Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and...

Annals of Pure and Applied Logic 114 (2002) 227–271www.elsevier.com/locate/apal

Substitutions of �01-sentences: explorations

between intuitionistic propositional logic andintuitionistic arithmetic

Albert VisserDepartment of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, Netherlands

Dedicated to Anne Troelstra on the occasion of his 60th birthday

Abstract

This paper is concerned with notions of consequence. On the one hand, we study admissibleconsequence, speci.cally for substitutions of �0

1-sentences over Heyting arithmetic (HA). On theother hand, we study preservativity relations. The notion of preservativity of sentences over agiven theory is a dual of the notion of conservativity of formulas over a given theory. We showthat admissible consequence for �0

1-substitutions over HA coincides with NNIL-preservativityover intuitionistic propositional logic (IPC). Here NNIL is the class of propositional formulaswith no nestings of implications to the left. The identical embedding of IPC-derivability (con-sidered as a preorder and, thus, as a category) into a consequence relation (considered as apreorder) has in many cases a left adjoint. The main tool of the present paper will be an al-gorithm to compute this left adjoint in the case of NNIL-preservativity. In the last section, weemploy the methods developed in the paper to give a characterization the closed fragment ofthe provability logic of HA. c© 2002 Elsevier Science B.V. All rights reserved.

MSC: 03B20; 03F40; 03F45; 03F55

Keywords: Constructive logic; Propositional logic; Heyting’s arithmetic; Schema; Admissiblerule; Consequence relation; Provability logic

Contents

1. Introduction1.1. Propositional logic1.2. Arithmetic1.3. Provability logic1.4. History and context1.5. Prerequisites

E-mail address: [email protected] (A. Visser).

0168-0072/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0168 -0072(01)00081 -1

228 A. Visser / Annals of Pure and Applied Logic 114 (2002) 227–271

2. Formulas of propositional logic3. Semi-consequence

3.1. Basic de.nitions3.2. Principles involving implication

4. Preservativity4.1. Basic de.nitions4.2. Examples of preservativity relations

4.2.1. The logic of a theory4.2.2. Admissible consequence4.2.3. A result of Pitts

4.3. Basic facts4.4. A survey of results on preservativity

4.4.1. Derivability for IPC4.4.2. Admissible consequence for IPC4.4.3. Admissible rules for �-substitutions over HA

5. Applications of Kripke semantics5.1. Some basic de.nitions5.2. The Henkin construction5.3. More de.nitions5.4. The Push Down Lemma

6. Robust formulas7. The NNIL-algorithm8. Basic facts and notations in arithmetic

8.1. Arithmetical theories8.2. A brief introduction to HA∗

9. Closure properties of �-preservativity9.1. Closure under B19.2. A closure rule for implication

10. On �-substitutions11. The admissible rules of HA12. Closed fragments

12.1. Preliminaries12.2. The closed fragment for HA∗

12.3. The closed fragment of PA12.4. The closed fragment of HA12.5. Comparing three functors12.6. Questions

AcknowledgementsAppendix A. Adjoints in preordersAppendix B. Characterizations and dependenciesAppendix C. Modal logic for �-preservativityReferences

1. Introduction

This paper is a study both of constructive propositional logic and of constructivearithmetic. In the case of arithmetic the focus is not on proof theoretical strength buton the ‘logical’ properties of arithmetical theories.

I will .rst explain the contents of the part of the paper that is concerned purelywith propositional logic. This part is independent of the arithmetical part and could bestudied on its own. After that, I will explain the arithmetical part.

A. Visser / Annals of Pure and Applied Logic 114 (2002) 227–271 229

1.1. Propositional logic

Intuitionistic propositional logic, IPC, diFers from classical propositional logic in thatit admits natural hierarchies of formulas. The .rst level of one of these hierarchies isformed by the NNIL-formulas. ‘NNIL’ stands for no nestings of implications to theleft. Thus (p→ (q→ (r ∨ (s→ t)))) is a NNIL-formula and ((p→ q)→ r) is not. NNILformulas are the analogues of purely universal sentences in the prenex normal formhierarchy for classical predicate logic, in view of the following characterization: moduloIPC-provable equivalence this class coincides with ROB the class of formulas that arepreserved under sub-Kripke-models. 1

Here ROB stands for robust. The characterization can be proved in at least two ways.One proof is presented in [49]. The idea of this proof is due to Johan van Benthem.Van Benthem’s proof is analogous to the classical proof of LoKs-Tarski theorem. Seealso [38]. Another proof is contained in the preprints [43, 44]. This second proof isthe one presented in this paper (see Section 7). The basic idea of the proof is that,given an arbitrary propositional formula A, one computes its best robust approximationfrom below, say A∗. The computation will terminate in a NNIL-formula. Since NNIL-formulas are trivially robust, it follows that robust formulas can always be rewritten toIPC-equivalent NNIL-formulas.

ReLecting on the proof, one sees that it is best understood in terms of the followingnotions. We de.ne IPC, ROB-preservativity as follows:

A .IPC;ROB B :⇔ ∀C ∈ ROB (C �IPC A⇒ C �IPC B):

The claim that A∗ is the best robust approximation of A, can be analyzed to meanthe following: A .IPC;ROB B⇔A∗ �IPC B. Thus (·)∗ is the left adjoint of the identicalembedding of �IPC considered as a preorder (and thus as a category) into .IPC;ROBconsidered as a preorder. As a spin-oF of our proof we obtain an ‘axiomatization’ of.IPC;ROB.

Preservativity can be viewed as dual to conservativity of sentences over a giventheory. Note that in the constructive context preservativity and conservativity cannotbe transformed into each other (modulo provable equivalence).

Another important notion of consequence is admissible consequence, discussed inthe next section. In the present paper we will study the two consequence notions,preservativity & admissibility, in tandem.

1 In fact, there is a way of strengthening the characterization, that makes NNIL a natural subclass of thepurely universal formulas of a speci.c theory in predicate logic. We can reformulate Kripke semantics in afamiliar way as a translation of IPC into a suitable theory in predicate logic, say T , of Kripke structures.One can characterize the formulas of IPC, modulo T -provable equivalence, as precisely the predicate logicalformulas in one free variable that are upwards persistent and preserved under bisimulation. The NNIL-formulas are, modulo provable equivalence, precisely the predicate logical formulas in one free variable thatare upward persistent, preserved under bisimulation and preserved under taking submodels. Thus, the NNIL-formulas are, modulo provable equivalence, precisely the purely universal formulas of T that correspond viathe translation to IPC-formulas. See [49].


1.2. Arithmetic

A �01 -formula is a formula in the language of arithmetic consisting of a block of

existential quanti5ers followed by a formula containing only bounded quanti.ers. A�0

1-formula is a formula in the language of arithmetic consisting of a block of uni-versal quanti5ers followed by a formula containing only bounded quanti.ers. We willoften write ‘�-formula’ and ‘�-formula’ for �0

1 -formula, �01 -formula, respectively. In

the classical context, �-formulas and �-formulas are interde.nable using only Booleanconnectives: a �-formula is the negation of �-formula and a �-formula is the negationof a �-formula. Constructively, under minimal arithmetical assumptions, a �-formulais still provably equivalent to the negation of a �-formula, but e.g. over Heyting’s arith-metic, we have that if a �-sentence is provably equivalent to any Boolean combinationof �-sentences, then it is either provable or refutable. 2

It is a common observation, that if we have proved some Boolean (or, perhaps,‘Brouwerian’) combination of �-sentences in Heyting arithmetic, (HA), then we oftenknow that a better Boolean combination of the same �-sentences is also provable.Moreover, this better Boolean combination can be found independent of the speci.c�-sentences under consideration.

Suppose, for example, that we have found that HA�¬¬S→ S. Then, using theFriedman translation, we may show that one also has HA� S ∨¬S.

In which sense is (S ∨¬S) better than (¬¬S→ S)? Well, we have

p ∨ ¬p �IPC ¬¬p→ p but not ¬¬p→ p �IPC p ∨ ¬p:

So the form of (S ∨¬S) is more informative than the form of (¬¬S→ S). Can wedo still better than (S ∨¬S)? Yes and no. Yes, since HA has the disjunction propertywe do have either HA� S or HA�¬S. Thus, for a speci.c S, we can .nd a furtherimprovement. No, if we demand that the improvement depend only on the form of theBoolean combination of �-sentences, in other words, if we want the improvement tobe uniform for arbitrary substitutions of �-sentences.

Analyzing these ideas, one arrives at the following explication of what one is look-ing for. Let P be a set of propositional variables. �P is the set of assignments of�-sentences to the propositional variables. Let f be such an assignment of �-sentences.We write fA for the result of substituting the fp’s for p in A. If we want to stressthe fact that we lifted f on P to a function on the full language of propositionallogic over P, we write [f] for the result of the lifting. Thus [f] is a substitution of�-sentences. We de.ne the relation of HA, �-admissible consequence as follows. LetA; B be IPC-formulas.

A|∼HA;�B ⇔ ∀f ∈ �P (HA � fA⇒ HA � fB):

2 One way to prove this is by using Corollary 9.2 in combination with the disjunction property of Heyting’sarithmetic.


Consider a propositional formula A. Let us call A’s desired best improvement as A∗.We will show that A∗ always exists and satis.es the following claim:

A|∼HA;�B ⇔ A∗ �IPC B:As the attentive reader would have guessed, (·)∗ of Section 1.1 is identical to (·)∗ ofSection 1.2. Thus, we also have |∼HA; � = .IPC;ROB.

1.3. Provability logic

Pure provability logic has, in my opinion, two great open problems. The .rst one isto characterize the provability logics of Wilkie and Paris’ theory I�0 +�1 and of Buss’theory S1

2. This problem has been extensively studied in [2, 40]. The second problemis to characterize the provability logic of HA. This problem is studied in [41–43, 19].

In the present paper we will present full characterizations for two fragments of theprovabilitity logic of HA. The .rst is the characterization of all formulas of the formA→ B, valid in the provability logic of HA, where A and B are purely propositional.

This characterization is due to Rosalie IemhoF. See her paper [18]. The second one isthe characterization of the closed fragment of the provability logic of HA. This is myold result, reported .rst in [43]. 3

In the appendix, I formulate a conjecture about the provability logic of HA. Part ofthis conjecture is that this logic is best formulated in the richer language containing abinary connective for �-preservativity.

1.4. History and context

The results reported in this paper took a long time from conception to publication.The main part of this research here was done between 1981 and 1984. The primaryobjects of this research were (i) the characterization of the closed fragment of HA and(ii) generalizing the results of Dick de Jongh on propositional formulas of one variable(see [6]). 4 The results of this research were reported in the preprint [43]. For variousreasons, however, the preprint was never published. In 1994 I completely rewrote thepaper, resulting in the preprint [44]. However, again, the paper was not published. Thepresent version, written in 2000, is an extensive update of the 1994 preprint.

To put the present paper in a broader context of research, let me brieLy sum upsome related work.

Firstly, there are many results on logics of theories (see Section 4.2.1). For simplepropositional logic there is De Jongh’s theorem saying that the propositonal logic of HA

3 The problem to characterize the closed fragment of the provability logic of PA was Friedman’s 35thproblem. See [9]. It was solved independently by van Benthem, Magari and Boolos.

4 The full generalization of de Jongh’s results had to wait till 1998 (see [47]). The crucial lemma, wasproved by Silvio Ghilardi, following a rather diFerent line of enquiry. See [12]. Ghilardi’s lemma charac-terizes two classes of propositional formulas, the projective formulas and the exact formulas as preciselythose with the extension property—a property formulated in terms of Kripke models. Guram Bezhanishvilishowed me that Ghilardi’s lemma can also be obtained by applying the work of R. Grigolia. See [15].


and related theories is precisely IPC. See e.g. [32, 43, 8], for some proofs. In the case ofe.g. HA+MP+ECT0, we get a stronger logic. See e.g. [11]. A precise characterizationof this logic is still lacking. For de Jongh’s theorem for Predicate Logic, see [5, 21,39]. Negative results for the case of intuitionistic predicate logic are contained in [26–28]. For the study of schemes in the classical predicate calculus, the reader is referredto [50].

Secondly, we have provability logic. Here the schematic languages are languages ofmodal logic. The reader is referred to the excellent textbooks [33, 3]. For the workin this paper, the development of interpretability logic is of particular interest. Seethe survey papers [20, 46]. Our central notion �-preservativity is closely related tothe notion of �0

1 -conservativity, studied in interpretability logic. This connection turnsout to be really useful: the expertise generated by research into Kripke models forinterpretability logic, is now used by Rosalie IemhoF in the study of the provabilitylogic of HA and of admissible rules. See [18, 19].

Thirdly, the work on axiom schemes can be generalized to the study of admissiblerules, exact formulas, uni.cation and the like. We refer the reader to [22,6, 7, 8, 29, 30, 12, 18]. A closely related subject is uniform interpolation. See [25, 13,14, 45].

