Gupta-Belnap's revision in an abstract setting

Edoardo Rivello Gupta-Belnap’s revision in anabstract setting

Abstract. Revision sequences of hypotheses are the kind of ordinal-length sequenceson which is based Gupta and Belnap’s book The revision theory of truth [2]. As shownby Visser [10], revision theories can be recast in a purely order-theoretic setting. Inthis framework, Visser provided an upper bound for the length of the revision sequencesneeded to carry out the original revision theories proposed by Herzberger and Gupta,independently, in 1982. In the present paper, I generalise Visser’s result and extend itsupper bound to Gupta and Belnap’s theory. My construction is worked out in an abstractsetting introduced in [9] and [8], making no use of the Axiom of Choice: this fact resultsin a more constructive proof which shows that the provided upper bound is an intrinsicproperty of the revision sequences, not depending on any assumptions made on the order-theoretic structure carried by the set of the hypotheses.

Keywords: Revision theories, revision sequences, transfinite sequences.

1. Introduction

Gupta and Belnap’s revision theory of definitions [2] is formalised by meansof revision sequences, a set-theoretical construction first exploited by Guptahimself [1] and by Herzberger [3], independently, as a tool in describing theirtheories of self-referential truth.

Revision sequences are ordinal-length iterations of a function (the revisionoperator) which, at the limit stages, obey to a constraint (the coherencecondition) stated in terms of the inferior limit of the sequence up to the limitstage. The values taken by the sequences are called “hypotheses”. The keynotion of revision, that of recurring hypothesis, is formalised by quantifyingover the collection of all revision sequences which are, as defined, properclasses. However, Gupta and Belnap show how to formalise their theory instandard set theory (ZF) by providing an equivalence between the notion ofrecurring hypothesis and that of reflexive hypothesis, the latter only requiringquantification over limit-length revision sequences, thus over sets. Finally,they recall McGee’s theorem [7], which provides an upper bound for thelength of the revision sequences we need in order to define reflexivity.

Presented by Name of Editor; Received December 1, 2005

Studia Logica (2006) 82: 1–20 c©Springer 2006

2 Edoardo Rivello

In Gupta and Belnap’s book the hypotheses are taken from an arbitraryspace of functions DX, where D (the domain) and X (the codomain) arenonempty sets. In this setting, McGee’s theorem is formulated as follows

Thm 1.1. [2, Thm. 5C.15, p. 176] Let h be a reflexive hypothesis. Letµ = max{|D| , |X| , ω}.1 Then the reflexivity of h is witnessed by a revisionsequence of length less than or equal to µ.

In [10], Visser recast revision in an order-theoretic framework, allowingthe hypotheses to be arbitrary objects (not necessary, functions) and onlyrequiring that the set P of the hypotheses carries the structure of a coherentcomplete partial order.2 In this more abstract setting, Visser reformulates therevision theories considered by Herzberger [3] and Gupta [1] and shows thatℵ+(P ) (the smallest cardinal bigger than max{|P | , ω}) provides an upperbound for the length of the sequences we need to define the notion of recurringhypotheses for these theories. In [9], I gave a new proof of Visser’s result byderiving it from intrinsic properties of the sequences involved, namely fromproperties which can be stated and proved without assuming any structureon the set P .

The main goal of the present paper is to extend Visser’s upper bound toGupta-Belnap’s revision theory preserving, on the same time, the improve-ments made by [9] on Visser’s original proof.

More in details, my proof that ℵ+(P ) provides a suitable upper boundeven for Gupta-Belnap’s revision (in Visser’s order-theoretic setting) willbe distinguished from the older ones by the following features. First, bothMcGee’s and Visser’s proofs of their respective upper bounds rely on set-theoretic assumptions which are known to be equivalent to weak forms ofthe Axiom of Choice: A Löwenheim-Skolem argument in the former caseand the regularity of ℵ+(P ) in the second. In contrast, the present proof willbe carry out entirely in Zermelo-Fraenkel set theory. This feature results in amore constructive proof and, as it is often the case when the Axiom of Choiceis avoided, in a more informative one. This leads to the second feature: I willidentify a minimal set of axioms, satisfied by Gupta-Belnap’s collection of allrevision sequences, which are sufficient to carry out the proof, so providing ageneralisation of the previous results. Thirdly, this axiomatisation of Gupta-Belnap’s revision theory will be laid down in the abstract framework of [9],showing that the upper bound can be stated and proved independently ofany structure supposed on the set of the hypotheses. We will refer to this

1|A| denotes the cardinality of a set A, ω denotes the least infinite ordinal.2See [11, p.191].

Gupta-Belnap’s revision in an abstract setting 3

fact by saying that the found upper bound is an “intrinsic” property of thecollections of transfinite iterations which obey the axioms, in the sense thatwe can disregard the nature of the values of that sequences.

This paper addresses two kinds of audience. Revision-theoretic definitionsare, in a sense, “the next step beyond inductive definitions” [4, p. 79], beingrevision sequences a straightforward generalisation of progressive iterationsand leading to sets of one complexity degree beyond that of the inductivelydefinable sets. Thus, set-theorists could be interested in my abstract versionof McGee’s theorem: This version can be read as an extension to revisionsequences of properties showed by progressive iterations. On the other hand,revision sequences are extensively used in mathematical philosophy in orderto investigate the concept of truth as well as other philosophical concepts.3

Formally-oriented philosophers could be interested in the interpretations ofthe mathematical aspects of Gupta-Belnap’s revision theory enlightened inthis paper. A fine-grained investigation of the notion of revision sequenceshould be useful to grasp the informal notion of revision process we needwhen we apply the formal theory to a philosophical analysis.

Taking in account these two intended audiences, the set-theoretical, in-trinsic part of the proof will be concentrated in Section 2 while, in Section 3, Iwill illustrate in details how Gupta-Belnap’s revision theory fits the abstractframework and I will give motivations and directions for the purely mathe-matical development of the former section. Accordingly, the philosophical-oriented reader could also directly jump to Section 3 and use Section 2 forreference.

Prerequisites Definitions and lemmata in this paper presuppose someknowledge of set theory, in fact of ordinal arithmetic, and little more. The rel-evant notions and facts about transfinite sequences are recalled in Section 2,mostly without the straightforward proofs. Notations, definitions and proofsof standard facts about ordinal arithmetic can be find in any textbook inset theory: In particular I refer to [5]. Throughout this paper, the usualnotations for the arithmetical operations of addition, multiplication and ex-ponentiation will always denote the corresponding operations on ordinals:No risk of confusion with cardinal arithmetic is possible.

