The $\Sigma_1$-Provability Logic of HA*

Abstract

For the Heyting Arithmetic HA, HA* is defined as the theory {A∣HA⊢A□}\{A\mid {\sf HA}\vdash A^{\Box}\}, where A□A^{\Box} is called the box translation of A. We characterize the Σ1\Sigma_1-provability logic of HA* as a modal theory iHσ∗{\sf iH}_\sigma^*.

Full text

The Σ1-Provability Logic of HA∗
Mohammad Ardeshir∗Mojtaba Mojtahedi†
Department of Mathematical Sciences Department of Mathematics,
Sharif University of Technology Statistics and Computer Science,
Tehran, Iran College of Sciences, University of Tehran
August 27, 2018
Abstract
For the Heyting Arithmetic HA,HA∗is dened [14,15] as the theory {A|HA ⊢A2}, where
A2is called the box translation of A(Denition 2.4). We characterize the Σ1-provability logic
of HA∗as a modal theory iH∗
σ(Denition 3.16).
Contents
1 Introduction 1
2 Denitions, conventions and basic facts 3
2.1 Denition of modal theories ................................ 5
2.2 HA∗and PA∗........................................ 5
3 Propositional modal logics 6
3.1 NNIL formulae and related topics ............................. 6
3.2 Box translation and propositional theories ........................ 9
3.3 Axiomatizing TNNIL-algorithm .............................. 11
4 The Σ1-Provability Logic of HA∗14
4.1 The Soundness Theorem .................................. 15
4.2 The Completeness Theorem ................................ 16
1 Introduction
This paper is a sequel of our previous paper [2], in which we characterized the Σ1-provability logic
of HA as a decidable modal theory iHσ(see Denition 3.16). Most of the materials of this paper are
from the paper mentioned above. Our techniques and proofs are very similar to those used there.
We use a crucial fact (Theorem 4.1 in this paper) proved in [2]. For the sake of self-containedness
as much as possible, we bring here some denitions from that paper.
For an arithmetical theory Textending HA, the following axiom schema is called the Completeness
Principle,CPT:
A→2TA.
∗mardeshir@sharif.ir
†http://mmojtahedi.ir/
1

Recall that by the work of Gödel in [5], for each arithmetical formula Aand recursively axiomatizable
theory T(like Peano Arithmetic PA), we can formalize the statement “there exists a proof in Tfor
A” by a sentence of the language of arithmetic, i.e. ProvT(⌜A⌝) := ∃xProofT(x, ⌜A⌝), where ⌜A⌝is
the code of A. Now, by interpreting 2Tby ProvT(⌜A⌝), the completeness principle for theory Tis
read as follows:
A→ProvT(⌜A⌝).
Albert Visser in [14,15] introduced an extension of HA,
HA∗:= HA +CPHA∗.
He called HA∗as a self-completion of HA. Moreover, he showed that HA∗may be dened as the
theory {A|HA ⊢A2}, where A2is called the box translation of A(Denition 2.4).
The notion of provability logic goes back essentially to K. Gödel [6] in 1933. He intended to
provide a semantics for Heyting’s formalization of intuitionistic logic IPC. He dened a translation,
or interpretation τfrom the propositional language to the modal language such that
IPC ⊢A⇐⇒ S4 ⊢τ(A).
Now the question is whether we can nd some modal propositional theory such that the 2
operator captures the notion of provability in Peano Arithmetic PA. Hence the question is to nd
some propositional modal theory T2such that:
T2⊢A⇐⇒ ∀∗ PA ⊢A∗
By ( )∗, we mean a mapping from the modal language to the rst-order language of arithmetic, such
that
•p∗is an arithmetical rst-order sentence, for any atomic variable p, and ⊥∗=⊥,
•(A◦B)∗=A∗◦B∗, for ◦ ∈ {∨,∧,→},
•(2A)∗:= ∃xProof PA (x, ⌜A∗⌝).
It turned out that S4 is not a right candidate for interpreting the notion of provability, since
¬2⊥is a theorem of S4, contradicting Gödel’s second incompleteness theorem (Peano Arthmetic
PA, does not prove its own consistency).
Martin Löb in 1955 showed [10] that the Löb’s rule (2A→A/A) is valid. Then in 1976,
Robert Solovay [12] proved that the right modal logic, in which the 2operator interprets the notion
of provability in PA, is GL. This modal logic is well-known as the Gödel-Löb logic, and has the
following axioms and rules:
• all tautologies of classical propositional logic,
•2(A→B)→(2A→2B),
•2A→22A,
• Löb’s axiom (L): 2(2A→A)→2A,
• Necessitation Rule: A/2A,
• Modus ponens: (A, A →B)/B.
2