Fourthly and .nally, there is the issue of formula classes, like NNIL, and theircharacterization. These matters are taken up in 5 e.g. [23, 38, 49, 4, 48].

1.5. Prerequisites

Some knowledge of [36, 37] is certainly bene.cial. At some places I will make useof results from [42, 8].

2. Formulas of propositional logic

In this section we .x some notations. We also introduce the central formula classNNIL.P;Q; : : : will be sets of propositional variables. p̃; q̃; r̃; : : : will be 5nite sets of

propositional variables. We de.ne L(P) as the smallest set S such that• P⊆S, �;⊥∈S,• If A; B∈S, then (A ∧B); (A ∨B); (A→B)∈S.sub(A) is the set of subformulas of A. By convention we will count ⊥ as a subformulaof any A. pv(A) is the set of propositional variables occurring in A.

We de.ne a measure of complexity �, which counts the left-nesting of → , asfollows:• �p := �⊥ := �� := 0,• �(A ∧B) := �(A ∨B) := max(�A; �B),• �(A→B) := max(�A+ 1; �B).

5 Ref. [49] may be considered as a companion paper of the present paper.


NNIL(P) := {A∈L(P) | �A61}. In other words, NNIL is the class of formulas withoutnestings of implications to the left. An example of a NNIL-formula is (p→ (q ∨ (s→t))) ∧ ((q ∨ r)→ s). It is easy to see that modulo IPC-provable equivalence each NNIL-formula can be rewritten to a NNIL0-formula, i.e. a formula in which as antecedents ofimplications only single atoms occur. For more information about NNIL, see [8, 49].For a generalization of the result of [49] to the case of predicate logic, see [48].

3. Semi-consequence

In this section we introduce the basic notion of semi-consequence relation. ‘.’ willrange over semi-consequence relations. Let � stand for derivability in IPC.

3.1. Basic de5nitions

Let B be a language (for propositional or predicate logic) and let T be a theoryin B. A semi-consequence relation on B over T is a binary relation on the set ofB-formulas satisfying:

A1. A�T B⇒A . B,A2. A . B and B . C⇒A . C,A3. C . A and C . B⇒C . (A ∧B).

The name ‘semi-consequence relation’ is ad hoc in this paper. We take

A≡B : ⇔A . B and B .A.

If we do not specify the theory corresponding to the semi-consequence relation, it isalways supposed to be over IPC. A further salient principle is

B1. A . C and B .C⇒ (A ∨B) . C.

A relation satisfying A1–A3 and B1 is called a nearly-consequence relation. Note that�T is a nearly-consequence relation over T .

We will use X � Y; X . Y for, respectively,∧X � ∨

Y and∧X .

∨Y , where X and

Y are .nite sets of formulas. Here∧ ∅ :=� and

∨ ∅ :=⊥. We treat implications sim-ilarly, writing (X →Y ) for (

∧X → ∨

Y ). We write X; Y for X ∪Y ; X; A for X ∪{A},etc.

Nearly-consequence relations over T can be alternatively described in the Genzenstyle as follows. Nearly-consequence relations are relations between (.nite sets of)formulas satifying the following conditions:

A1′. X �T Y ⇒X . Y .Thin X . Y ⇒X; Z . Y; U .Cut X . Y; A and Z; A . U⇒X; Z . Y; U .

We take the permutation rules to be implicit in the set notation. We leave it to thereader to check the equivalence of the Genzen style principles with A1–A3 plus B1.


We will be interested in adjoints involving semi-consequence relations. Some basicfacts concerning adjoints and preorders are given in Appendix A.

3.2. Principles involving implication

Principles involving implication play an important role in the paper. In fact, the mainreason for the complexity of, e.g. the admissible rules of IPC, the admissible rules of HAand the admissible rules of HA for �-substitutions is the presence of nested implicationsand the interplay of implications and disjunctions (in the antecedent).

Regrettably, the principles involving implication needed to characterize admissibleconsequence and �-preservativity are never going to win a beauty contest.

The principles introduced below, allow us to reduce the complexity of formulas.E.g. if a formula of the form (C→D)→ (E ∨F) is provable in IPC (HA), then also((C→D)→C) ∨ ((C→D)→E) ∨ ((C→D)→F) is provable in IPC (HA). The re-moval of the disjunction from the consequent makes further simpli.cation possible.

At this point, we will just introduce the principles in full generality. They willbecome more understandable in the light of their veri.cations (for various notions ofconsequence) in, e.g. Theorems 6.2 and 9.1.

To introduce the principles we need some syntactical operations. We de.ne theoperations [·](·) and {·}(·) on propositional formulas as follows:– [B]p :=p; [B]� :=�; [B]⊥ :=⊥,– [·](·) commutes with ∧ and ∨ in the second argument,– [B](C→D) := (B→ (C→D)).– {B}p := (B→p); {B}� :=�; {B}⊥ :=⊥,– {·}(·) commutes with ∧ and ∨ in the second argument,– {B}(C→D) := (B→ (C→D)).Note that [·](·) and {·}(·) do not preserve provable equivalence in the second argument.Note also that � B→ ([B]C↔C) and � B→ ({B}C↔C). De.ne [B]X := {[B]C |C ∈X }.

We will study the following principles for implication for semi-consequence relationson L(P).

B2. Let X be a .nite set of implications and let

Y := {C | (C → D) ∈ X } ∪ {B}:Take A :=

∧X . Then (A→B) . [A]Y .

B2′. Let X be a .nite set of implications and let

Y := {C | (C → D) ∈ X } ∪ {B}:Take A :=

∧X . Then (A→B) . {A}Y .

B3. A . B⇒ (p→A) . (p→B).It is easy to see that B2′ can be derived from B2 over A1–A3. Note that both B2, B2′

and B3 are non-ordinary schemes. Conditions B2 and B2′ involve .nite conjunctionsand disjunctions and B2 and B3 contain variables ranging over proposition letters.


(It is easy to see that one can replace B2′ by an equivalent scheme that makes nospecial mention of proposition letters. See also [18].) The fact that proposition lettersare not ‘generic’ in B2 and B3, reLects our interest in certain special substitutions,like substitutions of �-sentences.

If we spell out the conclusion of, e.g. B2, we get

(A→ B) .∨

{[A]C | (C → D) ∈ X or C = B}:

We give an example of a use of B2. Suppose . satis.es B2. Let

E := ((p→ q) ∧ (r → (s ∨ t))); F := (E → (u ∨ (p→ r))):

Then we have F . (p ∨ r ∨ u ∨ (E→ (p→ r))).

Remark 3.1. Rosalie IemhoF has shown that B2′ cannot be replaced by .nitely manyconventional schemes. This is an immediate consequence of the methods developed in[18]. I conjecture that a similar result holds for B2.

We say that a relation satisfying A1–A3, B1–B3 is a �-relation. We say that arelation satisfying A1–A3, B1, B2′ is an �-relation. We take I� to be the smallest�-relation, i.o.w I� is the semi-consequence relation axiomatized by A1–A3, B1–B3.Similarly, I� is the smallest �-relation.

4. Preservativity

In this section, we introduce preservativity relations which are special semi-consequence relations. In fact, we will treat some semi-consequence relations that arenot preservativity relations: we will also consider relations like provably deductiveconsequence. Treatment of these further relations is postponed till Section 12.

4.1. Basic de5nitions

Consider again any language B of propositional or predicate logic. If A∈L(P) andf : P→B, then fA is the result of substituting fp for p in A for each p∈P. If wewant to stress that we are speaking of f as L(P)→B we will use [f]. Let X ⊆B,let T be any theory in B and let F be a set of functions from P to B. We write• for A; B∈B; A .T;X B :⇔∀C ∈X (C �T A⇒C �T B),• for A; B∈L(P); A .T;X;F B :⇔∀f∈F fA .T;X fB.If Y ⊆B and F=YP, we will write .T;X; Y for .T;X;F. If X or F are singletons wewill omit the singleton brackets. If B=L(P) and T = IPC, we will often omit ‘T ’ inthe index. We will call .T;X;F; T; X;F-preservativity, etc. We will call the .T;X;F andthe .T;X preservativity relations. We will call the .T;X pure preservativity relations.

Clearly, .T;X is a semi-consequence relation over T and .T;X;F is a semi-consequencerelation over IPC. Below we will provide a number of motivating examples for ourde.nitions.


Remark 4.1. It is instructive to compare preservativity with conservativity (of sen-tences over a theory). De.ne

For A; B∈B; A .∗T;X B :⇔∀C ∈X (B�T C⇒A �T C).So A.∗T;X B means that T +B is conservative over T +A w.r.t. X . For classical theoriesT we haveX A .T;X B ⇔ ¬B .∗T;¬X ¬A, where ¬X := {¬C |C ∈X }.

Thus, in the classical case preservativity is a superLuous notion. In the constructivecase, however, the reduction given in X does not work. Note that conservativity asa relation between sentences over a theory is a natural extension of the notion ofconservativity between theories. There is no analogue of this for preservativity.

It should be noted that �02 -conservativity between theories is a central conceptual

tool for theory comparison. The only other notions of theory comparison of comparableimportance are relative interpretability and veri.able relative consistency.�0

1 -conservativity for RE extensions of Peano arithmetic, PA, in the language ofPA is equivalent to relative interpretability for these theories. In the constructive case,relative interpretability plays a much more modest role. Many translations that do notcommute with the logical connectives, play a signi.cant role. Moreover, the meta-mathematical properties of relative interpretability are much diFerent. We certainlydo not have anything resembling the connection between relative interpretability and�1-conservativity.�0

1 -conservativity of sentences over Heyting’s arithmetic, HA, is reducible to�-preservativity over HA byA .∗HA;�0

1B⇔¬B .HA;�0

1¬A.

As an auxiliary notion to prove metamathematical results, �-conservativity over HA isthe more useful notion, as will be illustrated in the rest of the paper.

4.2. Examples of preservativity relations

In this subsection, we provide examples of preservativity relations and we reviewsome results from the literature concerning these examples.

4.2.1. The logic of a theoryIf we take X :=B, then .T;X is simply equal to �T . More interestingly we consider

.T;B;F. It is easy to see that

A .T;B;F B ⇔ ∀f ∈ F fA �T fB:We de.ne• T;F := {A | ∀f∈F T �fA}.• T := T;BP . The theory T is the propositional logic of T .It is easy to see that

A .T;B;F B⇔ T;F � (A→ B)

T;F � A⇔� .T;B;F A


De Jongh’s Completeness Theorem for �-substitutions tells us that HA = HA; � = IPC.There are many diFerent proofs of De Jongh’s theorem, see e.g. [32] or [43] or [8].

4.2.2. Admissible consequenceIf we take X :={�}, then, for A; B∈B, we .nd

A .T;� B ⇔ (T � A⇒; T � B):

Thus .T;� is the relation of deductive consequence for T . Moreover, for A; B∈L(P),

A .T;�;F B ⇔ ∀f ∈ F (T � fA⇒ T � fB):

We .nd that .T;�;F is an admissible consequence for T w.r.t. F. We de.ne• A |∼T;F B :⇔A .T;�;FB,• A |∼T B :⇔A |∼T;B B.

We will provide more information about admissible consequence in Section 4.4.

4.2.3. A result of PittsSuppose Q⊆P. Let R :=P\Q. Consider .IPC;L(Q). By Theorem 5.3 below, in com-

bination with Lemma 4.2(2) below, .IPC;L(Q) is a nearly-consequence relation. AndrewPitts, in his [25], shows that for every A∈L(P), there is a formula (∀R A)∈L(Q),such that for all B∈L(P): A .IPC;L(Q) B⇔∀R A�B.

For a semantical treatment of Pitt’s result see [13] or [45]. Of related interest is thepaper [14].

4.3. Basic facts

We collect some technical facts about preservativity. A formula A of B is T -primeif, for any .nite set of B-formulas Z; A�T Z ⇒ ∃B∈Z A�T B. Note that if A isT -prime, then A0T ⊥. A class of formulas X is weakly T-disjunctive if every A∈Xis equivalent to the disjunction of a .nite set of T -prime formulas Y , with Y ⊆X .

For any class of formulas X , let disj(X ) be the closure of X under arbitrary dis-junctions. In the next lemma we collect a number of noteworthy small facts on preser-vativity.

Lemma 4.2. (1) .T;X is a semi-consequence relation over T . .T;X;F is a semi-consequence relation over IPC.

(2) Suppose X is weakly T -disjunctive; then .T;X and .T;X;F are nearly-consequencerelations.

(3) Suppose X is closed under conjunction; then

C ∈ X and A.T;X B⇒ (C → A) .T;X (C → B):

(4) Let range(F) be the union of the ranges of the elements of F. Suppose thatrange(F)⊆X and that X is closed under conjunction; then .T;X;F satis5es B3.

(5) Suppose A∈X . Then A.T;X B⇔A�T B.