2. Sequences

3See [6], for a survey.

4 Edoardo Rivello

Preliminaries on sequences and iterations A sequence (from a set P )is a function s such that its range, denoted by ran(s), is included in P andits domain (or its length), denoted by lh(s), is either a limit ordinal or theclass On of all ordinals. In the former case we will say that s is a limit-lengthsequence, in the latter we will say that the sequence is ordinal-length. We willuse Greek letters like σ to denote ordinal-length sequences. We will denotethe value of s at an ordinal α < lh(s) by s(α). Lim denotes the class of alllimit ordinals and Lim∗ = Lim∪{0}. The difference between two sets A andB will be denoted by A−B.

The elements of P will be called hypotheses4. Given a sequence s, ahypothesis p will be said cofinal in s if and only if for every α < lh(s) thereexists β < lh(s) such that α ≤ β ∧ s(β) = p. The set of all hypothesescofinal in s will be denoted by Cf(s).

We fix, throughout this paper, a function ρ : P → P , which will be calledthe revision operator. We say that a sequence s from P is an iteration ofρ whenever s(α + 1) = ρ(s(α)) for every α < lh(s). In the following, wewill often speak about iterations, without explicitly mentioning neither theoperator ρ nor the set of values P .

We define four operations and five binary relations between sequenceswhich will be useful in the proof of the main result (Theorem 2.18).

Def. 2.1. Given a sequence s and a limit ordinal γ < lh(s), the initialsegment of s at γ, denoted by s�γ, is the limit-length sequence t such that

(1) lh(t) = γ.

(2) t(ξ) = s(ξ), for every ξ < lh(t).

Def. 2.2. Given a sequence s and an ordinal α < lh(s), the final segment ofs at α, denoted by sα, is the limit-length (or ordinal-length) sequence t suchthat

(1) lh(t) = lh(s)− α.(2) t(ξ) = s(α+ ξ), for every ξ < lh(t).

Def. 2.3. Given two limit-length sequences s and t, the concatenation of sand t, denoted by s+ t, is the limit-length sequence u such that

4This terminology is borrowed from [2, p. 166], where the term hypothesis is reservedto functions from a given set D into a set X. In Section 3, the intended interpretation forthe set P will be the set of all partial functions from a set D into a set X. Accordingly,we will speak about partial or total hypotheses when this distinction will become relevant.See Section 3, for further details.


(1) lh(u) = lh(s) + lh(t).(2) u � lh(s) = s and u(α) = t(ξ) for every α such that lh(s) ≤ α < lh(u),

where ξ < lh(t) is unique such that α = lh(s) + ξ.

Def. 2.4. Given a limit-length sequence s and an ordinal α, the externalproduct of s and α, denoted by s · α, is the transfinite sequence t such that

(1) lh(t) = lh(s) · α.(2) t(β) = s(ξ), for every β < lh(t), where ξ < lh(s) is unique such that

β = lh(s) · η + ξ, for a unique η < α.

Remark 2.5. Let s, t be two sequences, γ a limit ordinal less than lh(s), α anordinal less than lh(s) and β any ordinal. If s and t are iterations then so ares �γ, sα, s+ t and s · β. The union of any chain of iterations is an iteration.Moreover,

(1) ran(s�γ) ⊆ ran(s).(2) ran(sα) ⊆ ran(s) and Cf(sα) = Cf(s).(3) ran(s+ t) = ran(s) ∪ ran(t) and Cf(s+ t) = Cf(t).(4) ran(s · β) = Cf(s · β) = ran(s).

Def. 2.6. Given two sequences s and t we say that

(a) t is a refinement of s if for all γ < lh(s) there exists δ < lh(t) such thatran(tδ) ⊆ ran(sγ).

(b) t is an exact refinement of s if for all γ < lh(s) there exists δ < lh(t) suchthat ran(tδ) = ran(sγ).

(c) s and t are cofinally interleaved if t is a refinement of s and s is a refine-ment of t.

(d) s and t are cofinally equivalent if there exists α < lh(s) and β < lh(t)such that tβ is an exact refinement of sα and sα is an exact refinementof tβ .

(e) s and t are eventually equivalent if there exists α < lh(s) and β < lh(t)such that sα = tβ .

Remark 2.7. Let s, t be two sequences and β any ordinal. If s and t arecofinally equivalent then they are cofinally interleaved, and if s and t areeventually equivalent then they are cofinally equivalent. Moreover,

(1) If β is a successor ordinal, then s and s · β are eventually equivalent.(2) If β is a limit ordinal, then s · β and s · ω are cofinally equivalent.(3) If t is a refinement of s then Cf(t) ⊆ Cf(s). Hence, if s and t are cofinally

interleaved then Cf(s) = Cf(t).

6 Edoardo Rivello

Limit rules The notion of “recurring hypothesis” is relative to a collectionC of ordinal-length iterations (of ρ). Given such a collection C , we denoteby Rec(C ) the set of all C -recurring hypotheses, defined (semi-formally) asfollows:

Rec(C ) =⋃{Cf(σ) | σ ∈ C }.

We will only be concerned with collections of iterations specified by meansof limit rules.5 Formally, a limit rule will be represented by a binary relationR between limit-length iterations and hypotheses. An iteration s will becalled an R-iteration if and only if R(s�γ, s(γ)) holds for every limit ordinalγ less than the length of s. Given a limit rule R, we denote by CR thecorresponding collection of all ordinal-length R-iterations and by TR theclass of all limit-length R-iterations.

Remark 2.8. If s is an R-iteration then so is s � δ for every δ ∈ Lim∩lh(s).The union of any chain of R-iterations is an R-iteration.

We lay down below a list of axioms for limit rules.

(R0) ∀s∃pR(s, p) (R is everywhere defined).6

(R1) ∀s, p (p ∈ Cf(s)→ R(s, p)).

(R2) ∀s, t, p if t is a refinement of s then R(t, p)→ R(s, p)).

Remark 2.9. From Axiom (R1) it follows p ∈ ran(s)→ R(s ·ω, p) for every sand p and from Axiom (R2) it follows that ran(s) ⊆ ran(t) and R(s, p) implyR(t · ω, p), for all s, t and p.

We also state the following two notions which will be used in the proofs.