Theorem. (Solovay-Löb) For any sentence Ain the language of modal logic, GL ⊢Aif and only if
for all interpretations ( )∗,PA ⊢A∗.
Now let we restrict the map ( )∗on the atomic variables in the following sense. For any atomic
variable p,(p)∗is a Σ1sentence. This translation or interpretation is called Σ1-interpretation. On
the other hand, let GLV =GL +CPa, where CPais the completeness principle restricted to atomic
variables, i.e., p→2p. Albert Visser [14] proved the following result:
Theorem. (Visser) For any sentence Ain the language of modal logic, GLV ⊢Aif and only if for
all Σ1interpretations ( )∗,PA ⊢A∗.
The question of generalizing Solovay’s result from classical theories to intuitionistic ones, such as
the intuitionistic counterpart of PA, well-known as HA, proved to be remarkably dicult and remains
a major open problem since the end of 70s [3]. For a detailed history of the origins, backgrounds and
motivations of the provability logic, we refer the readers to [3].
The following list contains crucial results about the provability logic of HA with arithmetical
nature:
• John Myhill 1973 and Harvey Friedman 1975. HA ⊬2HA (A∨B)→(2HA A∨2HA B), [11,4].
• Daniel Leivant 1975. HA ⊢2HA (A∨B)→2HA (2
.HA A∨2
.HA B), in which 2
.HA Ais a shorthand
for A∧2HA A, [8].
• Albert Visser 1981. HA ⊢2HA ¬¬ 2HA A→2HA 2HA Aand HA ⊢2HA (¬¬ 2HA A→2HA A)→
2HA (2HA A∨ ¬ 2HA A), [14,15].
• Rosalie Iemho 2001 introduced a uniform axiomatization of all known axiom schemas of the
provability logic of HA in an extended language with a bimodal operator . In her Ph.D.
dissertation [7], Iemho raised a conjecture that implies directly that her axiom system, iPH,
restricted to the normal modal language, is equal to the provability logic of HA, [7].
• Albert Visser 2002 introduced a decision algorithm for HA ⊢A, for all modal propositions A
not containing any atomic variable, i.e. Ais made up of ⊤,⊥via the unary modal connective
2HA and propositional connectives ∨,∧,→, [16].
• Mohammad Ardeshir and Mojtaba Mojtahedi 2014 characterized the Σ1-provability logic of
HA as a decidable modal theory [2], named there and here as iHσ. Recently, Albert Visser and
Jetze Zoethout [18] proved this result by an alternative method.
The authors of [1] found a reduction of the Solovay-Löb Theorem to the Visser Theorem only by
propositional substitutions [1]. This result is tempting to think of applying similar method for the
intuitionistic case. However it seems to us that there is no obvious way of doing such reduction for
the intuitionistic case, and it should be more complicated than the classical case.
In this paper, we introduce an axiomatization of a decidable modal theory iH∗
σ(see Denition
3.16) and prove that it is the Σ1-provability logic of HA∗. This arithmetical theory is dened [14,15]
as the theory {A|HA ⊢A2}, where A2is called the box translation of A(Denition 2.4). It is
worth mentioning that our proof of the Σ1-provability logic of HA∗is in some sense, a reduction to
the proof of the Σ1-provability logic of HA,only by propositional modal logic.
2 Denitions, conventions and basic facts
The propositional non-modal language contains atomic variables, ∨,∧,→,⊥and propositional modal
language is propositional non-modal language plus □. We use Aas a shorthand for A∧□A. For
simplicity, in this paper we use propositional language instead of propositional modal language.
IPC is the intuitionistic propositional non-modal logic over usual propositional non-modal language.
IPC□is the same theory IPC in the extended language of propositional modal language, i.e. its
3

language is propositional modal language and its axioms and rules are the same as the one in IPC.
Since we have no axioms for □in IPC□, it is obvious that □Afor each A, behave exactly like
an atomic variable inside IPC□. Note that nothing more than symbol of Aplays a role in □A.
The rst-order intuitionistic theory is denoted with IQC and CQC is its classical closure, i.e. IQC
plus the principle of excluded middle. We have the usual rst-order language of arithmetic which
has a primitive recursive function symbol for each primitive recursive function. We use the same
notations and denitions for Heyting’s arithmetic HA as in [13], and Peano Arithmetic PA is HA
plus the principle of excluded middle. For a set of sentences and rules Γ∪ {A}in propositional
non-modal, propositional modal or rst-order language, Γ⊢Ameans that Ais derivable from Γin
the system IPC,IPC□,IQC, respectively.
Denition 2.1. Suppose Tis an r.e arithmetical theory and σis a function from atomic variables
to arithmetical sentences. We extend σto all modal propositions A, inductively:
•σT(A) := σ(A)for atomic A,
•σTdistributes over ∧,∨,→,
•σT(□A) := P rT(⌜σT(A)⌝), in which P rT(x)is the Σ1-predicate that formalizes provability of
a sentence with Gödel number x, in the theory T.
We call σto be a Σ1-substitution, if for every atomic A,σ(A)be a Σ1-formula.
Denition 2.2. Provability logic of a suciently strong theory, Tis dened to be a modal proposi-
tional theory PL(T)such that PL(T)⊢Ai for arbitrary arithmetical substitution σT,T⊢σT(A).
If we restrict the substitutions to Σ-substitutions, then the new modal theory is PLσ(T).
Lemma 2.3. Let A(p1, . . . , pn)be a non-modal proposition with pi̸=pjfor all 0< i < j ≤n. Then
for every modal sentences B1, . . . , Bnwith Bi̸=Bjfor 0< i < j ≤nwe have:
IPC ⊢Ai IPC□⊢A[p1|□B1, . . . , pn|□Bn].
Proof. By simple inductions on complexity of proofs in IPC and IPC□.2
The following denition, the Beeson-Visser box-translation, is essentially from ([15, Def.4.1]).
Denition 2.4. For every proposition Ain modal propositional language, we associate a proposition
A□, called box-translation of A, dened inductively as follows:
•A□:= A∧□A, for atomic A, and ⊥□=⊥,
•(A◦B)□:= A□◦B□, for ◦ ∈ {∨,∧},
•(A→B)□:= (A□→B□)∧□(A□→B□),
•(□A)□:= □(A□).
The box-translation can be extended to rst-order arithmetical formulae A, as follows:
•(∀xA)□:= □(∀xA□)∧ ∀xA□,
•(∃xA)□:= ∃xA□.
Dene NOI (No Outside Implication) as set of modal propositions A, that any occurrence of →
is in the scope of some □. To be able to state an extension of Leivant’s Principle (that is adequate
to axiomatize Σ1-provability logic of HA) we need a translation on modal language which we name
it Leivant’s translation. We dene it recursively as follows:
•Al:= Afor atomic A, boxed Aor A=⊥,
4

•(A∧B)l:= Al∧Bl,
•(A∨B)l:= Al∨Bl,
•(A→B)lis dened by cases: If A∈NOI, dene (A→B)l:= A→Bl, else dene (A→
B)l:= A→B.
Denition 2.5. Minimal provability logic iGL , is same as Gödel-Löb provability logic GL , without
the principle of excluded middle, i.e. it has the following axioms and rules:
• theorem of IPC□,
•□(A→B)→(□A→□B),
•□A→□□A,
• Löb’s axiom (L): □(□A→A)→□A,
• Necessitation Rule: A/□A,
• Modus ponens: (A, A →B)/B.
2.1 Denition of modal theories
iK4is iGL without Löb’s axiom. Note that we can get rid of the necessitation rule by adding □Ato
the axioms, for each axiom Ain the above list. We wil l use this fact later in this paper. We list the
following axiom schemae:
• The Completeness Principle: CP := A→□A.
• Restricted Completeness Principle to atomic formulae: CPa:= p→□p, for atomic p.
• Leivant’s Principle: Le := □(B∨C)→□(□B∨C). [9]
• Extended Leivant’s Principle: Le+:= □A→□Al.
• Trace Principle: TP := □(A→B)→(A∨(A→B)). [15]
We dene theories iGLC := iGL +CP,H:= iGLC +TP,LLe := iGL +Le and LLe+:= iGL+Le++CPa.
Note that in the presence of CP and modus ponens, the necessitation rule is superuous. Later we
will nd Kripke semantics for iGLC and also we will see that iGLC and LLe+proves the same formulae
of restricted complexity (TNNIL).
2.2 HA∗and PA∗
HA∗and PA∗were rst introduced in [15]. These theories are dened as
HA∗:= {A|HA ⊢A□}and PA∗:= {A|PA ⊢A□}.
Visser in [15] showed that the provability logic of PA∗is H, i.e. H⊢Ai for all arithmetical
substitution σ,PA∗⊢σPA∗(A). That means that
PL(PA∗) = P Lσ(PA∗) = H.
Lemma 2.6. 1. For any arithmetical Σ1-formula A,HA ⊢A↔A□.
2. HA is closed under the box-translation, i.e., for any arithmetical formula A,HA ⊢Aimplies
HA ⊢A□, so HA ⊆HA∗.
5