(6) Suppose X ⊆Y and F⊆G; then .T;Y ⊆ .T;X and .T;Y;G ⊆ .T;X;F.(7) .T;disj(X ) = .T;X and; hence; .T;disj(X );F = .T;X;F.(8) Let id : P→L(P) be the identical embedding. Then .IPC; X; id = .IPC; X .(9) If T has the disjunction property; then |∼T and |∼T;F satisfy B1.

Proof. We treat (2) and (3).(2) Suppose X is weakly T -disjunctive and A.T;X C and B.T;X C. Let E be any

element of X . Suppose Y is a .nite set of T -prime formulas in X such that E isequivalent to the disjunction of Y . We have

E �T A ∨ B ⇒∨Y �T A ∨ B

⇒ ∀F ∈ Y F �T A ∨ B⇒ ∀F ∈ Y F �T A or F�TB⇒ ∀F ∈ Y F �T C⇒

∨Y �T C

⇒ E �T C

Condition B1 for .T;X;F is immediate from B1 for .T;X .(3) Suppose that X is closed under conjunction and that A.B. Let C; E ∈X and

suppose that E � (C→A). Then (E ∧C)�A and, hence, (E ∧C)�B. Ergo E � (C→B).

The next lemma is quite useful both to verify left adjointness and to show thatcertain classes of formulas are equal modulo IPC-provable equivalence.

Lemma 4.3. Let . be a semi-consequence relation over IPC and let ! : L(P)→L(P); let X ⊆L(P). Suppose that(1) A.!A;(2) !A�IPC A;(3) range(!)⊆X;(4) .⊆ .IPC; X .Then; we have:• A.B ⇔ A.IPC; X B⇔!A�IPC B;• X = range(!) modulo IPC-provable equivalence.

Proof. Under the assumptions of the lemma. We have

A.B ⇒ A .IPC; X B assumption (4)

⇒ !A .IPC; X B assumption (2); (4)

⇒ !A�IPC B Lemma 4:2(5)


⇒ !A . B

⇒ A . B assumption (1)

Consider any B∈X . We have B .!B. Hence, by assumption (4) and Lemma 4.2(5),B�IPC!B. Combining this with assumption (2), we .nd IPC�B↔!B.

We prove a basic theorem about admissible consequence. De.ne f ?g := [f] ◦ g. Itis easy to see that ? is associative and the identical embedding of P into L(P) isthe identity for ?.

Theorem 4.4. Suppose; T;F = IPC. Moreover; suppose that G⊆L(P)P and that;for any g∈G and f∈F; (f ?g)∈F. Then; |∼T;F ⊆ |∼IPC;G.

Proof. Suppose A |∼T;F B and � gA. Consider any f∈F. Clearly �T fgA, i.e. �T (f?g)A. Since (f?g)∈F, we .nd �T (f?g)B. Hence, for all f∈F, �T fgB and, hence,� gB.

Note that it follows that |∼IPC is maximal among the |∼T with T = IPC.

4.4. A survey of results on preservativity

In this section we give an overview of results concerning preservativity.

4.4.1. Derivability for IPCThe minimal preservativity relation over IPC is �IPC. There is an extension HA∗ of

HA (see in Section 8.2), such that �IPC = |∼HA∗ = |∼HA∗ ; �. This result was proved byde Jongh and Visser in their paper [8], adapting a result of Shavrukov (see [31] or[51]). Note that de Jongh’s theorem also follows that .HA;A;A = .HA;A; � =�IPC.

4.4.2. Admissible consequence for IPC|∼IPC is the maximal relation |∼T for theories T with T = IPC. This was shown in

Section 4.3. |∼IPC strictly extends �IPC. E.g. the independence of premiss rule

IP ¬p→ (q ∨ r)|∼IPC(¬p→ q) ∨ (¬p→ r)

is admissible but not derivable. We have the following facts:1. |∼IPC is decidable. This result is due to Rybakov. See the paper [29] or the book [30].

The result follows also, along a diFerent route, the results in [12] or, alternatively,the results in [18].

2. The embedding of �IPC into |∼IPC has a left adjoint, say (·)O. So we have

A|∼IPCB⇔ AO �IPC B:


This result is due to Ghilardi. See his paper [12]. (·)O does not commute withconjunction.

3. The following semi-consequence relations are identical:

|∼IPC; |∼HA; |∼HA; B�; .IPC;EX; I�:

We brieLy discuss the individual identities:• |∼IPC = |∼HA. This result is due to Visser. See his paper [47]. The result can be

obtained in a diFerent way via the results of [18] and the results of this paper. Wewill give the argument in Section 11.

• |∼IPC = |∼HA; B�. This result is due to de Jongh and Visser [8, Corollary 6:6].• |∼IPC = .IPC;EX. This result is immediate from the results in de Jongh and Visser’s

paper [8]. Here EX is the set of exact formulas. See e.g. [8] for a de.nition.Ghilardi’s result mentioned above in (2) strengthens this theorem, since, as it turnsout, the range of (·)O is precisely disj(EX).

• |∼IPC =I�. This result was obtained by IemhoF. See her paper [18].

4.4.3. Admissible rules for �-substitutions over HAObviously |∼HA; � extends |∼HA, and thus, ipso facto, |∼IPC. It follows from results to

be proved below that ¬¬p→p |∼HA; � p ∨¬p. However, we do not have ¬¬p→p |∼IPCp ∨¬p, as witnessed by the substitution p �→ ¬q.

We will connect |∼HA; �, i.e. HA-admissible consequence for �-substitutions, with theformula classes NNIL, ROB and f-ROB. NNIL is introduced in Section 2. ROB andf-ROB are de.ned in Section 6 below. We show that NNIL=ROB= f-ROB (mod IPC).The result was .rst proved in my unpublished paper [43]. We will present a versionof the proof in Section 7 of this paper. A diFerent proof, due to Johan van Benthem,is contained in [38]. A version of van Benthem’s proof is presented in [49].

We will further prove the following results:1. |∼HA; � = .HA; �;� = .IPC;NNIL =I�.2. The identical embedding of �IPC into |∼HA; � has a left adjoint (·)∗. I.o.w. we haveA |∼HA; � B⇔A∗ �IPC B.

These results were essentially already contained in [43].

5. Applications of Kripke semantics

To prepare ourselves for Section 6 we brieLy state some basic facts about Kripkemodels.

5.1. Some basic de5nitions

We suppose that the reader is familiar with Kripke models for IPC. Two goodintroductions are [36, 32]. Our treatment here is mainly intended to .x notations. A


model is a structure K= 〈K;6; �;P〉. Here K is a non-empty set of nodes, 6 isa partial ordering and � is a relation between nodes and propositional atoms in P,satisfying persistence: if k6k ′ and k �p, then k ′ �p. We call � the forcing relationof K. We use, e.g. PK for the set of propositional variables of K, �K for the forcingrelation of K, etc. A model K is a P-model if PK =P.

Consider K= 〈K;6; � ;P〉. We de.ne k �A for A∈L(P) in the standard way.K�A if, for all k ∈K , we have k �A.

A model K is 5nite if both KK and PK are .nite. A rooted model K is a structure〈K; b;6; � ;P〉, where 〈K;6; � ;P〉 is a model and where b∈K is the bottom elementw.r.t. 6.

For any k ∈K , we de.ne K[k] as the model 〈K ′; k;6′; � ′;P〉, where K ′ := ↑k :={k ′ | k6k ′} and where 6′ and � ′ are the restrictions of 6, respectively, � to K ′.(We will often simply write 6 and � for 6′ and � ′.)

5.2. The Henkin construction

A set X ⊆L(P) is adequate if it is non-empty and closed under subformulas (and,hence, contains ⊥). A set ' is X -saturated if:1. '⊆X ,2. ' 0 ⊥,3. If ' �A and A∈X , then A∈',4. If ' � (B ∨C) and (B ∨C)∈X , then B∈' or C ∈'.Let X be adequate. The Henkin model HX (P) is the P-model with, as nodes, theX -saturated sets � and as accessibility relation ⊆ . The atomic forcing in the nodes isgiven by '�p⇔p∈'. We have, by a standard argument, that '�A⇔A∈', for anyA∈X . Note that if X and P are .nite, then HX (P) is .nite. A direct consequenceof the Henkin construction is the Kripke Completeness Theorem. Let p̃ ⊇ pv(A),then

IPC � A ⇔ K�A; for all (.nite) p̃-models K:

5.3. More de5nitions

Let K be a set of P-models. MK is the P-model with nodes 〈k;K〉 for k ∈KK,K∈K and with ordering 〈k;K〉6〈m;M〉 :⇔K=M and k6Km: As atomic forcingwe de.ne 〈k;K〉�p :⇔ k �K p. In practice, we will forget the second components ofthe new nodes, pretending the domains to be already disjoint.

Let K be a P-model. BK is the rooted P-model obtained by adding a new bottomb to K and by taking b�p :⇔K�p. We put glue(K) :=BMK.

5.4. The Push Down Lemma

We will assume below that P is .xed. We will often notationally suppress it.


Theorem 5.1 (Push Down Lemma). Let X be adequate. Suppose � is X -saturatedand K��. Then glue(HX [�];K)��.

Proof. We show by induction on A∈X that b�A⇔A∈�. The cases of atoms, con-junction and disjunction are trivial. If (B→C)∈X and b� (B→C), then �� (B→C)and, hence, (B→C)∈�. Conversely, suppose (B→C)∈�. If b1B, we are easilydone. If b�B, then, by the Induction Hypothesis, B∈�, hence C∈� and, again by theInduction Hypothesis: b�C.

We say that � is (P-)prime if it is consistent and if, for every (C ∨D)∈L(P);�� (C ∨D) ⇒ ��C or ��D. A formula A is prime if {A} is prime.

Theorem 5.2. Suppose X is adequate and � is X -saturated. Then � is prime.

Proof. � is consistent by de.nition. Suppose ��C ∨D and �0C and �0D. Sup-pose further K��; K1C; M�� and M1D. Consider glue(HX (�);K;M). ByTheorem 5.1 we have b��. On the other hand, by persistence, b1C and b1D.Contradiction.

∼C exhibited next to a node means that C is not forced; this is not to be confusedwith ¬C exhibited next to a node, which means that ¬C is forced.

Theorem 5.3. Consider any formula A. The formula A can be written (modulo IPC-provable equivalence) as a disjunction of prime formulas C. Moreover; these C areconjunctions of implications and propositional variables in sub(A).

Proof. Consider a sub(A)-saturated �. Let ip(�) be the set of implications and atomsof �. It is easily seen that IPC�∧

ip(�)↔∧�. Take

D :=∨{∧

ip(�) |� is sub(A)-saturated and A ∈ �}:


Trivially, IPC�D→A. On the other hand if IPC0A→D, then by a standard construc-tion there is a sub(A)-saturated set ' such that A∈' and '0D.

Remark 5.4. Note that in the de.nition of D in Theorem 5.3, we can restrict ourselvesto the ⊆ -minimal sub(A)-saturated � with A∈�.

6. Robust formulas

In this section we study robust formulas. We will verify that .IPC;ROB is a �-relation,and thus I�⊆ .IPC;ROB.

We aim at the application of Lemma 4.3 to I� in the role of ., and ROB in the roleof X . The mapping ! needed for the application of Lemma 4.3, will be the mapping(·)∗ provided in Section 7.

Consider P-models K and M. We say that

K⊆M :⇔ ∃f : KK → KM(f is injective and f◦6K ⊆ 6M ◦fand ∀k ∈ KK; p ∈ P (k �K p⇔ f(k) �M p)):

Modulo the identi.cation of the elements of K with their f-images in M , clearly‘K⊆M’ means that K is a submodel of M. ROB is the class of formulas that ispreserved under submodels.

A submodel is full if the new ordering relation is the restriction of the old orderingrelation to the new worlds. We do not demand of our submodels that they are full.However, all our results work for full submodels too.

A ∈ ROB :⇔ A is robust :⇔ ∀M (M � A⇒ ∀K⊆M K � A):

We will let �; �′; *; : : : range over ROB. It is easy to see that NNIL⊆ROB. In Section 7we will see that modulo IPC-provable equivalence each robust formula is in NNIL.

Let us take as a local convention of this section that .:=.IPC;ROB. Clearly, we havethat P⊆ROB and that ROB is closed under conjunction. So it follows that . satis.esA1–A3 and B3. To verify B1, by Lemma 4.2(2), the following theorem suVces.

Theorem 6.1. ROB is weakly disjunctive.