Def. 2.10. We say that a limit rule R is eventually dependent if and onlyif R(s, p) ↔ R(t, p) holds for every p, whenever s and t are two eventuallyequivalent iterations.

We say that a limit rule R is cofinally dependent if and only if R(s, p)↔R(t, p) holds for every p, whenever s and t are two cofinally equivalent iter-ations.

By Remark 2.7, Axiom (R2) implies cofinal dependence and this latterimplies eventual dependence. The notion of eventual dependence will be

5Other ways of specifying a collection of revision sequences are discussed in the liter-ature on revision, but lie outside the scope of this paper. See, for instance, the notion offully varied revision sequence introduced in [2, p. 168].

6(R0) does not follow from (R1) since, when lh(s) is an ordinal injectable in P , we canhave Cf(s) = ∅.


useful since it is the weakest of the three, so we will able to use freely thefacts about eventual dependence which are collected in the following remark,also when we deal with the other two notions of dependence.

Remark 2.11. Let R be eventually dependent and let s, t ∈ TR. Let αbe any ordinal less than lh(s) and let β be any limit ordinal. Then, (a)R(s + t, p) ↔ R(t, p) and R(s, p) ↔ R(sα, p) hold for every p, and (b) sα

belongs to TR. Moreover, if R(s, s(0)) holds, then s ·β ∈ TR and if R(s, t(0))holds, then also s+ t ∈ TR.

The notion of cofinally dependent limit rule was introduced in [8] in orderto prove the following theorem (see below for the definitions of the notionsinvolved in the statement):

Thm 2.12. [8, Thm. 6.10] Let R be a cofinally dependent limit rule satisfyingAxiom (R0) and let p be any hypothesis. Then the following are equivalent:

(1) p is CR-recurring.

(2) p is CR-periodic.

(3) p is TR-indirectly reflexive.

Mimicking Gupta-Belnap’s Theorem 5C.13 [2, p. 174] about revisionsequences, the equivalence (1)↔ (3) shows that, whenever the limit rule R iscofinally dependent, the notion of CR-recurring hypothesis, which was definedabove by quantifying over ordinal-length R-iterations (thus, proper classes),can be formalised in ZF by quantifying over limit-length R-iterations (thus,sets), by using the technical notion of TR-indirect reflexivity. Our presentgoal is to provide also an upper bound for the length of the R-iterations weneed to consider in defining CR-recurrence. This will be done for the limitrules satisfying the axioms (R0)–(R2), a significant subclass of the cofinallydependent ones.

As an intermediate step, we will show that, for cofinally dependent limitrules satisfying Axiom (R1), the notion of TR-indirect reflexivity can bereplaced by another notion of reflexivity, called limit reflexivity, which willbe used later in estimating the upper bound.

Def. 2.13. A hypothesis p is TR-limit reflexive if and only if there exists anR-iteration s such that p = s(0) and R(s, s(0)) holds.

Lemma 2.14. Let R be a cofinally dependent limit rule satisfying Axiom (R1).Then, p is CR-recurring if and only if p is TR-limit reflexive.

8 Edoardo Rivello

To prove Lemma 2.14 we need to recall some facts about the notions ofperiodicity and reflexivity which was used in stating and proving Theorem2.12.7

Def. 2.15. A sequence s is periodic with period δ if and only if the followingequivalent conditions hold:

(a) There exists an ordinal δ > 0 such that s(α) = s(ξ) for every α < lh(s),where α = δ · η + ξ for unique η ≤ α and ξ < δ.

(b) There exists a sequence t such that s = t · η + t � ξ, where η ≤ lh(s) andξ < lh(t) are unique such that lh(s) = lh(t) · η + ξ.

Fact 2.16. Let σ be an ordinal-length sequence. Then

(1) σ is periodic if and only if there exists an initial segment s of σ such thatσ � α = s · η + s � ξ for every α ∈ On, where η ≤ α and ξ < lh(s) areunique such that α = lh(s) · η + ξ.

(2) If σ is periodic, then there exists a period δ of σ which is a limit ordinal.

(3) If σ is periodic, then Cf(σ) = ran(σ) = ran(σ � δ) = Cf(σ � η), where δ isany period of σ and η = δ · ω.

Def. 2.17. We say that a periodic limit-length sequence s is long if and onlyif δ · ω < lh(s), where δ denotes the least period of s.

A hypothesis p is TR-indirectly reflexive if and only if p occurs in a longperiodic R-iteration.

Proof of Lemma 2.14. Let p be CR-recurring. By Theorem 2.12, p is CR-periodic, namely p occurs in a periodic ordinal-length sequence σ ∈ CR ofperiod δ. By Fact 2.16, we can assume that δ is a limit ordinal. Let s = σ �δ,η = δ · ω, and t = σ �η. By Fact 2.16, p ∈ ran(s) = Cf(t). Suppose p = s(α),for some α < δ, and let u = tα, so that u(0) = p. Since η is limit, by theremarks 2.8 and 2.11 both t = σ � η and u belong to TR. By Fact 2.16,t = s · ω. Since p ∈ ran(s) = Cf(t), by Axiom (R1), R(t, p) holds. Thus, byRemark 2.11, also R(u, u(0)) holds.

On the other direction, suppose p is TR-limit reflexive, witnessed by anR-iteration s. Let γ = lh(s) and let t = s · (ω + ω). By Remark 2.11 andby Definition 2.15, t is a long periodic R-iteration. Since t0 = tγ = p, thehypothesis p is TR-indirectly reflexive. Thus, by Theorem 2.12, p is CR-recurring. �

7I refer the reader to [8] for further details on these notions.


Intrinsic version of Visser’s upper bound Our main goal is to find anupper bound for the length of the R-iterations which witness that a givenhypothesis p is TR-limit reflexive. This will be provided by the Hartogsordinal ℵ(P ),8 namely by the least ordinal non-injectable in P , wheneverR satisfies the axioms (R0)–(R2). More precisely, we want to prove thefollowing theorem:

Thm 2.18. Let R be a limit rule satisfying the axioms (R0)–(R2). Let ν(P ) =ωω if ℵ(P ) ≤ ω and ν(P ) = ℵ(P ) otherwise. Then p is CR-recurring if andonly if p is TR-limit reflexive witnessed by an R-iteration s of length lessthan ν(P ).