Proof. 1. See [15](4.6.iii).
2. See [15](4.14.i).
2
Lemma 2.7. For any Σ1-substitution σand each propositional modal sentence A, we have HA ⊢
σHA(A□)↔(σHA∗(A))□and hence
HA ⊢σHA(A□)i HA∗⊢σHA∗(A)
Proof. Use induction on the complexity of A. All the steps are straightforward. For the atomic case,
we use Lemma 2.6.1. 2
Remark 2.8. This lemma can be combined with the characterization of the Σ1-provability logic of
HA to derive directly a characterization of the Σ1-provability logic of HA∗:
Abelongs to the Σ1-provability logic of HA∗i A2belongs to the Σ1-provability logic of HA.
This means that we have a decision algorithm for the Σ1-provability logic of HA∗. The rest of this
paper is devoted to axiomatize the Σ1-provability logic of HA∗.
3 Propositional modal logics
3.1 NNIL formulae and related topics
The class of No Nested Implications in the Left,NNIL formulae in a propositional language was
introduced in [17] , and more explored in [16]. The crucial result of [16] is providing an algorithm that
as input, gets a non-modal proposition Aand returns its best NNIL approximation A∗from below,
i.e., IPC ⊢A∗→Aand for all NNIL formula Bsuch that IPC ⊢B→A, we have IPC ⊢B→A∗. In
the following we explain this algorithm and explain how to extend it to modal propositions.
To dene the class of NNIL propositions, let us rst dene a complexity measure ρon non-modal
propositions as follows:
•ρp =ρ⊥=ρ⊤= 0, where pis an atomic proposition,
•ρ(A∧B) = ρ(A∨B) = max(ρA, ρB),
•ρ(A→B) = max(ρA + 1, ρB),
Then NNIL ={A|ρA ≤1}.
Denition 3.1. We dene a measure complexity for modal propositions Das follows:
•I(D) := {E∈Sub(D)|Eis an implication that is not in the scope of a □},
•i(D) := max{|I(E)| | E∈I(D)}, where |X|is the number of elements of X,
•cD:= the number of occurrences of logical connectives which is not in the scope of a □,
•dD:= the maximum number of nested boxes. To be more precise,
–dD:= 0 for atomic D,
–dD:= max{dD1,dD2}, where D=D1◦D2and ◦ ∈ {∧,∨,→},
–d□D:= dD+ 1,
•oD:= (dD, iD, cD).
6

Note that the measure oDis ordered lexicographically, i.e., (d, i, c)<(d′, i′, c′)i d<d′or d=
d′, i < i′or d=d′, i =i′, c < c′.
Denition 3.2. For any two modal propositions A, B , we dene [A]Band [A]′B, by induction on
the complexity of B:
•[A]p= [A]′p=p, for atomic p,⊤and ⊥,
•[A](B1◦B2) = [A](B1)◦[A](B2),[A]′(B1◦B2) = [A]′(B1)◦[A]′(B2)for ◦ ∈ {∨,∧},
•[A](B1→B2) = A→(B1→B2),[A]′(B1→B2) = (A′∧B1)→B2, in which A′=
A[B1→B2|B2], i.e., replace each occurrence of B1→B2in Aby B2,
NNIL-algorithm
For each proposition A,A∗is produced by induction on complexity measure oAas follows. For
details see [16].
1. Ais atomic, take A∗:= A,
2. A=B∧C, take A∗:= B∗∧C∗,
3. A=B∨C, take A∗:= B∗∨C∗,
4. A=B→C, we have several sub-cases. In the following, an occurrence of Ein Dis called an
outer occurrence, if Eis not in the scope of an implication.
4.a. Ccontains an outer occurrence of a conjunction. In this case, we assume some formula
J(q)such that
•qis a propositional variable not occurred in A,
•qis outer in Jand occurs exactly once,
•C=J[q|(D∧E)].
Such Jobviously exists. Now set C1:= J[q|D], C2:= J[q|E]and A1:= B→C1, A2:= B→
C2and nally, dene A∗:= A∗
1∧A∗
2.
4.b. Bcontains an outer occurrence of a disjunction. In this case we suppose some formula
J(q)such that
•qis a propositional variable not occurred in A,
•qis outer in Jand occurs exactly once,
•B=J[q|(D∨E)].
Such Jobviously exists. Now set B1:= J[q|D], B2:= J[q|E]and A1:= B1→C, A2:= B2→
Cand nally, dene A∗:= A∗
1∧A∗
2.
4.c. B=Xand C=Yand X, Y are sets of implications or atoms. We have several
sub-cases:
4.c.i. Xcontains atomic p. Set D:= (X\ {p})and take A∗:= p→(D→C)∗.
4.c.ii. Xcontains ⊤. Dene D:= (X\ {⊤})and take A∗:= (D→C)∗.
4.c.iii. Xcontains ⊥. Take A∗:= ⊤.
4.c.iv. Xcontains only implications. For any D=E→F∈X, let
B↓D:= ((X\ {D})∪ {F}).
7