Proof. Consider any A∈ROB. We write A in disjunctive normal form DA as inTheorem 5.3 and Remark 5.4. Consider any disjunct C(�) of DA. Here, as inRemark 5.4, � is a ⊆ -minimal sub(A)-saturated set with A∈� and C(�) is theconjunction of the atoms and implications in �. We claim that C(�) is robust. Con-sider any model K⊆M�C(�). Trivially, M��. By the Push Down Lemma 5:1,glue(Hsub(A)[�];M)��. Hence, glue(Hsub(A)[�];M)�A. Now clearly,

glue(Hsub(A)(�);K)⊆ glue(Hsub(A)(�);M):


By the stability of A we get glue(Hsub(A)(�);K)�A. Consider

' := {G ∈ sub(A) | glue(Hsub(A)(�);K) � G}:

Clearly '⊆�. Moreover, ' is sub(A)-saturated and A∈'. By the ⊆ -minimality of� we .nd '=�. Hence, glue(Hsub(A)(�);K)�C(�) and so K�C(�). Ergo C(�) isrobust.

Theorem 6.2. . is closed under B2.

Proof. Let X be a .nite set of implications and let Y :={C |(C→D)∈X } ∪ {B}. LetA :=

∧X . We have to show: (A→B) . [A]Y . The proof is by contraposition. Consider

any H ∈ROB and suppose H 0 [A]Y . Let K= 〈K; b;6;�;P〉 be a rooted model suchthat K�H and K1

∨[A]Y , i.e. for all E ∈Y , we have K1 [A]E.

Let K′ :=K{A} := 〈K ′; b;6′;�′;P〉, be the full submodel of K on K ′ := {b} ∪{k ′ ∈K | k ′ �A}. Note that {k ′ ∈K | k ′ �A} is upward closed and that on {k ′ ∈K | k ′ �A} the old and the new forcing coincide. Moreover, on this class, for any G, we havethat [A]G is equivalent to G.

We claim that, for all F; b�′F ⇒ b� [A]F . The proof of the claim is by a simpleinduction on F . The cases of atoms, disjunction and conjunction are trivial. SupposeF is an implication and b�′F . Then certainly, for all k ∈K; (k �A⇒ k �′F). Sinceon the k with k �A; � and �′ coincide, we .nd b� (A→F), i.e. b� [A]F .

We return to the main proof. Remember that b�H and b1 [A]C, for all C with(C→D)∈X and b1 [A]B. We show that b�′H and b�′A and b1′B.

It is immediate that b�′H , since K′ is a submodel of K and H is robust.


Remember that A is the conjunction of the (C→D) in X . So it is suVcient toshow that, for each (C→D) in X , b�′(C→D). Consider any (C→D)∈X and anyk ′¿′b with k ′ �′C. Since b1 [A]C, we have, by the claim, b1′C. So k ′ �= b. But thenk ′ �A, hence k ′ �′A and, thus, k ′ �′(C→D). We may conclude that k ′ �′D and, hence,b�′(C→D).

From b1 [A]B and the claim we have immediately that: b1′B.

All the proofs in this section also work when we replace ROB by f-ROB, the set offormulas preserved by full submodels. Note that ROB⊆ f-ROB. Our result in Section 7will imply ROB= f-ROB=NNIL (modulo IPC-provable equivalence).

7. The NNIL-algorithm

In this section we produce the algorithm that is the main tool of this paper. Theexistence of the algorithm establishes the following theorem.

Theorem 7.1. For all A there is an A∗ ∈NNIL(pv(A)) such that AI�A∗ and A∗ �A.

Before proceeding with the proof of Theorem 7.1 we interpolate a corollary.

Corollary 7.2. (1) Let A and A∗ be as promised by Theorem 7:1. Then we have

AI�B ⇔ A .IPC,ROB B ⇔ A∗ �IPC B:

(2) NNIL=ROB= f-ROB (mod IPC).

Proof. It is immediate from the fact that both .IPC;ROB and .IPC; f-ROB are �-relationsthat I�⊆ .IPC;ROB and I�⊆ .IPC; f-ROB. Moreover, we have range((·)∗)⊆NNIL ⊆ROB⊆ f-ROB. Combining this with the result of Theorem 7.1, we can apply Lemma 4.3to obtain the desired results.

Corollary 7.2(2) was proved by purely model theoretical means by Johan van Benthem.See his [38] or, alternatively, [49]. The advantage of van Benthem’s proof is its relativesimplicity and the fact that the method employed easily generalizes. The advantage ofthe present method is the extra information it produces, like the axiomatization of.IPC;ROB and its usefulness in the arithmetical case, see Sections 9–12. It is an openquestion whether van Benthem’s proof can be adapted to the arithmetical case.

Remark 7.3. Combining Corollary 7.2 with the results of [49], we obtain the followingsemantical characterization of I�.

AI�B ⇔ ∀M(∀K⊆M K � A⇒M � B):


We proceed with the proof of Theorem 7.1. For the rest of this section we write. :=I�. For convenience we reproduce the axioms and rules of . here:A1. A�B⇒A . B,A2. A . B and B . C⇒A . C,A3. C . A and C . B⇒C . (A ∧B),B1. A . C and B . C⇒ (A ∨B) . C,B2. Let X be a .nite set of implications and let

Y := {C | (C → D) ∈ X } ∪ {B}:Take A :=

∧X . Then, (A→B) . [A]Y .

B3. A . B⇒ (p→A) . (p→B).

Proof of Theorem 7.1. We introduce an ordinal measure o of complexity on formulasas follows:• I(D) := {E ∈ sub(D) |E is an implication}.• iD := max{| I(E) | |E ∈ I(D)}.

(Here |Z | is the cardinality of Z).• cD := the number of occurrences of logical connectives in D.• oD :=! · iD + cD.Note that we count occurrences of connectives for c and types of implications, notoccurrences, for i. We say that an occurence of E in D is an outer occurrence if thisoccurrence is not in the scope of an implication.

The proof of Theorem 7.1 proceeds as follows. At every stage we have a formulaand associated with it certain designated subformulas, that still have to be simpli.ed.A step operates on one of these subformulas, say A. Either A will lose its designationand a number of its subformulas will get designated or A will be modi.ed, say to A′,and a number of subformulas of A′ will replace A among the designated formulas. Thesubformulas B replacing A among the designated formulas will satisfy oB¡oA. Thus,to every stage we can associate a multiset of ordinals below !2, viz., the multiset ofo-complexities of the designated formulas. A step will reduce one of the ordinals to anumber of strictly smaller ones. This means that we are doing a recursion over !!

2.

We will exhibit the properties of . that are used in each step between square brackets.�. Atoms: [A1] Suppose A is an atom. Take A∗ :=A./. Conjunction: [A1; A2; A3] Suppose A= (B ∧C). Clearly oB¡oA and oC¡oA.

Take A∗ :=B∗ ∧C∗. Clearly A∗ ∈NNIL(pv(A)). The veri.cation of the desired proper-ties is trivial.0. Disjunction: [A1; A2; B1] Suppose A= (B ∨C). Clearly oB¡oA and oC¡oA. Take

A∗ :=B∗ ∨C∗. Clearly A∗ ∈NNIL. The veri.cation of the desired properties is trivial.1. Implication: [A1; A2; A3; B2; B3] Suppose A= (B→C). We split the step into

several cases.11. Outer conjunction in the consequent: [A1; A2; A3] Suppose C has an outer

occurrence of a formula D ∧E. Pick any J (q) such that• q is a fresh variable,


• q occurs precisely once in J ,• q is not in the scope of an implication in J ,• C = J [q := (D ∧E)].

Let C1 := J [q :=D]; C2 := J [q :=E]. As is easily seen C is IPC-provably equivalent toC1 ∧C2. Let A1 := (B→C1) and A2 := (B→C2). Clearly A is IPC-provably equivalentto A1 ∧A2. We prove that oAi¡oA for i= 1; 2. Since it is clear that cAi¡cA, it issuVcient to show that iAi6iA. Since A and the Ai are implications we have to show that| I(Ai) |6| I(A) |. We treat the case that i= 1. It is suVcient to construct an injectivemapping from I(A1) to I(A). Consider any implication F in I(A1). If F =A1, we mapF to A. Otherwise F ∈ I(B) or F ∈ I(J ) or F ∈ I(D) (since q does not occur in thescope of an implication). In all three cases we can map F to itself. Since A1 cannotbe in I(B) or I(J ) or I(D), our mapping is injective. The case that i= 2 is similar.Set A∗ := (A∗1 ∧A∗2).

12. Outer disjunction in the antecedent: [A1;A2;A3] This case is completely anal-ogous to the previous one.

If A has no outer disjunction in the antecedent and no outer conjunction in theconsequent, then B is a conjunction of atoms and implications and C is a disjunctionof atoms and implications. It is easy to see, that applications of associativity, com-mutativity and idempotency to the conjunction in the antecedent or to the disjunctionin the consequent do not raise o. So we can safely write B=

∧X and C =

∨Y ,

where X and Y are .nite sets of atoms or implications. This leads us to the followingcase.

13. B is a conjunction of atoms and implications, C is a disjunction of atoms andimplications [A1;A2;A3;B2;B3].13:1. X contains an atom: [A1;A2;B3].13:1:1. X contains a propositional variable; say p: [A1;A2;B3] Consider D :=

∧(X \

{p}) and E := (D→C). Clearly �A↔ (p→E) and oE¡oA. Put A∗ := (p→E∗). Ev-idently (p→E∗) is in NNIL(pv(A)). We have E∗ �E by the Induction Hypothesis and,hence, (p→E∗)� (p→E)�A. We have E .E∗ by the Induction Hypothesis. It fol-lows by B3 that (p→E) . (p→E∗). From A� (p→E), we have, by A1, A . (p→E).Hence, by A2, A . (p→E∗).13:1:2. X contains � : [A1;A2] Left to the reader.13:1:3. X contains ⊥: [A1] Left to the reader.13:2. X contains no atoms: [A1;A2;A3;B2] This case—the last one—is the truly

diVcult one. To motivate its treatment, let us solve the diVculties one by one. We.rst look at an example.

Example 7.4. Consider (p→ q)→ r. Condition B2 gives us ((p→ q)→ r) . (p ∨ r).However, we do not have (p ∨ r)� ((p→ q)→ r). We can repair this by noting that((p→ q)→ r)� (q→ r) and (q→ r) ∧ (p ∨ r)� ((p→ q)→ r). So the full solution ofour example is as follows.


We have ((p→ q)→ r) . ((q→ r) ∧ (p ∨ r)), by(a) ((p→ q)→ r) . (p ∨ r) B2(b) ((p→ q)→ r)� (q→ r) IPC(c) ((p→ q)→ r) . (q→ r) A1(d) ((p→ q)→ r) . ((q→ r) ∧ (p ∨ r)) (a); (c); A3

Moreover, ((q→ r) ∧ (p ∨ r))� ((p→ q)→ r) and

((q→ r) ∧ (p ∨ r)) ∈ NNIL(pv((p→ q) → r)):

We implement this idea for the general case. There will be a problem with o, butwe will postpone its discussion until we run into it. A is of the form B→C, where Bis a conjunction of a .nite set of implications X and C is a disjunction of a .nite setof atoms or implications Y . For any D := (E→F)∈X , let

B ↓ D :=∧

((X \{D}) ∪ {F}):

Clearly o((B ↓D)→C)¡oA.Let Z := {E | (E→F)∈X }∪{C}. Put A0 :=

∨[B]Z . We will show that our problem

reduces to the question whether A∗0 exists. So for the moment we pretend it does. Wehave:(1) A .A0 B2(2) A0 . A∗0 assumption(3) A .A∗0 (1), (2), A2(4) ∀D∈X A� ((B ↓D)→C) IPC(5) ∀D∈X A . ((B ↓D)→C) (4);A1(6) ∀D∈X ((B ↓D)→C) . ((B ↓D)→C)∗ IH(7) ∀D∈X A . ((B ↓D)→C)∗ (5); (6);A2(8) A . (

∧{((B ↓D)→C)∗ | D∈X } ∧A∗0) (3); (7);A3

It is clear that (∧{((B ↓D)→C)∗ | D∈X } ∧A∗0)∈NNIL(pv(A)). We show that

∧{((B ↓ D) → C)∗ |D ∈ X } ∧ A∗0 � A:

It is suVcient to show∧{((B ↓D)→C) | D∈X } ∧ [B]E �A, for each E ∈Z . In case

E=C, we are immediately done by [B]C �B→C. Suppose (E→F)∈X for some F .Reason in IPC.

Suppose∧{((B ↓D)→C) | D∈X }, [B]E and B. We want to derive C. We have

((∧

(X \{E→F}) ∧F)→C), [B]E and B. From B we .nd∧

(X \{E→F}) and (E→F). From B and [B]E, we derive E. From E and (E→F) we get F . Finally, we inferfrom ((

∧(X \{E→F}) ∧F)→C),

∧(X \{E→F}) and F the desired conclusion C.

Here ends our discussion inside IPC.So the only thing left to do is to show that A∗0 exists. If iA0 = 0, we are easily done.

Suppose iA0¿0. If for each E in Z with iE¿0, we would have i([B]E)¡iA, it wouldfollow that iA0¡iA. Hence we would be done by the Induction Hypothesis.