The general strategy to prove Theorem 2.18 can be sketched as follows.Since the right-to-left direction trivially follows from Lemma 2.14, we haveonly to prove the opposite direction. Lemma 2.14 yields the existence ofa sequence s which witnesses p to be TR-limit reflexive. We want to showthat, given such a witness s, there exists another witness t of length less thanν(P ). The existence of t will be proved constructively, building on a carefulinspection of the relevant information carried by s. Since s is an iteration, itcan be entirely recovered from the sequence of its limit values: Indeed, givenan x occurring at some limit stage γ of s, the portion of s between γ and γ+ω(the next limit stage) is forced to be equal to the ω-iteration of ρ starting withx, which does not depend on the limit rule R. This simple remark suggeststo look at the wellordering W(s) of such values (instead of considering theirsequence). If we success in constructing from W(s) an R-iteration t which“resemble” enough s to be a witness of TR-limit reflexivity, we will also havea control on the length of t, due to the construction. This length will dependon that of W(s). And since this latter is obviously bounded by ℵ(P ), wehave a good reason to hope that also the length of t will be bounded by anordinal ν(P ) which is a function of P , hence by an ordinal which will be thesame for every R-iteration from P .

To construct t we will define a general operation 〈A, R, ρ〉 7→ sA of un-folding of a wellordering which, given any wellordering of elements of P ,returns an R-iteration of ρ from P . Then we will prove that when A is thewellordering of the limit values of an R-iteration which witnesses the TR-reflexivity of p = s(0), then sA is the requested witness of length less thanν(P ). The definition of the unfolding operation will also involve a general op-eration 〈s,R〉 7→ sR of expansion of a sequence which, given any limit-lengthsequence s from P , returns another limit-length sequence sR from P with

8Assuming the Axiom of Choice, ℵ(P ) = ℵ+(P ).

10 Edoardo Rivello

the same limit values wellordered in the same way. So let us start dealingwith wellorderings of elements of P , in general, and with the wellorderingW(s) of the limit values of a sequence s, in particular.

Wellorderings and sequences We fix some notations about wellorderedsets.9 A wellordering from P is any wellordered set A = 〈A,<A〉 such thatA ⊆ P . For a ∈ A, a<A will denote both the set {b ∈ A | b <A a} andthe wellordering 〈a<A, <A〉. We write B v A when A end extends B (or Bis an initial segment of A), that is, when B ⊆ A is wellordered by <A anda <A b→ a ∈ B for every b ∈ B. We write B @ A for B v A ∧ B 6= A.

Def. 2.19. Given any sequence s, define

• W (s) = {s(γ) | γ ∈ lh(s) ∩ Lim∗} (the set of the limit values of s).• µs(x) = min{γ < lh(s) | γ ∈ Lim∗ ∧ s(γ) = x}, for x ∈W (s).• µs(γ) = min{δ < lh(s) | δ ∈ Lim∗ ∧ s(δ) = s(γ)}, for γ ∈ lh(s) ∩ Lim∗.• M(s) = {γ ∈ lh(s) ∩ Lim∗ | µs(γ) = γ}.

We will use the same letter µ to denote both functions µs : W (s) →lh(s) ∩ Lim∗ and µs : lh(s) ∩ Lim∗ → lh(s) ∩ Lim∗, since they can be easilyidentified by the argument. When the sequence s is clear from the context,we will also drop the subscript s, simply writing µ(x) and µ(γ).

For x ∈ W (s) and γ ∈ lh(s) ∩ Lim∗, s(µ(x)) = x, µ(s(γ)) = µ(γ) andµ(µ(γ)) = µ(γ), hence µ : W (s) → M(s) is a bijection whose inverse iss�M(s).

We say that W (s) is wellordered by first limit occurrence (denoted by /)when it is endowed with the wellordering induced on W (s) by the bijections�M(s) together with the natural wellordering of M(s) as a set of ordinals.It follows that s �M(s) is the unique isomorphism between 〈M(s), <〉 andW(s) = 〈W (s), /〉.Remark 2.20. Let s, t be two sequences, γ a limit ordinal less than lh(s), αan ordinal less than lh(s) and β any. Then

(1) For every B @ W(s) there exists η < lh(s) such that W(s � η) = B. Inparticular, since B = b<W(s) for a unique b ∈W (s), the least such η is theordinal µs(b).

(2) If s ⊆ t then W(t) end extends W(s).(3) W(s�γ) = {a ∈ W(s) | µs(a) < γ}.

9See [9, Section 3.4] for a bit more.


(4) W(sα) ⊆ W(s).

(5) W(s+ t) =W(s) ∪W(t).

(6) W(s · β) =W(s).

Moreover, W(u) =⋃{W(u′) | u′ ∈ C} whenever C is a chain of sequences

and u =⋃C, hence W(u) is the unique wellordering of W (u) which end

extends each W(u′), for u′ ∈ C.

Expansion of a sequence

Def. 2.21. Let ∗ denote a fixed set not belonging to P . Let s be any limit-length sequence from P and let R be any limit rule. Define by ω-recursion:

(i) sR0 = s, aR,s(0) = ∗ and a < x if and only if a ∈ W (s) and x = ∗ ora, x ∈W (s) ∧ a <W(s) x.

(ii) aR,s(n + 1) = minW(s) {a ∈ W (s) | a < aR,s(n) ∧ R(sRn, a)}, if thereexists a ∈ W (s) such that a < aR,s(n) and R(sRn, a) holds. Otherwise,aR,s(n+ 1) = aR,s(n).

(iii) sR(n+1) = sRn + (sRn)µ(aR,s(n+1)) · ω if aR,s(n+ 1) < aR,s(n). Otherwise,sR(n+1) = sRn.

(iv) n̄ = min{n < ω | aR,s(n+ 1) = aR,s(n)}, aR,s = aR,s(n̄) and sR = sRn̄.

The relation < is a wellordering of the setW (s)∪{∗}. Moreover, aR,s(n+1) ≤ aR,s(n) for every n < ω, so it must exist n < ω such that aR,s(n+ 1) =aR,s(n). Thus n̄ is well defined.

Lemma 2.22. Let s be a limit-length sequence from P and let R be any limitrule. Then

(1) ran(sR) = ran(s) and W(sR) =W(s).

(2) sRn ⊆ sR(n+1) for every n ∈ ω.

(3) There exists k < ω such that lh(sR) is a limit ordinal less than lh(s) · ωk.

(4) Let R be cofinally dependent. If s ∈ TR then sRn ∈ TR for every n ∈ ω.In particular, sR = sRn̄ ∈ TR.