Let Z:= {E|E→F∈X} ∪ {C}and A0:= [B]Z:= {[B]E|E∈Z}. Now if oA0<oA,
we take
A∗:= {((B↓D)→C)∗|D∈X} ∧ A∗
0,
otherwise, rst set A1:= [B]′Zand then take
A∗:= {((B↓D)→C)∗|D∈X} ∧ A∗
1
We can extend ρto all modal language with ρ(□A) := 0. The class of NNIL propositions may be
dened for propositional modal language as well, i.e. we call a modal proposition Ato be NNIL□, if
ρ(A)≤1(for extended ρ). We also dene two other classes of propositions:
Denition 3.3. TNNIL (Thoroughly NNIL)is the smallest class of propositions such that
•TNNIL contains all atomic propositions,
• if A, B ∈TNNIL, then A∨B , A ∧B, □A∈TNNIL,
• if all →occurred in Aare contained in the scope of a □(or equivalently A∈NOI)and
A, B ∈TNNIL, then A→B∈TNNIL.
Finally we dene TNNIL−as the set of all the propositions like A(□B1, . . . , □Bn), such that
A(p1, . . . , pn)is an arbitrary non-modal proposition and B1, . . . , Bn∈TNNIL.
We can use the same algorithm with slight modications treating propositions inside □as well.
First we extend Denition 3.2 to capture modal language.
Denition 3.4. For any two modal propositions A, B , we dene [A]Band [A]′Bby induction on
the complexity of B. We extend Denition 3.1 by the following item:
•[A]2B1= [A]′2B1:= □B1.
It is clear that we are treating a boxed formula as an atomic variable.
NNIL2-algorithm
We use NNIL-algorithm with the following changes to produce a similar NNIL-algorithm for modal
language.
1. Ais atomic or boxed, take A∗=A.
4. An occurrence of Ein Dis called an outer occurrence, if Eis neither in the scope of an implication
nor in the scope of a boxed formula.
4. c(i). Xcontains atomic or boxed formula p. We set D:= (X\ {p})and take A∗:= p∗→(D→
C)∗.
Remark 3.5. In fact, we have two ways to nd out NNIL□approximation of a modal proposition.
First: simply apply NNIL2-algorithm to a modal proposition Aand compute A∗.
Second: let B1, . . . , Bnbe all boxed sub-formulae of Awhich are not in the scope of any other
boxes. Let A′(p1, . . . , pn)be unique non-modal proposition such that {pi}1≤i≤nare fresh atomic
variables not occurred in Aand A=A′[p1|B1, . . . , pn|Bn]. Let ρ(A) := (A′)∗[p1|B1, . . . , pn|Bn].
Then it is easy to observe that IPC2⊢ρ(A)↔A∗.
The above dened algorithm is not deterministic, however by the following Theorem, we know
that A∗is unique up to IPC2equivalence. The notation A▷IPC2,NNIL2B(A,NNIL2-preserves B)
from [16], means that for each NNIL2modal proposition C, if IPC2⊢C→A, then IPC2⊢C→B,
in which A, B are modal propositions.
Theorem 3.6. For each modal proposition A,NNIL2algorithm with input Aterminates and the
output formula A∗, is an NNIL2proposition such that IPC2⊢A∗→A.
Proof. See [2, The. 4.5]. 2
8

TNNIL-algorithm
Here we dene A+as TNNIL-formula approximating A. Informally speaking, to nd A+, we rst
compute A∗and then replace all outer boxed formula □Bin Aby □B+. To be more accurate,
we dene A+by induction on dA. Suppose that for all Bwith dB < dA, we have dened B+.
Suppose that A′(p1, . . . , pn)and □B1, . . . , □Bnsuch that A=A′[p1|□B1, . . . , pn|□Bn]where A′is
a non-modal proposition and p1, . . . , pnare fresh atomic variables (not occurred in A). It is clear
that dBi<dAand then we can dene A+:= (A′)∗[p1|□B+
1, . . . , pn|□B+
n].
Lemma 3.7. For any modal proposition A,
1. for al l Σ1-substitution σwe have HA ⊢2σHA (A)↔2σHA (A+)and hence HA ⊢σHA(A)i
HA ⊢σHA(A+).
2. iGL ⊢A1→A2implies iGL ⊢A+
1→A+
2, and iK4 ⊢A1→A2implies iK4 ⊢A+
1→A+
2.
3. iGL ⊢A1↔A2implies iGL ⊢A+
1↔A+
2, and iK4 ⊢A1↔A2implies iK4 ⊢A+
1↔A+
2.
Proof. See [2, Corollary 4.8]. 2
TNNIL2-algorithm
Corollary 3.8. There exists a TNNIL2-algorithm such that for any modal proposition A, it halts
and produces a proposition A−∈TNNIL2such that IPC□⊢A+→A−.
Proof. Let A:= B(□C1, . . . , □Cn), and B(p1, . . . , pn)is non-modal. apparently such Bexists. Then
dene A−:= B(□C+
1, . . . , □C+
n). Now denition of A+implies A+= (A−)∗and hence Theorem 3.6
implies that A−has desired property. 2
Lemma 3.9. For each modal proposition Aand Σ1-substitution σ,HA ⊢σHAA↔σHAA−.
Proof. Use denition of (.)−and Lemma 3.7.1. 2
Remark 3.10. Note that iGLC ⊢A↔Bdoes not imply iGLC ⊢A+↔B+. A counter-example is
A:= ¬¬pand B:= ¬(¬p). We have A+=A∗=pand iGLC ⊢B+↔(□¬p→p). Now one can
use Kripke models to show iGLC ⊬¬¬p→(□¬p→p).
Remark 3.11. In the NNIL2-algorithm, if we replace the operation (·)∗by (·)†, and change the
step 1 to
1. A†:= A, if Ais atomic, and (□B)†:= □B†,
then the new algorithm also halts, and for any modal proposition A, we have iK4 ⊢A†↔A+.
3.2 Box translation and propositional theories
Following Visser’s denition of the notion of a base in arithmetical theories [15], we dene
Denition 3.12. A modal theory Tis called to be closed under box-translation if for every propo-
sition A,T⊢Aimplies T⊢A□.
Proposition 3.13. For arbitrary subset Xof {CP,CPa,L},iK4+Xis closed under box-translation.
Proof. The proof can be carried out in three steps:
1. For any proposition Arst we show that IPC□⊢Aimplies iK4 ⊢A□. This can be done by a
routine induction on the length of the proof in IPC. Note that for any axiom Aof IPC, we have
iK4 ⊢A□. As for the rule of modus ponens, suppose that IPC□⊢Aand IPC□⊢A→B. By
induction hypothesis, then iK4 ⊢A□and iK4 ⊢(A□→B□)∧□(A□→B□)and so iK4 ⊢B□.
9