We study some examples. These examples show that we cannot generally hope toget i([B]E)¡iA. The examples will, however, suggest a way around the problem: we


produce a logical equivalent, say Q, of [B]E, such that iQ¡iA. So we replace A0

by the disjunction R of the Q’s, which is logically equivalent and has iR¡iA0. PutA∗0 :=R∗.

Example 7.5. Consider (p→ q)→ (r→ s). Take E :=C = (r→ s). We have [B]E=[p→ q](r→ s) = (p→ q)→ (r→ s). So no simpli.cation is obtained. However, [B]Eis IPC-provably equivalent to ((r ∧ (p→ q))→ s), which has lower i. By A1;A2 wecan put ([B]E)∗ := ((r ∧ (p→ q))→ s)∗. We could have applied the reduction beforegoing to [B]E. We did not choose to do so for reasons of uniformity of treatment.

Example 7.6. Consider (((p→ q)→ r)→ s). Take E := (p→ q). We .nd [B]E=[(p→ q)→ r](p→ q) = (((p→ q)→ r)→ (p→ q)). [B]E is IPC-provably equivalent to((p ∧ (q→ r))→ q), which has lower i.

Consider [B]E for E ∈Z = {E | (E→F)∈X } ∪ {C}. [B]E is the result of replacingouter implications (G→H) of E by (B→ (G→H)). If there is no outer implicationin E, we .nd that [B]E=E and i([B]E) = 0. In this case [B]E is implication free andhence a fortiori in NNIL. We can put ([B]E)∗ :=E. Suppose E has an outer implication.

Consider any outer implication (G→H) of E. We .rst show i(B→ (G→H)) 6 i(B→C). We de.ne an injection from I(B→ (G→H)) to I(B→C). Any implicationoccurring in I(B) or I(G→H) can be mapped to itself in I(B→C). None of theseimplications has as image (B→C), since (G→H)∈ I(B) ∪ I(C). So we can send(B→ (G→H)) to (B→C).

The next step is to replace (B→ (G→H)) by an IPC-provably equivalent formula Kwith lower value of i. Let B′ be the result of replacing all occurrences of (G→H) in Bby H and let K := ((G ∧B′)→H). Clearly, K is provably equivalent to (B→ (G→H)).We show that i(K)¡i(B→ (G→H)).

We de.ne a non-surjective injection from I(K) to I(B→ (G→H)). Consider animplication M in I(K). If M =K we send it to (B→ (G→H)). Suppose M �=K , i.e.M ∈ I(B′) ∪ I(G) ∪ I(H). A subformula N of I(B) ∪ I(G) ∪ I(H) is a predecessor ofM if M is the result of replacing all occurrences of (G→H) in N by H . Clearly,M has at least one predecessor and any predecessor of M must be an implication.Send M to one of its shortest predecessors. Clearly our function is injective, sincetwo diFerent implications cannot share a predecessor. Finally (G→H) cannot be inthe range of our injection. If it were, we would have M =H , but then H would be ashorter predecessor. A contradiction.

Replace every outer implication in [B]E by an equivalent with lower value of i. Theresult is the desired Q.

Remark 7.7. The algorithm given with our proof is non-deterministically speci.ed.However by Corollary 7.2 the result is unique modulo provable equivalence. I did nottry to make the algorithm optimally eVcient.


In [43] a simple adaptation of the NNIL algorithm is given for .IPC;�. Here thealgorithm computes not a value in NNIL, but a value in {�;⊥}. Thus we obtainan algorithm that checks for provability. It is easy to use the present algorithm forthis purpose too, since there is a p-time algorithm to decide IPC-provability of NNIL0

formulas, i.e. formulas with only propositional variables in front of arrows. The outputsof our algorithm are in NNIL0. It has been shown by Richard Statman (see [34]) thatchecking whether a formula is IPC-provable is p-space complete. This puts an absolutebound on what an algorithm like ours can do.

Example 7.8 (Sample computations). We use some obvious short-cuts.

(¬¬p→ p) → (p ∨ ¬p) ≡(p→ (p ∨ ¬p)) ∧ ( [¬¬p→ p]¬¬p ∨ [¬¬p→ p]p ∨ [¬¬p→ p]¬p ) ≡¬¬p ∨ p ∨ ¬p ≡( (⊥ → p) ∧ ( [¬p]p ∨ [¬p]⊥ ) ) ∨ p ∨ ¬p ≡p ∨ ¬p

((p→ q) → r) → s ≡(r → s) ∧ ( [(p→ q) → r](p→ q) ∨ [(p→ q) → r]s ) ≡(r → s) ∧ ( ((p ∧ (q→ r)) → q) ∨ s ) ≡(r → s) ∧ ( (p→ ((q→ r) → q)) ∨ s ) ≡(r → s) ∧ ( (p→ ((r → q) ∧ ( [q→ r]q ∨ [q→ r]q))) ∨ s ) ≡(r → s) ∧ ((p→ q) ∨ s):

8. Basic facts and notations in arithmetic

8.1. Arithmetical theories

The arithmetical theories T considered in this paper are RE theories in A, the lan-guage of HA. These theories all extend i-S1

2, the intuitionistic version of Buss’ theoryS1

2. Another salient theory is intuitionistic elementary arithmetic, i-EA := i-I�0 + Exp;the constructive theory of �0-induction with the axiom expressing that the exponen-tiation function is total. i-EA is .nitely axiomatizable; we stipulate that a .xed .niteaxiomatization is employed. We will use T for the formalization of provability in T .In the present paper, we are mostly interested in extensions of HA, so we will notworry too much about details concerning weak theories.

Suppose A is a formula. TA means ProvT (t(̃x)), where ProvT is the arithmetizationof the provability predicate of T and where x̃ is the sequence of variables occurring inA and t(̃x) is the term ‘the GWodelnumber of the result of substituting the GWodelnumbersof the numerals of the x̃’s for the variables in A’. Suppose e.g. that 11, 12 and 15 code,respectively, ‘(’ , ‘)’ and ‘=’, that ∗ is the arithmetization of the syntactical operation


of concatenation and that num is the arithmetization of the numeral function. Underthese stipulations e.g. T (x= x) means: ProvT (11 ∗ num(x) ∗ 15 ∗ num(x) ∗ 12). 6

Suppose T extends i-EA. We write Tn for the theory axiomatized by the .nitelymany axioms of i-EA, plus the axioms of T which are smaller than n in the standardGWodelnumbering. We write ProvT; n for the formalization of provability in Tn. We con-sider ProvT; n as a form of restricted provability in T . The following well known factis quite important:

Theorem 8.1. Suppose T is an RE extension of HA in the language of HA. Then T isessentially re?exive (veri5ably in HA). I.e. for all n and; for all A-formulas A withfree variables x̃; T � ∀̃x ( T; nA→A). Using UC for ‘the universal closure of’; evenHA�∀x ∀A∈ FORA T UC ( T; xA→A).

Proof (Sketch). The proof is roughly as follows. Ordinary cut-elimination for con-structive predicate logic (or normalization in case we have a natural deduction system)can be formalized in HA. Reason in HA. Let a number x and a formula A be given.Introduce a measure of complexity on arithmetical formulas that counts both the depthof quanti.ers and of implications. Find y such that both the axioms of Tx and A havecomplexity ¡y. We can construct a truth predicate Truey for formulas of complexity¡y. We have T UC (Truey A→A). Reason inside T . Suppose we have T; xA. By cutelimination we can .nd a Tx-proof p of A in which only formulas of complexity ¡yoccur. We now prove by induction on the subproofs of p, that all subconclusions ofp are Truey. So A is Truey. Hence we obtain A.

8.2. A brief introduction to HA∗

In this subsection we describe the theory HA∗. This theory will be employed in theproof of Theorem 10.2. In Section 12 we will study the closed fragment of HA∗.

The theory HA∗ was introduced in [42]. HA∗ is to Beeson’s fp-realizability, intro-duced in [1], as Troelstra’s HA+ ECT0 is to Kleene’s r-realizability. This means thatfor a suitable variant of fp-realizability HA∗ is the set of sentences such that their fp-translations are provable in HA. The theory HA∗ is HA plus the Completeness Principlefor HA∗. The Completeness Principle can be viewed as an arithmetically interpretedmodal principle. The Completeness Principle viewed modally is

C � A→ A:

The Completeness Principle for a speci.c theory T is

C[T ] � A→ TA:

6 Our notational convention introduces a scope ambiguity. Fortunately by standard metamathematical results,we know that as long as the terms we employ stand for T -provably total recursive functions the diFerentreadings are provably equivalent. In this paper we will only employ terms for p-time computable functions,so the ambiguity is harmless.


We have HA∗ =HA+ C[HA∗]. 7 We brieLy review some of the results of [42, 8].• Let A be the smallest class closed under atoms and all connectives except im-

plication, satisfying A∈�1 and B∈A⇒ (A→B)∈A. Note that modulo provableequivalence in HA all prenex formulas (of the classical arithmetical hierarchy) arein A. HA∗ is conservative w.r.t. A over HA. Note that NNIL0(�)⊆A.

• There are in.nitely many incomparable T with T =HA+ C[T ]. However if T =HA+ C[T ] veri5ably in HA, then T =HA∗.

• Let KLS := Kreisel–Lacombe–Shoen.eld’s Theorem on the continuity of the eFectiveoperations. We have HA∗ �KLS→ HA∗⊥. This immediately gives Beeson’s resultthat HA 0 KLS [1].

• Every prime RE Heyting algebra H can be embedded into the Heyting algebra ofHA∗. This mapping is primitive recursive in the enumeration of the generators ofH w.r.t which H is RE. The mapping sends the generators to �-sentences [8].

We insert some basic facts on the provability logic of HA∗. These materials will beonly needed in Section 12. Clearly, HA∗ satis.es the LWob conditions.L1. �A⇒ � A,L2. � (A→B)→ ( A→ B),L3. � A→ A,L4. � ( A→A)→ A.i-K is given by IPC+ L1;L2. i-L is i-K+ L3;L4. We write i-K{P} for the extension ofi-K with some principle P. Note that i-L{C} is valid for provability interpretations inHA∗.

A principle closely connected to C is the Strong LWob Principle:

SL � ( A→ A) → A:

As a special case of SL we have �¬¬ ⊥.

Theorem 8.2. i-L{C} is interderivable with i-K{SL}.

Proof. Condition L4 is immediate from SL. We show “i-K{SL} �C”. Reason in i-K{SL}. Suppose A. It easily follows that (A ∧ A)→ (A ∧ A), and, hence, by SL, thatA ∧ A. We may conclude that A.

We show “i-L{C} �SL”. Reason in i-L{C}. Suppose ( A→A). It follows, by C, that( A→A) ∧ ( A→A). Hence, by LWob’s Principle, ( A→A) ∧ A and thus A.

9. Closure properties of �-preservativity

In this section we verify two closure properties of �-preservativity over HA in HA.

7 The natural way to de.ne HA∗ is by a .xed point construction as: HA plus the Completeness Principlefor HA∗. Here it is essential that the construction is veri.able in HA.


9.1. Closure under B1

We will show that HA veri.es that �-preservativity is closed under B1 for any sub-stitutions. We produce both proofs known to us. The .rst one employs q-realizability.This form of realizability is a translation from A to A, due to Kleene. It is de.nedas follows:• x qP :=P, for P atomic,• x q (A ∧B) := ((x)0 qA ∧ (x)1 qB);• x q (A ∨B) := (((x)0 = 0→ (x)1 qA) ∧ ((x)0 �= 0→ (x)1 qB));• x q (A→B) :=∀y (y qA→∃z ({x}y" z ∧ z qB)) ∧ (A→B);• x q∃y A(y) := (x)0 qA((x)1),• x q∀y A(y) :=∀y∃z ({x}y" z ∧ z qA(y)).The following facts can be veri.ed in HA:(1) HA� x qA→A;(2) for any A∈� and any set {y1; : : : ; yn} with FV(A)⊆y1; : : : ; yn, we can .nd an

index e such that HA�A→∃z({e}(y1; : : : ; yn)" z ∧ z qA);(3) Suppose B1; : : : ; Bn; C have free variables among {y1; : : : ; ym} and that {x1; : : : ; xn}

is disjoint from {y1; : : : ; ym}. Suppose B1; : : : ; Bn �HA C. Then, for some e, we have:

x1qB1; : : : ; xnqBn �HA ∃ z({e}(x1; : : : ; xn; y1; : : : ; ym) " z ∧ z q C):

The proofs are all simple inductions. For details the reader is referred to [35, pp. 188–202]. We will now show that �-preservativity satis.es B1. It is easily seen that ourproof is veri.able in HA. We reason it as follows:(�) A .HA; � C assumption(/) B .HA; � C assumption(0) S ∈� and HA� S→ (A ∨B) assumption(1) x q S �HA ∃z ({e}(x)" z ∧ z q (A ∨B)) 0; e provided by (3)(>) S �HA ∃z ({e}(f)" z ∧ z q (A ∨B)) 1; f provided by (2)(?) S ∧∃z ({e}(f)" z ∧ (z)0 = 0)�HA A >(@) S ∧∃z ({e}(f)" z ∧ (z)0 �= 0)�HA B >(A) S ∧∃z ({e}(f)" z ∧ (z)0 = 0)�HA C ?; �(B) S ∧∃z ({e}(f)" z ∧ (z)0 �= 0)�HA C @; /(C) S �HA ∃z ({e}(f)" z ∧ (z)0 = 0) ∨

∃z ({e}(f)" z ∧ (z)0 �= 0) >(D) S �HA C A; B; CThe second proof employs a translation proposed by Dick de Jongh. For later use ourde.nition is slightly more general than is really needed for the problem at hand. LetC be an A-sentence and let n be a natural number. We write, locally, n for HA; n.De.ne a translation [C]n(·) as follows:• [C]nP :=P for P atomic,• [C]n(·) commutes with ∧; ∨;∃,• [C]n(A→B) := ([C]nA→ [C]nB) ∧ n(C→ (A→B));• [C]n∀y A(y) :=∀y[C]nA(y) ∧ n(C→∀y A(y)):


Let us .rst make a few quick observations, that make life easier. We write [C]n' :={[C]nD |D∈'}. We have

(‡i) HA� [C]nA→ n(C→A):(‡ii) HA� [C]n((A→B) ∧ (A′ →B′)) ↔

([C]nA→ [C]nB) ∧ ([C]nA′ → [C]nB′)∧n(C → ((A→ B) ∧ (A′ → B′))).