(5) If, further, R satisfies Axiom (R1), then aR,s ≤W(s) a ↔ R(sR, a) andR(s, a)→ R(sR, a), for every a ∈W (s).

12 Edoardo Rivello

Proof. (1) By Remark 2.5 and Remark 2.20. a(2) By definition of concatenation sRn ⊆ sR(n+1), for every n ∈ ω. a(3) Either sR = sRn̄ is s itself or is of the form sR = t+ tµ(aR,s(n̄)) ·ω, for

some t. We prove by induction that lh(sRn) ≤ lh(s) · ωn+1, for every n ∈ ω.lh(sR0) = lh(s) < lh(s) · ω. Assume, by the inductive hypothesis, lh(sRn) ≤lh(s) · ωn+1. Then lh(sR(n+1)) ≤ lh(sRn) · ω ≤ lh(s) · ωn+1 · ω = lh(s) · ωn+2.Hence lh(sR) = lh(sRn̄) < lh(s) · ωk, with k = n̄+ 2. a

(4) Suppose that R is cofinally dependent and that s ∈ TR. We willshow by induction that sRn ∈ TR for every n < ω. lh(sR(n+1)) = lh(sRn) +lh((sRn)µ(aR,s(n+1)))·ω is a limit ordinal. sR(n+1) is a ρ-iteration by the induc-tive hypothesis and Remark 2.5. By definition of aR,s(n+1), R(sRn, aR,s(n+1)) holds. By Remark 2.11, also R((s(Rn))µ(aR,s(n+1)), aR,s(n + 1)) holds.Since (s(Rn))µ(aR,s(n+1))(0) = aR,s(n + 1), from Remark 2.11 it follows that(s(Rn))µ(aR,s(n+1)) · ω ∈ TR so, by Remark 2.11 again, sR(n+1) ∈ TR. a

(5) Let a ∈W (s). On one direction, suppose aR,s ≤W(s) a. By definitionof aR,s, this implies sR 6= s, so sR = t + tµ(aR,s) · ω, for some t. Since{sRn | n ∈ ω} is a chain, s ⊆ t. Thus aR,s ≤W(s) a implies a ∈ ran(sµ(aR,s)) ⊆ran(tµ(aR,s)). Hence, by Axiom (R1), R(tµ(aR,s) · ω, a) holds and, by Remark2.11, R(sR, a) holds too. On the other direction, suppose R(sR, a). SinceaR,s = aR,s(n̄) and sR = sRn̄, aR,s = aR,s(n̄+ 1) ≤W(s) a. a

Suppose a ∈W (s) and R(s, a). Then aR,s ≤W(s) aR,s(1) ≤W(s) a. HenceR(sR, a) holds by the previous argument. �

Unfolding of a wellordering Given any x ∈ P , we denote by xy theiteration of ρ of length ω generated by x, namely, lh(xy) = ω, xy(0) = xand xy(n+ 1) = ρ(xy(n)) for every n < ω.Remark 2.23. If s is an iteration of ρ, then ran(s) =

⋃{ay | a ∈ W (s)}.

Thus, if s and t are two iterations of ρ then W (s) = W (t)→ ran(s) = ran(t).

Def. 2.24. Let A be a wellordering from P and let R be any limit rule.By recursion on the length of A we define a transfinite sequence sA, theunfolding of A, as follows:

s∗A =

{ ⋃{sC | C @ A} if A has no maximum.

sB + by if b = maxA and B = b<A.

sA = (s∗A)R

Remark 2.25. The unfolding of the empty wellordering is the empty sequence,and sC ⊆ s∗A ⊆ sA, for every C @ A. Hence s∗A and sA are well-definedsequences.


Proof. s∗A ⊆ sA by Lemma 2.22. We prove that, for every C @ A, bothsC and s∗A are sequences such that sC ⊆ s∗A, by induction on the length ofA. If A has no maximum, then {sC | C @ A} is a chain by the inductivehypothesis. So s∗A is a sequence and sC ⊆ s∗A for each C such that C @ A.If b = maxA, let B = b<A. By the inductive hypothesis, sB is a sequenceand sB ⊆ sB + by = s∗A. If C @ B then, by the inductive hypothesis on A,sC ⊆ sB ⊆ s∗A. �

Lemma 2.26. Let A be any wellordering from P .

(1) ran(sA) =⋃{ran(ay) | a ∈ A}.

(2) W(sA) = A.(3) sA is a limit-length ρ-iteration.(4) Let µ = µs∗A . For all c ∈ A, µ(c) = lh(sC), where C = c<A.

(5) For all a, b ∈ A, a ≤A b implies b ∈ ran((s∗A)µ(a)).(6) Let ℵ(P ) be the Hartogs ordinal for P . If ℵ(P ) ≤ ω then lh(sA) < ωω;

otherwise lh(sA) < ℵ(P ).

Proof. By induction on the length of A.(1) If A has no maximum then, by the inductive hypothesis and by

Lemma 2.22, ran(sA) = ran(s∗A) =⋃{ran(sB) | B @ A} =

⋃{⋃{ay | a ∈

B} | B @ A} =⋃{ay | a ∈ A}. If b = maxA then, by the inductive hypoth-

esis and by Lemma 2.22, ran(sA) = ran(sB) ∪ ran(by) =⋃{ran(ay) | a ∈

B} ∪ ran(by) =⋃{ran(ay) | a ∈ A}. a

(2) If A has no maximum then, by the inductive hypothesis, by Lemma2.22 and by Remark 2.20, W(sA) = W(s∗A) =

⋃{W(sB) | B @ A} =⋃

{B | B @ A} = A. If b = maxA then, by the inductive hypothesisand by Lemma 2.22, W(sA) =W(sB) ∪W(by) = B ∪ {b} = A. a

(3) If A has no maximum, then s∗A is a limit-length ρ-iteration by theinductive hypothesis and by Remark 2.5. If b = maxA, then s∗A is a limit-length ρ-iteration by Remark 2.5, since by is a ρ-iteration. Hence sA = (s∗A)R

is a limit-length ρ-iteration by Lemma 2.22(2). a(4) Let c ∈ A, C = c<A and γ = lh(sC). If c = maxA then c = s∗A(γ) by

definition of sA. Otherwise, there exists b ∈ A such that c <A b. Let B = b<A.Since c ∈ B, γ = lh(sc<B