2. Next observe that
(□A→□□A)□=□A□→□□A□
and also
iK4 ⊢[(□(A→B)∧□A)→□B]□↔(□(A□→B□)∧□A□)→□B□.
3. We observe that for any axiom A∈X,iK4 +X⊢A□.
2
The following two lemmas will be used in the proof of Theorem 3.18.
Lemma 3.14. For any modal propositions A, A′and B, and any propositional modal theory T
containing IPC□,
1. iK4 +□A□⊢([A]B)□↔([A□]B□).
2. T⊢A↔A′implies T⊢[A]B↔[A′]B.
Proof. Proof of both parts are by induction on the complexity of B:
1. The only non-trivial case is when Bis an implication. Let B:= C→D. By Denitions 3.4
and 2.4,
([A](C→D))□=(A□→((C□→D□)∧□(C□→D□)))
and also
[A□](C→D)□= (A□→(C□→D□)) ∧□(C□→D□).
Now it is easy to observe that
iK4 +□A□⊢([A](C→D))□↔([A□](C→D)□).
2. Similar to the rst item.
2
Notation. In the sequel of paper, we use A≡Bas a shorthand for iK4 ⊢A↔B.
Lemma 3.15. Let A=B→Cbe a modal proposition such that B=Xand C=Y, where X
is a set of implications and Yis a set of atomic, boxed or implicative propositions. Then
(A□)+≡
E→F∈X
□(E→F)□+→(B↓D→C)□+|D∈X∧([B]Z)□+
where Z={E|E→F∈X}∪{C}.
Proof. To simplify notations, Let us indicate
• the sets of all atomic and boxed propositions by At and Bo, respectively,
•X′:= {E□→F□|E→F∈X},
•Z′:= Z□={E□|E→F∈X}∪{C□},
•B′:= X′,
• for any I⊆Y,CI:= E→F∈I□(E□→F□)∨E∈I∩At □E∨E→F∈Y\I(E□→F□)
∨E∈(Y\I)∩At E∨E∈Bo∩YE□,
• and ZI:= {E□|E→F∈X}∪{CI}.
10

By repeated use of distributivity of conjunction over disjunction, which is valid in IPC , we have
(3.1) C□≡
I⊆Y
CIand Z□≡
I⊆Y
ZI
Note that A□= (B□→C□)∧□(B□→C□), and then by denition of (·)+,
(A□)+= (B□→C□)+∧□(B□→C□)+.
Now we compute the left conjunct:
B□→C□+=
I⊆YB□→CI+
(3.2)
≡
I⊆Y

E→F∈X
□E□→F□+→ 
E→F∈XE□→F□→CI+

(3.3)
≡
E→F∈X
□(E→F)□+→
I⊆YB′→CI+
(3.4)
≡
E→F∈X
□(E→F)□+→
I⊆YB′↓D′→CI+|D′∈X′∧[B′]ZI+
(3.5)
≡
E→F∈X
□(E→F)□+→B′↓D′→C□+|D′∈X′∧([B′]Z′)+
(3.6)
and hence
(3.7)
A□+≡
E→F∈X
□(E→F)□+→(B↓D→C)□+|D∈X∧([B′]Z′)+
Note that 3.2 and 3.3 hold by NNIL□-algorithm, 3.4 holds by properties of IPC□,3.5 holds by TNNIL-
algorithm, 3.6 holds by TNNIL-algorithm and equation 3.1, and nally equation 3.7 is derived from
3.6 by deduction in iK4 and TNNIL-algorithm. Now it is enough to show that the last formula is
equivalent to the following one in iK4:
(3.8) 
E→F∈X
□(E→F)□+→(B↓D→C)□+|D∈X∧([B]Z)□+
To show this, it is enough to show
iK4 ⊢
E→F∈X
□(E→F)□+→([B]Z)□+↔([B′]Z′)+.
Then by Lemma 3.7.2, it is enough to show iK4 ⊢E→F∈X□(E→F)□→(([B]Z)□↔[B′]Z′).
Since E→F∈X□(E→F)□≡□B□, then it is enough to show iK4 +□B□⊢([B]Z)□↔[B′]Z′.
Now, by Lemma 3.14.1, we have iK4 +□B□⊢([B]Z)□↔[B□]Z□. Hence we should show iK4 +
□B□⊢[B□]Z□↔[B′]Z′. We have Z′=Z□and iK4 +□B□⊢B□↔B′. Then by Lemma 3.14.2,
iK4 +□B□⊢[B□]Z□↔[B′]Z′.2
3.3 Axiomatizing TNNIL-algorithm
In this section, we introduce the axiom set Xsuch that iK4 +X⊢(A□)−↔A□. Note that we may
simply choose X:= {(A□)−↔A□|Ais arbitrary proposition}. However, we want to reduce Xto
some smaller ecient set of formulae.
11