Similarly for conjunctions of more than two implications.(‡iii) HA� [C]n∀y∀z A(y; z) ↔ (∀y∀z [C]nA(y; z) ∧ n(C→∀y∀z A(y; z))). Similarly

for larger blocks of universal quanti.ers.(‡iv) HA� [C]n∀y (A(y)→B(y)) ↔

∀y ([C]nA(y) → [C]nB(y)) ∧ n(C → ∀y (A(y) → B(y))):(‡v) HA� [C]n∀y¡z A(y) ↔ ∀y¡z [C]nA(y).(‡vi) For S ∈�, HA� S↔ [C]nS.

(‡vii) ' �HA; n A⇒ [C]n' �HA [C]nA (veri.ably in HA).(‡v) is immediate from the well-known fact that

HA � ∀y ¡ z nA(y) → n∀y ¡ z A(y):

(‡vi) is immediate from (‡v).

Proof. (‡vii) The proof is by induction on the proof witnessing ' �HA; n A. We treattwo cases.Case 1: '= ∅ and A is an induction axiom, say for B(x), of HAn. Clearly [C]nA is

HA-provably equivalent to:

( ( [C]nB0 ∧∀x ([C]nBx → [C]nB(x + 1)) ∧n(C → ∀x (Bx → B(x + 1))) ) →

(∀x [C]nBx ∧ n(C → ∀x Bx)) ) ∧n(C → A) :

We have HA� nA, and, hence, HA� n(C→A). So it follows that

HA � n(C → ∀x (Bx → B(x + 1))) → n(C → ∀x Bx):Moreover (as an instance of induction for [C]nBx),

HA � ([C]nB0 ∧ ∀x ([C]nBx → [C]nB(x + 1))) → ∀x [C]nBx:

Combining these we .nd the promised: HA� [C]nA.Case 2: Suppose that A= (D→E) and that the last step in the proof was

';D �HA;n E ⇒ ' �HA;n D→ E:

From ';D�HA; n E, we have, by the Induction Hypothesis, [C]n', [C]nD�HA; n [C]nEand, hence, [C]n' �HA; n [C]nD→ [C]nE. Moreover, for some .nite '0 ⊆', we have'0; D�HA; n E. Let B be the conjunction of the elements of '0. We .nd [C]n' �HA; n


n(C→B) and �HA; n n(B→ (D→E)). Hence, [C]n' �HA; n n(C→ (D→E)). We mayconclude

[C]n' �HA;n ([C]nD→ [C]nE) ∧ n(C → (D→ E)):

Here ends our proof of (‡vii).

We now prove our principle. As is easily seen the argument can be veri.ed in HA.(�) A .HA; � C assumption(/) B .HA; � C assumption(0) S ∈� and HA� S→ (A ∨B) assumption(1) for some n S �HA; n (A ∨B) 0(>) [�]nS �HA ([�]nA ∨ [�]nB) 1; (‡vii)(?) S �HA ( nA ∨ nB) >; (‡vi); (‡i)(@) HA� nA→A Theorem 8.1(A) HA� nA→C �; @(B) HA� nB→B Theorem 8.1(C) HA� nB→C /; B(D) HA� (S→C) ?; A; C

9.2. A closure rule for implication

To formulate our next closure rule it is convenient to work in a conservativeextension of HA. Let A+ be A extended with new predicate symbols (including the0-ary case) for �-formulas. Let f be some assignment of �-formulas of A of theappropriate arities to the new predicate symbols. We extend f to A+ in the obviousway. De.ne [·]◦n(·) and <·=(·) :A+ →A as follows:• <A=P := [A]◦nP :=fP for P atomic,• <A=(·) and [A]◦n(·) commute with ∧, ∨ and ∃;• <A=(B→C) :=f(A→ (B→C)),

[A]◦n(B→C) := nf(A→ (B→C)):• <A=∀x B(x) :=f(A→∀x B(x)),

[A]◦n∀x B(x) := nf(A→∀x B(x)):We have, for B in A+, extending the numbering of ‡-principles of Section 9.1:(‡viii) The following implications are derivable in HA: [fA]n(fB)→ [A]◦nB, [A]◦nB→

<A=B and <A=B→(A→B).(‡ix) [A]◦nB is provably equivalent to a �-formula.

The proof of (‡viii) is an easy induction on B. Condition (‡ix) is trivial.

Theorem 9.1. Suppose X is a 5nite set of implications in A+ and let B be inA+. Say A :=

∧X . Let Y := {C | (C→D)∈X } ∪ {B}. We have; veri5ably in HA;

f(A→B) .HA; �∨<A=Y .

Before giving the proof of Theorem 9.1, we .rst draw a corollary.


Corollary 9.2. The following facts are veri5able in HA:1. .HA; �;� is closed under B2.2. .HA; �;A is closed under B2′.3. �8 of Appendix C is HA-valid for �-preservativity.

We leave the easy veri.cation of the corollary to the reader. We proceed with theproof of Theorem 9.1.

Proof. To avoid heavy notations we suppress ‘f’. In the context of HA we assumethat an A+-formula is automatically translated via f to the corresponding A-formula.Let S be a �-sentence (of A). Suppose S �HA (A→B). It follows that, for some n,S �HA; n (A→B) and, hence, by (‡vii) [A]nS �HA [A]n(A→B). By (‡ii) and (‡vi) we.nd

S �HA(∧

{[A]nC → [A]nD | (C → D) ∈ X } ∧ n(A→ A))→ [A]nB;

and so

S �HA∧

{[A]nC → [A]nD | (C → D) ∈ X } → [A]nB:

It follows by (‡viii) that

S �HA∧

{[A]◦nC → [A]nD | (C → D) ∈ X } → [A]◦nB:

Since HA is a subtheory of PA, we get

S �PA∧

{[A]◦nC → [A]nD | (C → D) ∈ X } → [A]◦nB:

By classical logic, we get S �PA∨

[A]◦nY . Remember that PA is, veri.able in HA,conservative over HA w.r.t. �2-sentences (see [10]). Since (S→ ∨

[A]◦nY ) is �2, weget S �HA

∨[A]◦nY . Ergo, by (‡viii), S �HA

∨<A=Y .

Before closing this section we insert some remarks on the proof.

Remark 9.3. (1) The above proof was obtained after analyzing an argument in [6].(2) The step involving conservativity of PA over HA uses the GWodel–Friedman trans-

lation. Closer inspection reveals that the argument at hand just requires the Friedmantranslation. We give a sketch of the proof below. It follows that our results general-ize to all essentially reLexive RE extensions T of HA that are closed both under theFriedman and the de Jongh translation.

(3) It seems to me that, from a suVciently abstract perspective, it shouldbecome clear that our present proof is just a variant of the proof of Theorem 6.2.Thus, {G | n(A→G)} in the present proof would correspond to the gray part of theKripke model in the picture below Theorem 6.2. [A]n(·) would correspond to the oper-ation of adding the bottom-node in the picture (via SmoryKnski’s operation) to the graypart. The detour via PA would correspond to the fact that our Kripke model argument


is essentially classical. The special behavior of �-sentences under translation wouldcorrespond with the fact that whether a �-sentence is forced or not (in a model ofHA) is dependent just on the node under consideration and not on other nodes. Let mepose it as a problem to further develop such a perspective.

We prove the claim of (2): we show that the double negation translation can beeliminated from the proof of Theorem 9.1. We pick up the proof from the point wherewe have proved:

(a) S �HA∧{[A]◦nC→ [A]nD | (C→D)∈X }→ [A]◦nB.

Let H :=∨

[A]◦nY . The Friedman translation (E)H of an arithmetical formula E is(modulo some details to avoid variable-clashes) the result of replacing all atomic for-mulas P in E by (P ∨H). One easily shows:

(b) HA�E⇒HA� (E)H ,(c) HA�H→ (E)H ,(d) for S ∈�, HA� (S)H ↔ (S ∨H).By (a), (b) and (d) we have(e) S ∨H �HA∧{(([A]◦nC) ∨H)→ ([A]nD)H | (C→D)∈X }→ (([A]◦nB) ∨H).By (e) and propositional logic we .nd:

(f) S �HA∧{H→ ([A]nD)H | (C→D)∈X }→H .

So by (f) and (c):(g) S �HAH .

Hence, by (g) and (‡viii), S �HA∨<A=Y .

10. On �-substitutions

In this section we prove our main results on �-substitutions. Given the results wealready have, this is rather easy.

Theorem 10.1. .HA; �;� is; veri5ably in HA; a �-relation.

Proof. Every preservativity relation satis.es A1–A3, and, hence, so does .HA; �;�.Moreover, .HA; �;� satis.es B1 in virtue of our result of Section 9.1. .HA; �;� satis-.es B2 by Corollary 9.2. .HA; �;� satis.es B3 by Lemma 4.2(4). It is easy to see thatall the steps are veri.able in HA.

Let R be any arithmetically de.nable relation. We write RHA for the relation givenby: mRHAn :⇔ HA�mRn.

Theorem 10.2. The following relations are coextensive:(a) I�; (b) .HAHA; �;�; (c) .HA; �;�; (d) |∼HAHA; �; (e) |∼HA; �:

Proof. By Theorem 10.1, .HAHA; �;� is a �-relation. So, (a)⊆ (b).


Since HA is sound, we .nd (b)⊆ (c). By Lemma 4.2(6), (c)⊆ .HA;�; � = (e). Sincethe argument for Lemma 4.2(6) is just universal instantiation, it is veri.able in HA.Hence (b)⊆ (d).

By the soundness of HA, we have (d)⊆ (e).We show that (e)⊆ (a), i.e. |∼HA; �⊆I�. Suppose not AI�B. Then, by Corollary 7.2

(1), A∗ 0IPC B. A∗ is IPC-provably equivalent to a disjunction of prime NNIL-formulas.It follows that for some such disjunct, say C; C 0IPC B. The Heyting algebra H

axiomatized by C is prime and RE. By the embedding theorem proved in [8] (seeSection 8.2 of this paper) there is an f∈�P, such that HA∗ �fC and HA∗ 0fB. SincefC ∈NNIL(�), we have by the NNIL(�)-conservativity of HA∗ over HA (proved in[42]; see also Section 8.2 for the statement of the full result), that HA�fC. Since HAis a subtheory of HA∗, we have HA0fB. Since IPC�C→A, it follows that HA�fAand HA 0fB. We may conclude: not A |∼HA; � B.

Remark 10.3. (1) A propositional formula A is HA; �-exact iF there is a substitutionf∈�P; such that HA�fB⇔A�IPC B; i.o.w. if A axiomatizes the ‘theory of f’, i.e. HA;{f}. See also [8]. The proof of Theorem 10.2 establishes also that the primeNNIL-formulas are precisely the HA; �-exact formulas.

(2) By Theorem 10.2, |∼HAHA; � is a �-relation. Note, however, that it is not HA-provably a �-relation. The reason is that closure under B1 is not veri.able. To provethis, suppose HA veri.es B1. Let R be the standard Rosser sentence. So R satis.esHA�R↔ HA¬R6 HAR. Let R⊥ := HAR¡ HA¬R. We have (a) HA� HAR→ HA⊥and (b) HA� HAR⊥ → HA⊥. This is simply the Formalized Rosser Property. More-over, we have (c) HA� HA⊥→ (R ∨R⊥). This is the formalization in HA of theinsight that if ⊥ is witnessed then both R and ¬R will be witnessed; one mustbe witnessed .rst, which means that either R or R⊥. From (c), it follows that: (d)HA� HA HA⊥→ HA(R ∨R⊥). On the other hand, from (a), (b) and the assumed ver-i.able closure under |∼HAHA; �, we .nd that (e) HA� HA(R ∨R⊥)→ HA⊥. Combining(d) and (e), we get HA� HA HA⊥→ HA⊥. Quod non, by LWob’s Theorem.