). By the inductive hypothesis and by Remark 2.25,c = s∗B(γ) = s∗A(γ). Suppose, towards a contradiction, that there exists alimit δ < γ such that s∗A(δ) = c. By Remark 2.25, sC ⊂ s∗A, hence δ < γimplies sC(δ) = s∗A(δ) = c. Hence c ∈ W (sC). By (2), W (sC) = C = c<A,thus c /∈ W (sC): Contradiction. Therefore, γ is the least ordinal such thats∗A(γ) = c, namely µ(c) = γ. a

14 Edoardo Rivello

(5) By (4), a ≤A b implies µ(a) ≤ µ(b), hence b ∈ ran((s∗A)µ(a)). a(6) By Lemma 2.22, lh(sA) < lh(s∗A) · ωk, for some k < ω.By induction on the length of A we will show that there exists j < ω

such that lh(sA) < ωlh(A)+j .Claim: There exists m < ω such that lh(s∗A) ≤ ωlh(A)+m.Proof of Claim. If A = ∅ then lh(s∗A) = lh(∅) = 0 < 1 = ω0 = ωlh(A)+0.If b = maxA then lh(s∗A) = lh(sB+by) = lh(sB)+lh(by) ≤ ωlh(B)+j+ω ≤

ωlh(A)+j + ω ≤ ωlh(A)+j · ω = ωlh(A)+j+1.IfA has no maximum then lh(A) is limit and, by the inductive hypothesis,

for every B @ A there exists k < ω such that lh(sB) ≤ ωlh(B)+k ≤ ωlh(A).Hence lh(s∗A) = sup{lh(sB) | B @ A} ≤ ωlh(A). a

By the claim, lh(sA) < lh(s∗A) · ωk ≤ ωlh(A)+m · ωk = ωlh(A)+j , withj = m+ k.

By definition, lh(A) < ℵ(P ). If ℵ(P ) ≤ ω then lh(A) < ω, so lh(sA) <ωlh(A)+j < ωω. Otherwise, ℵ(P ) is limit, so |lh(A) + j| ≤ lh(A) + j < ℵ(P ).Since ℵ(P ) is an initial ordinal greater than ω, by Schoenflies’ identity10∣∣ωlh(A)+j

∣∣ = max{ω, |lh(A) + j|} < ℵ(P ), hence ωlh(A)+j < ℵ(P ). Thuslh(sA) < ωlh(A)+j < ℵ(P ). �

Lemma 2.27. Let s be a limit-length ρ-iteration and let A =W(s). Then

(1) W(sA) =W(s).(2) ran(sA) = ran(s).

Proof. (1) By Lemma 2.26.(2), W(sA) = A =W(s). a(2) By Lemma 2.26.(3), sA is a limit-length ρ-iteration. Hence, by (1)

and Remark 2.23, ran(sA) = ran(s). �

Lemma 2.28. Let R be a limit rule which satisfies the axioms (R0)–(R2) andlet s ∈ TR. Let A =W(s). Then

(1) R(s, p)→ R(sA, p), for every p ∈ P .(2) sA ∈ TR.

Proof. By induction on the length of A, using Remark 2.20.(1) Suppose that b = maxA and let γ = µs(b) and B = b<A.If sA = s∗A then lh(s) = γ + ω. For, suppose on the contrary γ +

ω < lh(s). Since γ + ω ∈ Lim, s(γ + ω) ∈ W (s). s∗A and s � (γ + ω) areeventually equivalent. Hence R(s�(γ+ω), s(γ+ω)) implies R(s∗A, s(γ+ω)),contradicting sA = s∗A. Hence s = s � γ + by and sA = s∗A = sB + by

10If β > 0 then∣∣ωβ∣∣ = max{ω, |β|} [5, Theorem 2.11, p. 126].


are eventually equivalent, so R(s, p) implies R(sA, p), for every p ∈ P , sinceAxiom (R2) implies that R is cofinally dependent.

If sA 6= s∗A, let c = aR,s∗A and µ = µs∗A .Claim: c ≤A s(δ) for every δ limit such that γ < δ < lh(s).Proof of the claim. By transfinite induction.If δ = γ + ω then s∗A and s � δ are eventually equivalent, so R(s � δ, s(δ))

implies R(s∗A, s(δ)). Thus, by Lemma 2.22, R(sA, s(δ)) and c ≤A s(δ).If δ = η + ω, with η limit such that γ < η < lh(s), then by the in-

ductive hypothesis c ≤A s(η). By Lemma 2.26(5), s(η)y ⊆ ran((s∗A)µ(c))hence, by Axiom (R2) and Remark 2.9, from R(s � δ, s(δ)), which impliesR(s(η)y, s(δ)), it follows R((s∗A)µ(c) · ω, s(δ)) and R(sA, s(δ)). Thus, byLemma 2.22, c ≤A s(δ).

If δ is a limit of limits then, by the inductive hypothesis, a ≤A s(η) forevery η limit such that γ < η < δ. By Lemma 2.26(5), ran(sηδ ) ⊆ ran((s∗A)µ(c))for every η limit such that γ < η < δ. Hence, by Axiom (R2) and Remark2.9, R(s�δ, s(δ))→ R(sηδ , s(δ))→ R((s∗A)µ(c) · ω, s(δ))→ R(sA, s(δ)). Thus,by Lemma 2.22, c ≤A s(δ). a

By the claim, c ≤A s(δ) for every δ limit such that γ < δ < lh(s) hence,by Lemma 2.26(5), ran(sγ+ω) ⊆ ran((s∗A)µ(c)). Thus, by Axiom (R2) andRemark 2.9, R(s, p)→ R(sγ+ω, p)→ R((s∗A)µ(c) · ω, p)→ R(sA, p), for everyp ∈ P . a

Suppose that A has no maximum. We want to show that s is a refinementof sA. Then R(s, p)→ R(sA, p) will follow from Axiom (R2).

Let β < lh(sA) = sup{lh(sC) | C @ A}. Let B be the least initial segmentof A such that β < lh(sB). Let B = b<A and γ = µs(b). Let α be suchthat γ ≤ α < lh(s). So α = δ + n for some δ limit with n < ω ands(α) ∈ ran(s(δ)y).

If γ ≤ µs(δ) ≤ δ then s(δ) /∈ B. By Lemma 2.27, W (sB) = B andW (sA) = A = W (s), so s(δ) ∈ W (sA) −W (sB). By Remark 2.25, s(δ) =sA(δ′) for some δ′ limit such that β < lh(B) ≤ δ′. So s(α) ∈ (ran(sA)β).