We use some modal variant of Visser’s ▶σin [16]. It is exactly the same as the relation ▶in [2]
(sec. 4.3) except for item B2, which is a little bit dierent:
• B2′. Let Xbe a set of implications, B:= Xand A:= B→C. Also assume that
Z:= {E|E→F∈X}∪{C}. Then A▶[B]Z,
The relation ▶∗is dened to be the smallest relation on modal propositional sentences satisfying:
• A1. If iK4 ⊢A→B, then A▶∗B,
• A2. If A▶∗Band B▶∗C, then A▶∗C,
• A3. If C▶∗Aand C▶∗B, then C▶∗A∧B,
• A4. If A▶∗B, then 2A▶∗2B,
• B1. If A▶∗Cand B▶∗C, then A∨B▶∗C,
• B2. Let Xbe a set of implications, B:= Xand A:= B→C. Also assume that Z:=
{E|E→F∈X}∪{C}. Then A∧□B▶∗[B]Z,
• B3. If A▶∗B, then p→A▶∗p→B, in which pis atomic or boxed.
A▶◀∗Bmeans A▶∗Band B▶∗A.
Denition 3.16. We dene
iH∗
σ:= iGL +CP +{□A→□B|A▶∗B}.
Note that the Σ1-provability logic of HA is proved in [2] to be
iHσ:= iGL +CPa+Le++{□A→□B|A▶B},
in which CPais the Completeness Principle restricted to atomic propositions.
Lemma 3.17. For any propositional modal sentences A, B,A▶∗Bimplies A□▶∗B□.
Proof. It is clear that A▶∗Bi there exists a Hilbert-type sequence of relations {Ai▶∗Bi}0≤i≤n
such that An=A, Bn=Band for each i≤n,Ai▶∗Biis an instance of axioms A1 or B2, or it is
derived by making use of some previous members of sequence and some of the rules A2-A4 or B1 or
B3. Hence we are authorized to use induction on the length of such sequence for A▶∗Bto show
A□▶∗B2. The only non-trivial steps are axioms A1 and B2. Suppose that A▶∗Bis an instance
of A1, i.e. iK4 ⊢A→B. Then by Proposition 3.13, we have iK4 ⊢A□→B2and hence again by
A1, A□▶∗B□, as desired.
For treating B2, suppose that A:= B→C,B=X,Xis a set of implications and Z:= {E|E→
F∈X}∪{C}. We must show (A∧2B)□▶∗([B]Z)□. Dene X′:= {E2→F2|E→F∈
X}, B′:= X′. Hence by B2, B′→C□∧2B′▶∗[B′]Z□. Note that we have □B′≡□B□and
also iK4 +□B′⊢B′↔B□.
Now by using properties of ▶∗(A1-A3) and Lemma 3.14(2), we can deduce (B□→C□)∧2B□▶∗
[B□]Z□. Then Lemma 3.14(1) implies (B□→C□)∧2B□▶∗([B]Z)□. Then by A1 and A2, we
can deduce ((B→C)∧2B)□▶∗([B]Z)□, as desired. 2
The following theorem is analogous to the Theorem 4.18 in [2]:
Theorem 3.18. For any modal proposition A,A□▶◀∗(A□)+.
Before proving this theorem, we state a corollary.
Corollary 3.19. iH∗
σ⊢A□↔(A□)−.
12

Proof. Let A□=B(□C1,□C2, . . . , □Cn)where B(p1, . . . , pn)is a non-modal proposition. It isn’t
hard to observe that for each 1≤j≤n,iK4 ⊢□Cj↔□C□
j. By denition of (A□)−, we have
(A□)−=B(□C+
1, . . . , □C+
n). Now by Lemma 3.7, we can deduce that iK4 ⊢B(□C+
1, . . . , □C+
n)↔
B(□(C□
1)+, . . . , □(C□
n)+). Then Theorem 3.18 implies that iH∗
σ⊢□(C□
i)+↔□C□
i. Hence iH∗
σ⊢
(A□)−↔A□.2
Proof. (Theorem 3.18)We prove by induction on o(A□). Suppose that we have the desired result
for each proposition Bwith o(B□)<o(A□). We treat Aby the following cases.
1. (A1) Ais atomic. Then (A□)+=A□, by denition, and result holds trivially.
2. (A1-A4, B1) A=□B, A =B∧C, A =B∨C. All these cases hold by induction hypothesis.
In boxed case, we use of induction hypothesis and A4. In conjunction, we use of A1-A3 and
in disjunction we use A1,A2 and B1.
3. A=B→C. There are several sub-cases. similar to denition of NNIL-algorithm, an occur-
rence of a sub-formula Bof Ais said to be an outer occurrence in A, if it is neither in the
scope of a 2nor in the scope of →.
(c).i.(A1-A3) Ccontains an outer occurrence of a conjunction. We can treat this case using
induction hypothesis and TNNIL-algorithm.
(c).ii.(A1-A3) Bcontains an outer occurrence of a disjunction. We can treat this case by
induction hypothesis and TNNIL-algorithm.
(c).iii. B=Xand C=Y, where X, Y are sets of implications, atoms and boxed formulae.
We have several sub-cases.
(c).iii.α.(A1-A4, B3) Xcontains atomic variables. Let pbe an atomic variable in X. Set
D:= (X\ {p}). Then
(A□)+≡[(p∧□p)→(D□→C□)+]
≡[(p∧□p)→((D→C)□)+]
On the other hand, we have by induction hypothesis and A1,A2 and B3, that
[(p∧□p)→((D→C)□)+]▶◀∗(p∧□p)→((D→C)□)
which by use of A4 implies:
2[(p∧□p)→((D→C)□)+]▶◀∗2[(p∧□p)→((D→C)□)]
And by use of A1-A3 we have
[(p∧□p)→((D→C)□)+]▶◀∗[(p∧□p)→((D→C)□)]
Finally by A1 and A2 we have : (A□)+▶◀∗A□.
(c).iii.β.(A1-A4, B3) Xcontains boxed formula. Similar to the previous case.
(c).iii.γ.(A1, A2) Xcontains ⊤or ⊥. Trivial.
(c).iii.δ.(A1-A4, B2, B3) Xcontains only implications. This case needs the axiom B2 and it
seems to be the interesting case.
By Lemma 3.15,
(A□)+≡
E→F∈X
□(E→F)□+→(B↓D→C)□+|D∈X∧([B]Z)□+.
13

Then by induction hypothesis, A1-A4 and B3 we have:
(A□)+▶◀∗
E→F∈X
□(E→F)□→(B↓D→C)□|D∈X∧([B]Z)□
▶◀∗□B→{B↓D→C|D∈X} ∧ [B]Z□
We show that for each E∈Z,
(*) iK4 ⊢({(B↓D)→C|D∈X} ∧ [B]E)→A.
If E=C, we are done by IPC□⊢[B]C→(B→C). So suppose some E→F∈X. We
reason in iK4. Assume {(B↓D→C|D∈X},[B]Eand B. We want to derive C. We
have ((X\ {E→F})∧F)→C,[B]Eand B. From Band [B]E, we derive E. Also from
B, we derive E→F, and so F. Hence we have (X\ {E→F})∧F, which implies C, as
desired.
Now (*) implies
iK4 ⊢
G
  