11. The admissible rules of HA

The results of Rosalie IemhoF [18] imply the following theorem.

Theorem 11.1 (IemhoF). |∼IPC =I�. In other words; the admissible rules of IPC areaxiomatized by A1–A3; B1; B2′.

From IemhoF’s result, we derive the following theorem.

Theorem 11.2. The following relations are coextensive:

(a) I�; (b) .HAHA;�;A; (c) .HA;�;A; (d) |∼HAHA; (e) |∼HA; (f ) |∼IPC:


Proof. We .rst show that (a)⊆ (b). It is clearly suVcient to show that .HA; �;A is HA-veri.able an �-relation. Closure under A1–A3 is trivial. Moreover, .HA; �;A satis.es B1in virtue of our result of Section 9.1. .HA; �;A satis.es B2 by Corollary 9.2.

By the soundness of HA, we have (b)⊆ (c) and (d)⊆ (e).By Lemma 4.2(6), (c)⊆ .HA;�;A = (e). Since Lemma 4.2(6) is veri.able in HA, we

also have (b)⊆ (d).Since, by de Jongh’s theorem, HA = IPC, we .nd, by Theorem 4.4, that (e)⊆ (f ).Finally, by Theorem 11.1, (f )⊆ (a).

12. Closed fragments

In this section, we study the closed fragments of the provability logics of threetheories. The .rst is the theory HA∗. The second is Peano Arithmetic, PA. The thirdis HA. We will .rst introduce some de.nitions and prove some basic facts.

12.1. Preliminaries

Consider the language of modal propositional logic without propositional variables.Let us call this language L ;0. So L ;0 is given as follows:

A ::= ⊥|�|(A ∧ A) | (A ∨ A) | (A→ A)| A:Let T be any theory extending, say, i-S1

2, the intuitionistic version of Buss’ S12. What

is relevant here is just that arithmetization of metamathematics should be possible inT and that T veri.es the LWob conditions. We can map the formulas of L ;0,say viaeT , to the language of T , by stipulating that eT commutes with all the propositionalconnectives including � and ⊥ and that

eT A := TeTA := ProvT (gn(eTA)):

The closed fragment CT of the provability logic of T is the set of A in L ;0 such thatT � eTA.

An important class, DF, of closed formulas is formed by the degrees of falsity �⊥,where � ranges over !+ :=! ∪ {∞}. Here:• 0⊥ :=⊥,• n+1⊥ := n⊥,• ∞⊥ :=�,

We will often use the evident mappings df : � �→ �⊥ and $ : �⊥ �→ �.To link the closed fragments to the framework of this paper, we will study purely

propositional theories associated with these fragments. The language of these proposi-tional theories is the usual language of propositional logic with as ‘propositional atoms’the degrees of falsity. These atoms function more as constants than as propositionalvariables. We call this language BDF. So BDF is given as follows:

A ::= �⊥ | (A ∧ B) | (A ∨ B) | (A→ B):


We will consider theories F in BDF, which satisfy (at least) intuitionistic propositionallogic and the up axiom:

up � n⊥ → n+1⊥:

The minimal such theory will be called UP. So UP is axiomatized by IPC and up. Wewill say that F is DF-irre?exive if �⊥�F /⊥ ⇒ �6 /. It is easy to show that UPis DF-irreLexive. Note that subtheories of DF-irreLexive theories are DF-irreLexive.

An arithmetical theory T would have associated it to the propositional theory T;eT :={A∈BDF |T � eTA}. (Strictly speaking eT here is the restriction of eT as de.ned aboveto BDF.)

We will write �T⊥ for eT ( �⊥). We will say that T is DF-sound if, for no n∈!,

T � nT⊥.

Lemma 12.1. Suppose T is DF-sound. Then; T;eT is DF-irre?exive.

Proof. Suppose we have �⊥� T;eT/⊥. We assume, to obtain a contradiction, that

/¡�. Then, by de.nition, �T⊥�T /

T⊥. It follows that /+1T ⊥�T /

T⊥ and /∈!. ByLWob’s Rule, we obtain T � /

T⊥. But this is impossible, by DF-soundness. Ergo, �6 /.

The fundamental structure in the study of closed fragments is !+ :=! ∪ {∞}={0; 1; 2; : : : ;∞} equipped with the obvious partial order. !+ with partial order can beextended to a Heyting algebra. The ordering .xes the operations of the Heyting algebrauniquely. We have:

• � :=∞; ⊥ := 0,• (� ∧ /) := min(�; /).• (� ∨ /) := max(�; /),

• (�→ /) :={∞ if �6 /;/ otherwise:

Abuse !+ for, partial order and Heyting algebra. We will put �+∞=∞+ �=∞.We will repeatedly use the following convenient property: �+ n6 / + n⇒ �6 /.

The map df : � �→ �⊥ can be considered as a functor from !+, considered as acategory, to a theory F considered as the category of the preorder �F.

We will be interested in theories where df has a right adjoint @F, i.e. where we have

df(�) �F A ⇔ �6 @FA:

Lemma 12.2. Suppose F is DF-irre?exive and @F exists. Then; @F �⊥= �.

Proof. We have, for DF-irreLexive F,

06 @F �⊥ ⇔ 0⊥ �F �⊥ ⇔ 06 �:


We will write @T for @ T;eT . We can ‘lift’ @F from BDF to L ;0 using the auxiliaryfunction (·)aux : L ;0 →BDF as follows:

• (·)aux commutes with the propositional connectives including � and ⊥,• ( A)aux := df@FAaux:

We can now de.ne �F :L ;0 →!+, by �FA := @FAaux. We show that, for DF-irreLexive F; @F and �F coincide on BDF-formulas.

Lemma 12.3. Suppose F is DF-irre?exive. Let A∈BDF. Then �FA= @FA.

Proof. Suppose F is DF-irreLexive. First we show, by induction on n, that �F n⊥= n.For n= 0, this is immediate. We have

�F k+1⊥ = @F( k+1⊥)aux

= @F df@F( k⊥)aux

= @F df�F k⊥= @F df(k)

= @F k+1⊥= k + 1:

It now follows that, for A∈BDF, we have Aaux =A. The theorem follows directlyfrom this.

Finally, de.ne, for A; B in BDF:

A|∼provT B :⇔ T � T eTA→ T eTB:

So |∼prov is a form of provable deductive consequence. Note that if T is �-sound,then T;eT = {A∈BDF | �|∼provT A}. We will write the induced equivalence relation of|∼prov as ≈prov.

The following theorem is the main lemma for the subsequent development.

Theorem 12.4. Let H be a function from BDF to !+. We put IA := HA⊥. Note thatH and I are interde5nable. Let T be a theory containing i-S1

2. Let F be a BDF-theorycontaining IPC and up. We assume the following:Z1. T is DF-sound.Z2. F is a T -sound; i.e. F�A⇒ T � eTA; for A∈BDF.Z3. IA�F A.Z4. A |∼provT IA.

Then we have

(1) F= T;eT .(2) I is the left adjoint of the embedding of �F into |∼provT . In other words, we have:

IA�F B⇔ A |∼provT B.


(3) The full subcategory of �F obtained by restriction to DF is equivalent to |∼provT .This subcategory is isomorphic to !+.

(4) The full subcategory of |∼provT obtained by restriction to DF is a skeleton of |∼provT .This subcategory is isomorphic to !+.

(5) H( �⊥) = � and H= 1F.(6) We have, for any A∈L ;0 :(a) CT �A↔ Aaux;(b) CT �A⇔ �FA=∞.

Proof (Under the assumptions of the theorem). We prove (1). By Z2, F isT -sound. We show that it is also T -complete. Suppose T � eTA. Then T � TeTA.By Z4, T � TeTIA. By Z1, it follows that IA=�. By Z3, we may conclude thatF�A.

We prove (2). Note that from Z3,4 it follows that IC ≈provT C. We have

IA �F B⇒ eTIA �T eTB⇒ TeTIA �T T sfa eTB

⇒ TeTA �T TeTB

⇒ TeTIA �T TeTIB

⇒HA+ 1 6 HB+ 1

⇒HA6 HB

⇒IA �F IB⇒IA �F B

We prove (3). The equivalence of the restriction of �F to DF and |∼provT , is immediatefrom Lemma A.1(5). Since, by Z1, T is DF-sound, clearly F is DF-irreLexive. Hencethe restriction of �F to DF is isomorphic to !+.

We prove (4). The restriction of |∼provT to DF is, by Lemma A.1(4) and Z1, equivalentto |∼provT . Trivially, the restriction is isomorphic to !+.

We prove (5). The .rst part is immediate from the assumptions of the theorem. Thesecond part follows by

df� �F A⇔Idf� �F A⇔ df�| ∼provT A

⇔ df�| ∼provT IA

⇔ �6 HA

Since right adjoints are unique modulo natural isomorphism, it follows that H= 1F.We prove (6). The proof of (6)(a) is by induction on A, using that, for B∈BDF,

CT � B↔ IB. Condition (6)(b) is immediate from (6)(a).

We close this section with a lemma.


Lemma 12.5. Let B∈BDF. Then we haveHA � eHA B↔ eHA∗ B↔ ePA B:

Proof. It is well known that both HA∗ (see Section 8.2) and PA are HA-provably�0

2-conservative over HA. It follows that HA� HA⊥ ↔ HA∗⊥ ↔ PA⊥. The lemmafollows by induction on B.

12.2. The closed fragment for HA∗

In Section 8.2 we introduced HA∗ and its provability logic. We saw that in this logicat least the principles of the constructive version of LWob’s Logic plus the Strong LWobPrinciple SLP, � ( A→A)→A, hold.

The propositional theory UP∗ is the BDF-theory, axiomatized by the rules of IPCplus• up � n⊥→ n+1⊥,• slp � ( n+1⊥→ n⊥)→ n⊥.We de.ne a mapping HHA∗ := <·= from BDF to !∪{∞} as follows:• <⊥= := 0, <�= :=∞,• <A ∧B= := min(<A=; <B=),• <A ∨B= := max(<A=; <B=),• <A→B= := (<A=→ <B=).De.ne further IHA∗A :=A♦ := <A=⊥. We will verify Z1–Z4 of Theorem 12.4 for HA∗,UP∗ and <·=.

Clearly HA∗ is DF-sound. So we have Z1. Moreover, UP∗ is HA∗-sound. thus wehave Z2. It is easy to see, by an induction on the subformulas of A, that for A in BDF:

$ UP∗ � A↔ A♦:

The only non-trivial step is the case of implication which uses slp. It follows that <·=de.nes an equivalence between the category of UP∗ and !+.

It is now easily seen that $ immediately implies both Z3 and Z4. It follows thatUP∗ = HA∗ ;eHA∗ . Moreover, since, evidently (·)♦ is naturally isomorphic to the identityfunctor on �UP∗ , we .nd that provably deductive consequence and derivable conse-quence coincide for HA∗, i.e. |∼provHA∗ =�UP∗ .

12.3. The closed fragment of PA

Friedman’s 35th problem was to give a decision procedure for the closed fragmentof the provability logic of PA. See [9]. It was independently solved by van Benthem,Boolos and Magari. To present the solution in our style, we .rst introduce the theoryUPc. This is a classical propositional logic in the language BDF with the principle up.By a simple model theoretic argument one can show that UPc = PA;ePA . This fact willalso follow from the application of Theorem 12.4 below. Note that, trivially, UPc isPA-sound.


Consider any A in BDF. In UPc we can rewrite A to a normal form nf(A) as follows.First, we rewrite A to conjunctive normal form, obtaining a conjunction of disjunctionsof degrees of falsity and negations of degrees of falsity. The disjunctions of degreesof falsity and negations of degrees of falsity can be rewritten to implications fromconjunctions of degrees of falsity to disjunctions of degrees of falsity. Finally, usingup, we can contract conjunctions of degrees of falsity and disjunctions of degreesof falsity to single degrees of falsity. Thus, we obtain a formula nf(A) of the form∧i (

�i⊥→ /i⊥). We de.ne IPA := (·)♥ := (·)♦ ◦ nf. Here (·)♦ is the functor fromSection 12.2.

We verify Z1–Z4 of Theorem 12.4 for PA, UPc and (·)♦. Z1 and Z2 are trivial. Wehave• UPc �A↔ nf(A),• (nf(A))♦ �UPc nf(A)�UPc A.The second item is simply by inspecting the computation of (·)♦ on a formula of thespecial form nf(A). From the above, we have Z3. Moreover, we have

PA ePA A �PA PA ePA nf(A) (1)

�PA PA ePA (nf(A))♦: (2)

The step labeled (2) is by a simple direct computation: .rst bring the conjunctionsoutside the box and then apply LWob’s theorem to the conjuncts (if appropriate). Thuswe have proved Z4.