If µs(δ) < γ < δ then let C = W(s � δ). Since s ∈ TR, R(s � δ, s(δ))holds. Since µs(δ) < δ, s(δ) ∈ C. Hence, by the inductive hypothesis,R(sC , s(δ)). Let µ = µs∗C and c = aR,s∗C . By Lemma 2.22, c ≤C s(δ). Hence,by Lemma 2.26(5), s(δ) ∈ ran((s∗C)

µ(c)). R(sC , s(δ)) and s(δ) ∈ C implysC 6= s∗C , hence sC is of the form t + tµ(c) · ω for some t such that s∗C ⊆ t.Hence s(δ) ∈ ran((s∗C)

µ(c)) implies s(δ) ∈ ran(tµ(c)). Thus, s(δ) is cofinal insC , hence there exists δ′ < lh(sC) such that β < δ′ and sA(δ′) = sC(δ

′) = s(δ).Therefore, s(α) ∈ (ran(sA)β). a

(2) By Lemma 2.22 it is enough to show that s∗A ∈ TR.

16 Edoardo Rivello

If A has no maximum then s∗A =⋃{sB | B @ A}, so s∗A ∈ TR by the

inductive hypothesis and Remark 2.8.If b = maxA, then s∗A = sB + by, where B = b<A and by is the ω-

sequence generated by ρ and b. By the inductive hypothesis, sB ∈ TR so,by Lemma 2.11, it only remains to show that R(sB, b) holds. Let γ = µs(b).B =W(s�γ) so, by (1), R(s�γ, s(γ)) implies R(sB, b). �

Let p be a TR-limit reflexive hypothesis and let this fact be witnessedby an R-iteration s. If R is a limit rule satisfying the axioms (R0)–(R2)then, by the lemmata 2.26(6) and 2.28, sA ∈ TR, lh(sA) < ν(P ), sA(0) =s(0) = p and R(sA, p) holds since R(s, p) holds, where A =W(s). And this,together with the right-to-left direction of Lemma 2.15, completes our proofof Theorem 2.18.

3. Gupta-Belnap’s revision and Visser’s theorem

In this section we will use Theorem 2.18 in order to prove that ℵ(P ) is asuitable upper bound for the length of the revision sequences we need forformalising Gupta-Belnap’s notion of recurring hypothesis [2, p. 174] in anorder-theoretic setting like that used in [11].11

For the rest of this paper we assume that the set of the hypotheses Pcarries the structure of a partially ordered set, denoted by P = 〈P,�〉. ForX ⊆ P and p ∈ P , p is an upper bound of X in P if q � p whenever q ∈ X,and p is a lower bound of X in P if p � q whenever q ∈ X. lub(X) andglb(X) will denote the least upper bound and the greatest lower bound of Xin P, respectively, when they exist.

The key notion in doing a revision process is the inferior limit of a se-quence.

Def. 3.1 (Inferior limit). Let s be any sequence from P . Assuming that allglb’s and lub’s in the following formula exist, define

liminf(s) = lub{glb(ran(sα)) | α < lh(s)}.

In this paper we will always assume that the liminf operation is definedon every sequence from P .12

11See [9] for more details and a discussion on how the original Gupta-Belnap’s notion ofhypothesis and other settings for revision fit into the present one.

12Visser’s notion of coherent complete partial order (ccpo) grants this requirement. See[9] for details.


The main reason for studying, in Section 2, the notion of recurring hypoth-esis was that, in most applications of revision and notably in Gupta-Belnap’srevision theory of truth, the hypotheses in P can be identified with functionsfrom some domain of objects D (sentences, in the case of truth) and that abasic output of the revision process is the classification of the objects of Dinto stable and unstable ones. Informally, the stable objects are those that“survive” the revision process, in the sense that they eventually receive thesame value by the hypotheses involved in the process, and it turns out that,formally, the set of the elements which are stable under a revision theory Ctogether with their stable values will coincide with the intersection of all C -recurring hypotheses. Let us denote by stab(C ) the set of all objects whichare stable with respect to some collection C of revision sequences. Hence, inthe order-theoretic setting, the notion of C -stability reduces to the notion ofC -recurrence by the relation stab(C ) = glb(Rec(C )).13

A hypothesis p ∈ P is said to be s-coherent if and only if liminf(s) � p.An iteration s of the revision operator ρ is a revision sequence if and onlyif s(γ) is (s � γ)-coherent for every limit ordinal γ less than the length of s.Gupta-Belnap’s revision theory is defined as the collection B of all ordinal-length revision sequences.

Gupta-Belnap’s revision theory can be identified14 with the revision the-ory B = CB specified by the limit rule B, where B simply coincides withthe coherence condition: B(s, p) ↔ lim inf(s) � p. Then it is easy to checkthe following, fundamental, fact:

Fact 3.2. The coherence condition satisfies the axioms (R0)-(R2) for limitrules.

Proof. Let s be any iteration of the revision operator ρ.(R0) Trivial, taking p = lim inf(s). a(R1) Let p ∈ Cf(s). By the definition of inferior limit, if Cf(s) 6= ∅ then

lim inf(s) � glb(Cf(s)). Hence lim inf(s) � p. a(R2) Let t be a refinement of s. By the definition of inferior limit, it

follows lim inf(s) � lim inf(t). Hence, lim inf(t) � p implies lim inf(s) �p. �

From this fact, the results proved in Section 2 immediately yield thefollowing:

13More precisely, in the order-theoretic setting we define stab(C ) = glb{lim inf(σ) | σ ∈C } and it can be shown that this set corresponds to the sets of stabilities in Gupta-Belnap’ssense. See [9].

14See below for a discussion.

18 Edoardo Rivello

Thm 3.3. Let B be the Gupta-Belnap’s limit rule and let p be any hypothesis.Then the following are equivalent:

(a) p is reflexive.(b) There exists a limit-length revision sequence s such that p = s(0) and p

is s-coherent (by Lemma 2.14).(c) p is reflexive and its reflexivity is witnessed by a revision sequence s whose

length is less than ℵ(P ) (by Theorem 2.18. When P is finite we shouldtake ωω instead of ℵ(P )).

