic will be passed on tothe formal theory TR that interprets it. However, the p0/a0 paradox relies, in particular, ona distinctive feature of Russell’s substitutional theory that has previously been mentioned:the nominalization of wffs. This feature has not been fully retained in R (see footnote 10),so that the paradox cannot arise. How R could be more faithful to the full principle ofnominalization of wffs (as in Russell’s original substitutional theory), without giving riseto the p0/a0 paradox, is an open question.



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paradoxes de la logique’, Revue de Metaphysique et de Morale 14 (1906),627–650.

——— [1973]: ‘On the substitutional theory of classes and relations’, in [Lackey,1973], pp. 165–189. Manuscript received by the London Mathematical Soci-ety on 24 April, 1906.

van Benthem, Johan, and Natasha Alechina [1997]: ‘Modal quantificationover structured domains’, in Maarten de Rijke, ed., Advances in IntensionalLogic, vol. I, pp. 1–27. Dordrecht: Kluwer.



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