Remark 12.6. There is an alternative way to prove (2) of the above argument.

PA � PA ePA A→ PA ePA nf(A) (3)

→ HA eHA nf(A) (4)

→ HA∗ eHA∗ nf(A) (5)

→ HA∗ eHA∗ (nf(A))♦ (6)

→ HA eHA (nf(A))♦ (7)

→ PA ePA (nf(A))♦ (8)

Step (3) is shown simply by classical logic. Step (4) uses .rst the HA-veri.able �02-

conservativity of PA over HA and, then Lemma 12.5. Step (5) is by Lemma 12.5.Step (6) uses the results of Section 12.2. Step (7) employs the HA-veri.able �0

2-conservativity of HA∗ over HA. Step (8) is by Lemma 12.5.

Of course, this second argument is a silly way of proving (2). It is more diVcult, ituses more theory and it is less general (if we replace PA by other classical theories).However, it uncovers an analogy with the argument we are going to give for the caseof HA.


We have veri.ed assumptions Z1–Z4 of Theorem 12.4. The theorem gives us thedesired characterization of the closed fragment of PA.

Our argument, which is just a form of presenting the classical argument, for thevan Benthem–Magari–Boolos result works for all DF-sound RE extensions of Buss’sS1

2. 8 In contrast, the proof of Solovay’s Arithmetical Completeness Theorem onlyworks as far as we know if Exp is present.

12.4. The closed fragment of HA

De.ne IHA := (·)♣ := (·)♦ ◦ (·)∗. Here (·)∗ is the functor described in Section 7.We will verify Z1–Z4 of Theorem 12.4 with UP in the role of F. Z1 and Z2 areimmediate. We have

(A∗)♦ �UP A∗ (9)

�UP A (10)

Step (9) uses the fact that A∗ ∈NNIL and a simple induction of the complexity ofNNIL-formulas. Step (10) is immediate from the results of Section 7. Thus we haveproved Z3. Finally, we have

HA � HA eHA A→ HA eHA A∗ (11)

→ HA∗ eHA∗ A∗ (12)

→ HA∗ eHA∗ (A∗)♦ (13)

→ HA eHA (A∗)♦ (14)

Step (11) follows from Theorem 10.2 in combination with the results of Section 7.Step (12) uses Lemma 12.5. Step (13) employs the results of Section 12.2. Step (14)is by the HA-veri.able �0

1-conservativity of HA∗ over HA and by Lemma 12.5. Thuswe have veri.ed Z4.

Example 12.7 (Sample computations). Often the characterization of (·)♣ by A♣ =@UPA⊥, provides a quick way to compute A♣. First, one guesses the best approxi-

mation from below in DF of A, then one proves, e.g. using a Kripke model, that theconjectured approximation is indeed best. In the examples below we use our algorithm.We write ≈ for ≈provHA .1. Consider A :=¬ ⊥→ ⊥. We .nd that A∗ = ⊥ ∨ ⊥. Hence, A ≈ 2⊥.2. Consider A= (¬¬ ⊥→ ⊥)→ ( ⊥ ∨¬ ⊥). Let B :=¬¬ ⊥→ ⊥. We com-

pute A∗.

A≈ ( ⊥ → ( ⊥ ∨ ¬ ⊥)) ∧([B]¬¬ ⊥ ∨ [B] ⊥ ∨ [B]¬ ⊥)

8 However, as we pointed out, the argument of Remark 12.6 does not work in general.


≈ ( ⊥ → ( ⊥ ∨ ¬ ⊥)) ∧ (¬¬ ⊥ ∨ ⊥ ∨ ¬ ⊥)

≈ ( ⊥ → ( ⊥ ∨ ¬ ⊥)) ∧ ( ⊥ ∨ ⊥ ∨ ¬ ⊥)

Applying (·)♦, we .nd that A ≈ ⊥.3. Consider A= (¬¬ ⊥→ ⊥)→ ( ⊥ ∨¬ ⊥). Let B :=¬¬ ⊥→ ⊥. We com-

pute A∗, using some shortcuts involving UP-principles.

A≈ ( ⊥ → ( ⊥ ∨ ¬ ⊥)) ∧([B]¬¬ ⊥ ∨ [B] ⊥ ∨ [B]¬ ⊥)

≈¬¬ ⊥ ∨ ⊥ ∨ ¬ ⊥≈ ⊥ ∨ ⊥ ∨ ¬ ⊥

Applying (·)♦, we .nd A ≈ 2⊥.

12.5. Comparing three functors

In this last subsection of closed fragments, we compare the functors IHA, IHA∗ andIPA. By our previous results, it suVces to compare @HA, @HA∗ and @PA. We have

�6 @HAA⇔ df(�) �UP A⇒ df(�) �UPc A⇔ �6 @PAA

Ergo, @HAA 6 @PAA. By a similar argument, we .nd @HAA 6 @HA∗ A. Now considerthe following formulas:

E�;/;0 := (¬¬ �⊥ ∧ /⊥) ∨ ((¬¬ �⊥ → �⊥) ∧ 0⊥):

Then we have, for �6 / and �6 0, that @HAE�;/; 0 = �, @HA∗E�;/; 0 = /, and @PAE�;/; 0 = 0.This shows that

{〈�; /; 0〉 ∈ !+3|∃A ∈ BDF @HAA = �; @HA∗A = /; @PAA = 0}= {〈�; /; 0〉 ∈ !+3|�6 /; �6 0}:

12.6. Questions

1. What are the possible T;eT for DF-sound RE extensions T of i-S12? It is clear that

they must be RE, DF-irreLexive BDF-theories extending UP.2. Characterize the closed fragments of HA+ ECT0, HA+MP and HA+ ECT0 +MP.

It seems to me that closer inspection of the results of the present paper shouldreveal that the closed fragment of HA + ECT0 is the same as the closed fragmentof HA.

3. Is there a T of the appropriate kind where |∼provT does not have a left adjoint?


Acknowledgements

In various stages of research I bene.tted from the work, the wisdom and=or theadvice of: Johan van Benthem, Guram Bezhanishvili, Dirk van Dalen, Rosalie IemhoF,Dick de Jongh, Karst Koymans, Jaap van Oosten, Piet Rodenburg, Volodya Shavrukov,Rick Statman, Anne Troelstra and Domenico Zambella. Lev Beklemishev spotted a sillymistake in the penultimate version of the paper. I am grateful to the anonymous refereefor his=her helpful comments. I thanks Sander Hermsen for making the pictures for thispaper.

Appendix A. Adjoints in preorders

The consequence relations we consider are preorders. A preorder is a non-empty setwith a reLexive transitive relation. It is a special kind of category, viz., a categorywhere, between any two objects, we have at most one arrow. In the present paper, wewill use some facts from category theory and some facts speci.c for preorders. Wecollect most of these facts in the following lemma. The easy proofs are left to thereader.

Lemma A.1. Suppose 6 is a preorder on C and that & is a preorder on D. Say; "is the induced equivalence relation of 6 and ≡ is the induced equivalence relation of&. Let L :D → C be the left adjoint function to R :C → D; that is Ld6c⇔d&Rc;for every d∈D and every c∈C. Then the following holds:1: LRc6 c and d&RLd.2: L and R are order preserving; in other words; L and R are morphisms of preorders;and thus functors of the corresponding categories.

3: Rc≡RLRc and Ld"LRLd.4: Suppose that R is a surjection on objects. Then; for any d∈D; d≡RLd.5: Suppose R is a surjection on objects. Let X be the range of L. Let 60 be the fullsub-preorder of 6 obtained by restricting 6 to X . Then; 60 is equivalent to &.This means that; for x∈X; x " LRx and; for d∈D; d≡RLd.

6: L ◦ R is a uniquely determined modulo " by X := range(L).7: Suppose we have sums on the preorders 6 and &; i.e. there are binary functions

∨ and + such that• c1 6 c′ and c26c′ iI (c1 ∨ c2) 6 c′;• d1 & d′ and d2 &d′ iI (d1 + d2) & d′.Then L(d+ e)"Ld ∨Le.

Let . be a semi-consequence relation on L(P). Both 〈L(P);�〉 and 〈L(P); .〉 canbe considered as preorders. Condition A1 tells us that the identical mapping ' :A �→Ais an embedding from 〈L(P);�〉 to 〈L(P); .〉. In this paper, we are interested in caseswhere ' has a left adjoint L. If such a left adjoint of the embedding functor exists,


we call 〈L(P);�〉 a re?ective subcategory of 〈L(P); .〉. The functor L is calledthe re?ector. (See [24] for more on these notions.) Note that ' is also surjective onobjects.

Suppose . is a nearly-consequence relation. Condition B1 tells us that ∨ is a sumfor the category .. Since left adjoints commute with sums, by Lemma A.1 (7), wehave: � L(A ∨B)↔ (LA ∨LB).

Appendix B. Characterizations and dependencies

Consider any consistent RE theory T , extending HA, in the language of HA. Weintroduce some notions, closely related to �-preservativity for T . De.ne, suppressingthe index for T , the following consequence relations:

Provable Deductive Cons: A .pdc B :↔ ( A→ B)Uniform Deductive Cons: A .udc B :↔ ∀x∃y ( x A→ yB)Strong Uniform Deductive Cons: A .sudc B :↔ ∀x ( x A→ xB)Uniform Provably Deductive Cons: A .updc B :↔ ∀x∃y ( xA→ yB)

We prove the Orey–HKajek characterization for �-preservativity and Uniform ProvableDeductive Consequence.

Theorem B.1 (Orey–H Kajek). T proves that the following are equivalent:(i) A .� B; (ii) ∀x ( xA→B), (iii) A .updc B.

Proof. Reason in T .(i)→ (ii): Suppose A .� B and consider any x. We have, by Theorem 8.1, ( xA

→A) and ( xA)∈�, hence ( xA→B).(ii)→ (iii): From ( xA→B), we have that, for some y; y( xA→B) and hencey( xA→B). Ergo, ( y xA→ yB). Clearly, for any u, ( uA→ y uA). We may

conclude ( xA→ yB). Hence, A .updc B.(i)← (iii): Suppose A.updcB and (S→A). It follows that, for some x; x(S→A).

Moreover, (S→ xS). So (S→ xA). Hence, for some y, (S→ yB). Ergo, by re-Lection, (S→B).

Theorem B.2. We have T �A .� B→A .udc B.

Proof. Reason in T . Suppose A .� B. Consider any x. We can .nd a y such that wehave y( xA→B). Suppose xA. Clearly, for any u; uA→ y uA. It follows that yB.

Theorem B.3 (Orey–H Kajek for HA∗). We have

HA∗ � A .HA∗ ; � B↔ A .udc,HA∗ B:


Proof. Immediate by the principle C, Theorems B.1 and B.2.

Theorem B.4. HA∗ � ( HA∗A→B)→A.HA∗ ; �B.

Proof. Reason in HA∗. Suppose A→B. It follows that ∀x ( xA→B). Hence, ∀x ( xA→B). Thus we may conclude, A .� B.

Appendix C. Modal logic for �-preservativity

Consider the language of modal propositional logic with a unary modal operatorand a binary operator .. We de.ne arithmetical interpretations in the language of

HA in the usual manner, interpreting as provability in HA and . as �-preservativityover HA. We state the principles valid in HA for this logic, known at present. Withthe exception of �4 and �8 these principles are the duals of the principles of theinterpretability logic ILM. For a discussion of this duality, see [46].L1. � A⇒� A.L2. � (A→B)→ ( A→ B):L3. � A→ A.L4. � ( A→A)→ A.�1. � (A→B)→A . B.�2. � A . B ∧B . C→A . C.�3. � C . A ∧C . B→C . (A ∧B).�4. � A . C ∧B . C→ (A ∨B) . C.�5. � A . B→ ( A→ B).�6. � A . A.�7. � A . B→ ( C→A) . ( C→B).�8. Let X be a .nite set of implications and let

Y := {C | (C → D) ∈ X } ∪ {B}:

Take A :=∧X . Then, � (A→B) . {A}Y . Here (A→B) . {A}Y is short for

(A→B) .∨{A}Y and {C}D is de.ned as follows:

– {C}p := (C→p); {C}⊥ :=⊥, {C}� :=�,– {C} E := E; {C}(E .F) = (C→ (E .F)); {C}(E→F) := (C→ (E→F)),– {C}(·) commutes with ∧ and ∨ .

The veri.cation of L1–L4, �1–�3, �5–�7 is routine. For �4 see Theorem 9.1 and for�8 see Theorem 9:2.

From our principles Leivant’s principle can be derived. (This is one of the Stellingenof [21].)

Le � (A ∨B)→ (A ∨ B).We leave this as an exercise to the reader. It is open whether our axioms are arith-matically complete.


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