To accommodate in this order-theoretic setting the standard presentationof Gupta-Belnap’s revision theory given in [2] we only need to observe a fewthings. In [2], the hypotheses are the functions from a set D into a set Xand the revision operator ρ is a function from this set of hypotheses DX intoDX. We identify our partially ordered set P with the set P of all partialfunctions from D to X ordered by inclusion. Hence the “hypotheses” inGupta-Belnap’s sense are the total functions, namely the maximal elementsof P, which constitute a subset H ⊂ P . Accordingly, the limit rule B can beidentified in the order-theoretic setting with the condition:

B(s, p)↔ lim inf(s) � p ∧ p ∈ H.

To apply Theorem 3.3 to B defined as above we can check that Fact 3.2 holdsby essentially the same proof. In particular, since for every partial functionp there exists a total function h extending p, Axioms (R0) is satisfied by B.

The equivalence between (a) and (b) above is a direct strengthening ofGupta-Belnap’s equivalence between recurring and reflexive hypotheses [2,Thm. 5C.13, p. 175]. Indeed, it implies that if the reflexivity of a hypothesish is witnessed by some sequence s where h occurs at stages 0 and α, weare free to assume that α is a limit ordinal (possibly, changing the witnesssequence).

On the other hand, the equivalence between (a) and (c) extends Visser’supper bound [11, Remark 59, p. 220] from revision theories like Herzberger’sand Gupta’s to every revision theory satisfying the axioms (R0)–(R2) and,in particular, to Gupta-Belnap’s revision theory. The proof of Theorem 3.3can be formalised in Zermelo-Fraenkel set theory without Choice.

Final remarks Gupta-Belnap’s revision theory B and Herzberger’s re-vision theory H are among the most well-studied revision theories. Theyare also representative of two opposite ways of handling the revision pro-cesses at the limit stages: In [8], I called them the deterministic and the


non-deterministic ones, respectively, and I refer to that work for details.Essentially, a deterministic revision theory allows one and only one revi-sion sequence for each starting hypothesis, while a non-deterministic one ismore liberal and Gupta-Belnap’s, in particular, allows all possible revisionsequences.

Both revision theories B and H are based on cofinally dependent limitrules. In [8] I showed that cofinal dependence is enough to yield the so-called “periodicity property”: Every revision sequence in H (B) is cofi-nally equivalent to a periodic revision sequence in H (B, respectively). Inthis paper I have shown that even Visser’s upper bound, established in [10]for Herzberger’s revision in the order-theoretic setting, is shared by Gupta-Belnap’s revision. However, my proofs of the upper bound in the two casessuggest that the results rely on different reasons. In [9], I proved the upperbound for Herzberger’s revision as a consequence of the cofinal invarianceof the limit rule which is, roughly speaking, the result of being cofinally de-pendent and deterministic. In contrast, to obtain the same result for Gupta-Belnap’s revision, we have to consider the axioms (R0)–(R2) which imply thecofinal dependence of the limit rule B but which also capture strong closureproperties of the collection B = CB.

In both cases, however, we used similar techniques to prove the upperbound. Starting with a witness sequence s (of periodicity, in Herzberger’scase, and of reflexivity, in Gupta-Belnap’s one) we worked on the insightthat the relevant information about s must be already encapsulated in thewellordering A = W(s) of the limit values of s and that s itself only addssome redundancy. Then we built from s a new sequence s′ with the samewellordering A, yet following a more efficient procedure which allows us tocontrol the length of s′. The reason why the procedure works are different inthe two cases. Essentially, at each stage of the unfolding process we have tochoose an element a ∈ A which satisfies the limit rule: In the deterministiccase, whenever such an a exists it is unique while, in Gupta-Belnap’s revision,we can choose the minimum one.

Summarising: The results proved in this paper (Lemma 2.14 and Theo-rem 2.18) extend to Gupta-Belnap’s revision theory, formulated in the order-theoretic setting, an upper bound which was already established in [10] forHerzberger-like revision theories, in the same setting, and by McGee andBurgess (independently) for Gupta-Belnap’s revision in a setting essentiallyequivalent to that of [2]. Loosely speaking, the results can be synthesised asfollows: If either C is deterministic or is sufficiently closed under operationson iterations from P , then ℵ(P ) is an upper bound for the length of the re-vision sequences we need in order to formalise in ZF the notion of recurring

20 Edoardo Rivello

hypothesis. My proof of this fact makes no use of the Axiom of Choice andis carried out in the same abstract setting previously investigated in [9] and[8].

References

[1] Gupta, Anil, ‘Truth and paradox’, Journal of Philosophical Logic, 11 (1982), 1,1–60.

[2] Gupta, Anil, and Nuel Belnap, The Revision Theory of Truth, A Bradford Book,MIT Press, Cambridge, MA, 1993.

[3] Herzberger, Hans G., ‘Notes on naive semantics’, Journal of Philosophical Logic,11 (1982), 1, 61–102.

[4] Kühnberger, Kai-Uwe, et al., ‘Comparing Inductive and Circular Definitions:Parameters, Complexity and Games’, Studia Logica, 81 (2005), 79–98.

[5] Lévy, Azriel, Basic Set Theory, Springer-Verlag, Berlin, 1979.[6] Löwe, Benedikt, ‘Revision forever!’, in Henrik Schärfe, et al., (eds.), Conceptual

Structures: Inspiration and Application, 14th International Conference on ConceptualStructures, ICCS 2006, Aalborg, Denmark, July 16–21, 2006, no. 4068 in LectureNotes in Artificial Intelligence, Springer-Verlag, 2006, pp. 22–36.

[7] McGee, Vann, Truth, Vagueness, and Paradox. An Essay on the Logic of Truth,Hackett Publishing Company, Indianapolis, Cambridge, 1991.

[8] Rivello, Edoardo, ‘Periodicity and reflexivity in revision sequences’, Studia Logica,To appear.

[9] Rivello, Edoardo, ‘Cofinally invariant sequences and revision’, Studia Logica, 103(2015), 3, 599–622.

[10] Visser, Albert, ‘Semantics and the Liar paradox’, in Handbook of PhilosophicalLogic, vol. 4, Kluwer Academic Publishers, 1989.

[11] Visser, Albert, ‘Semantics and the Liar paradox’, in Handbook of PhilosophicalLogic, vol. 11, Kluwer Academic Publishers, 2004. 2nd ed. of [10].

Edoardo RivelloDepartment of MathematicsUniversità di TorinoVia Carlo Alberto, 10Torino, [email protected]

Gupta-Belnap's revision in an abstract setting

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