({(B↓D→C|D∈X} ∧ [B]Z)→A
Then by Proposition 3.13, we have iK4 ⊢(G□∧B2)→C2. This implies iK4 ⊢(B2→
(G□∧B2)) →(B2→C2), and hence iK4 ⊢(B2→G2)→(B2→C2). Then because
B2→2B2, we have iK4 ⊢(2(B2)→G2)→(B2→C2). Hence by necessitation, we derive
iK4 ⊢(□B→({(B↓D→C|D∈X} ∧ [B]Z))□→A□. Hence (A□)+▶∗A□.
To show the other way around, i.e., A□▶∗(A□)+, by Proposition 3.13, it is enough to show
A▶∗□B→{B↓D→C|D∈X} ∧ [B]Z
or equivalently
A∧2B▶∗{B↓D→C|D∈X} ∧ [B]Z
We have IPC2⊢A→{B↓D→C|D∈X}, and hence by A1, A∧2B▶∗{B↓D→C|D∈X}.
On the other hand, A∧2B▶∗[B]Z, which by A3, implies
A∧2B▶∗{B↓D→C|D∈X} ∧ [B]Z
2
4 The Σ1-Provability Logic of HA∗
In this section we will show that iH∗
σis the provability logic of HA∗for Σ1-substitutions.
Before we continue with the soundness and completeness theorem, let us state the main theorem
from [2] that plays a crucial role in the sequel of this paper.
Theorem 4.1. Let A∈TNNIL2be a modal proposition such that iGLC ⊬A. Then there exists some
arithmetical Σ1-substitution σsuch that HA ⊬σHA(A).
Proof. For the rather long proof of this fact, see [2], Theorems 4.26 and 5.1. 2
14

4.1 The Soundness Theorem
Let us dene some notions from [16]. We call a rst-order sentence A,Σ-preserves B(AT ,Σ1B),
if for each Σ1-sentence C, if T⊢C→A, then T⊢C→B. For modal propositions Aand B,
we dene AT,Σ1,Σ1Bi for each arithmetical Σ-substitution σT, we have σT(A)T,Σ1σT(B).
For arbitrary modal sentences A, B, the notation A|∼T,Σ1Bmeans that T⊢σT(A)implies T⊢
σT(B), for arbitrary Σ1-substitution σT. All the above relations with a superscript of HA, means
“an arithmetical formalization of that relation in HA ”, for example, AHA
HA∗,Σ1Bmeans HA ⊢
“AHA∗,Σ1B”.
Lemma 4.2. 1. For each rst-order sentences A, B,AHA
HA∗,Σ1Bi A□HA
HA,Σ1B□,
2. For each propositional modal A, B,AHA
HA∗,Σ1,Σ1Bi A□HA
HA,Σ1,Σ1B□.
Proof. To prove part 1, use Lemma 2.6.1 and denitions of HA
HA∗,Σ1and HA
HA,Σ1.
To prove part 2, note that AHA
HA∗,Σ1,Σ1Bi for all Σ-substitution σ,σHA∗(A)HA
HA∗,Σ1σHA∗(B)
i for all Σ-substitution σ,σHA∗(A)□HA
HA,Σ1σHA∗(B)□(by previous part) i for all Σ-substitution
σ,σHA(A□)HA
HA,Σ1σHA(B□)i A□HA
HA,Σ1,Σ1B□.2
Lemma 4.3. HA
HA,Σ1is closed under B1.
Proof. See [16], 9.1. 2
Corollary 4.4. HA
HA∗,Σ1is closed under B1.
Proof. Immediate corollary of Lemma 4.2 and 4.3.2
Lemma 4.5. HA
HA,Σ1,Σ1satises A1-A4, B1, B2′and B3.
Proof. Proof of closure under A1-A4 and B3 is straightforward. Closure under B1 is by Lemma 4.3.
For a proof of case B2′, see [16].9.2. 2
Corollary 4.6. HA
HA∗,Σ1,Σ1satises B2.
Proof. Let A, B, C, X, Z be as stated in dening B2. We must prove A∧□BHA
HA∗,Σ1,Σ1[B]Z. Hence
by Lemma 4.2, it is enough to show (A∧□B)□HA
HA,Σ1,Σ1([B]Z)□. Let X′:= {E□→F□|E→F∈
X}, B′:= X′, C ′:= C□, Z′:= {E□|E→F∈X} ∪ {C′}. Now Because HA
HA,Σ1,Σ1satises B2′
(Lemma 4.5), we have (B′→C′)HA
HA,Σ1,Σ1[B′]Z′. Note that Z□=Z′and IPC□⊢(B′∧□B′)↔B□.
Hence by Lemma 3.14.2, iK4 +□B′⊢[B′]Z′↔[B□]Z□. Also by Lemma 3.14.1, iK4 +□B′⊢
[B□]Z□↔([B]Z)□. So iK4 +□B′⊢[B′]Z′↔([B]Z)□. Now because HA
HA,Σ1,Σ1satises A1, we
have □B′HA
HA,Σ1,Σ1[B′]Z′↔([B]Z)□. Now one can easily observe that because HA
HA,Σ1,Σ1is closed
under A1-A3, we can deduce (B′→C′)∧□B′HA
HA,Σ1,Σ1([B]Z)□. This by using A1-A3implies
((B→C)∧□B)□HA
HA,Σ1,Σ1([B]Z)□. Hence by Lemma 4.2.2, (B→C)∧□BHA
HA,Σ1,Σ1[B]Z, as
desired. 2
Corollary 4.7. HA
HA∗,Σ1,Σ1is closed under B3.
Proof. Let pbe atomic or boxed and assume some A, B such that AHA
HA∗,Σ1,Σ1B. Then by Lemma
4.2.2, A□HA
HA,Σ1,Σ1B□. Because HA
HA,Σ1,Σ1satises B3, we get p□→A□HA
HA,Σ1,Σ1p□→B□. Now
by A4, □[p□→A□]HA
HA,Σ1,Σ1
□[p□→B□], which implies (p→A)□HA
HA,Σ1,Σ1(p→B)□. Now by
Lemma 4.2.2, p→AHA
HA∗,Σ1,Σ1p→B, as desired. 2
Lemma 4.8. We have the following inclusions:
▶∗⊆HA
HA∗,Σ1,Σ1⊆ |∼HA
HA∗,Σ1
15

Proof. The second inclusion is a trivial. We only prove the rst inclusion. We show that HA
HA∗,Σ1,Σ1is
closed under A1-A4 and B1-B3. One can observe that HA
HA∗,Σ1,Σ1is closed under A1-A4 and we leave
this to the reader. Closure under B1, B2 and B3 is by Corollaries 4.4,4.6 and 4.7, respectively. 2
Theorem 4.9. (Soundness)iH∗
σis sound for Σ1-arithmetical interpretations in HA∗, i.e. iH∗
σ⊆
PLσ(HA∗).
Proof. We show that for arbitrary Σ-substitution, σHA∗, and for any A, if iH∗
σ⊢A, then HA∗⊢
σHA∗(A). This can be done by induction on the complexity of iH∗
σ⊢A. All inductive steps clearly
holds, except for the axioms □A→□Bwith A▶∗B. This case is a direct consequence of Lemma
4.8.2
4.2 The Completeness Theorem
Theorem 4.10. Σ1-arithmetical interpretations in HA∗are complete for iH∗
σ, i.e.
PLσ(HA∗)⊆iH∗
σ
Proof. We prove the Completeness Theorem contra-positively. Let iH∗
σ⊬A(p1, . . . , pn). Then
iH∗
σ⊬A□and hence by Corollary 3.19,iH∗
σ⊬(A□)−. This, by Theorem 3.6, implies iH∗
σ⊬((A□)−)∗
and hence iH∗
σ⊬(A□)+, and a fortiori, iGLC ⊬(A□)+. Hence by Theorem 4.1, there exists some
Σ1-substitution σ, such that HA ⊬σHA((A□)+). Hence by Lemma 3.7.1, HA ⊬σHA(A□)and by
Lemma 2.7,HA∗⊬σHA∗(A).2
Corollary 4.11. For any modal proposition A,iH∗
σ⊢Ai iHσ⊢A2.
Proof. By Theorems 4.9 and 4.10 and Lemma 2.7.2
Corollary 4.12. iH∗
σis decidable.
Proof. A proof can be given either with inspections in the proof of the Completeness Theorem (4.10)
or by using the decidability of iHσ[2] and Corollary 4.11.2
Open problems
1. The statement of Corollary 4.11 is purely propositional. However, our proof of this corollary is
based on Theorem 4.10, that has arithmetical theme. A tempting problem is to nd a direct
propositional proof for this corollary. Then we can derive Theorem 4.10.
2. We conjecture that the full provability logic of HA∗is the logic iH∗, axiomatized as follows
iH∗:= iGL +CP +{2A→2B:A▶∗
αB},
in which the relation ▶∗
αis dened as the smallest relation satisfying:
• A1. If iK4 ⊢A→B, then A▶∗
αB,
• A2. If A▶∗
αBand B▶∗
αC, then A▶∗
αC,
• A3. If C▶∗
αAand C▶∗
αB, then C▶∗
αA∧B,
• A4. If A▶∗
αB, then 2A▶∗
α2B,
• B1. If A▶∗
αCand B▶∗
αC, then A∨B▶∗
αC,
• B2. Let Xbe a set of implications, B:= Xand A:= B→C. Also assume that
Z:= {E|E→F∈X}∪{C}. Then A∧□B▶∗
α{B}Z,
• B3. If A▶∗
αB, then 2C→A▶∗
α2C→B.
16

The notation {A}(B), for modal propositions Aand B, is dened inductively:
•{A}(2B) = 2Band {A}(⊥) = ⊥.
•{A}(B1◦B2) = {A}(B1)◦ {A}(B2), for ◦ ∈ {∨,∧},
•{A}(B) = A→Bfor all of the other cases, i.e. when Bis atomic variable or implication.
And {A}Γ, for a set Γof modal propositions, is dened as B∈Γ{A}(B).
References
[1] M. Ardeshir and M. Mojtahedi, Reduction of provability logics to Σ1-provability logics, Logic
Journal of IGPL 23 (2015), no. 5, 842–847.
[2] , The Σ1-Provability Logic of HA, Annals of Pure and Applied Logic 169 (2018), no. 10,
997–1043.
[3] S. Artemov and L. Beklemishev, Provability logic, in Handbook of Philosophical Logic (D. Gab-
bay and F. Guenthner, eds.), vol. 13, Springer, 2nd ed., 2004, pp. 189–360.
[4] H. Friedman, The disjunction property implies the numerical existence property, Proc. Nat.
Acad. Sci. U.S.A. 72 (1975), no. 8, 2877–2878. MR 0379141 (52 #47)
[5] K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Sys-
teme I, Monatsh. Math. Phys. 38 (1931), no. 1, 173–198.
[6] , Eine interpretation des intuitionistischen aussagenkalkuls, Ergebnisse eines mathema-
tischen Kolloquiums 4(1933), 39–40, English translation in: S. Feferman etal., editors, Kurt
Gödel Collected Works, Vol. 1, pages 301-303. Oxford University Press, 1995.
[7] R. Iemho, Provability logic and admissible rules, Ph.D. thesis, University of Amsterdam, 2001.
[8] D. Leivant, Absoluteness in intuitionistic logic, Ph.D. thesis, University of Amsterdam, 1975.
[9] , Absoluteness of intuitionistic logic, Mathematical Centre Tracts, vol. 73, Mathematisch
Centrum, Amsterdam, 1979.
[10] M. Löb, Solution of a Problem of Leon Henkin, Journal of Symbolic Logic 20 (1955), no. 2,
115–118.
[11] J. Myhill, A note on indicator-functions, Proceedings of the AMS 39 (1973), 181–183.
[12] R. M. Solovay, Provability interpretations of modal logic, Israel J. Math. 25 (1976), no. 3-4,
287–304.
[13] A. S. Troelstra and D. van Dalen, Constructivism in mathematics. Vol. I, Studies in Logic and
the Foundations of Mathematics, vol. 121, North-Holland Publishing Co., Amsterdam, 1988,
An introduction.
[14] A. Visser, Aspects of diagonalization and provability, Ph.D. thesis, Utrecht University, 1981.
[15] , On the completeness principle: a study of provability in Heyting’s arithmetic and
extensions, Ann. Math. Logic 22 (1982), no. 3, 263–295.
[16] , Substitutions of Σ0
1sentences: explorations between intuitionistic propositional logic and
intuitionistic arithmetic, Ann. Pure Appl. Logic 114 (2002), no. 1-3, 227–271, Commemorative
Symposium Dedicated to Anne S. Troelstra (Noordwijkerhout, 1999).
17

[17] A. Visser, J. van Benthem, D. de Jongh, and G. R. R. de Lavalette, NNIL, a study in intu-
itionistic propositional logic, Modal logic and process algebra (Amsterdam, 1994), CSLI Lecture
Notes, vol. 53, CSLI Publ., Stanford, CA, 1995, pp. 289–326.
[18] A. Visser and J. Zoethout, Provability Logic and the Completeness Principle, ArXiv e-prints
(2018).
18