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arXiv:2403.19525v2 [math.LO] 29 Mar 2024
Projectivity meets Uniform Post-Interpolant:
Classical and Intuitionistic Logic
Mojtaba Mojtahedi 1Konstantinos Papafilippou 2
Ghent University
Abstract
We examine the interplay between projectivity (in the sense that was introduced by
S. Ghilardi) and uniform post-interpolant for the classical and intuitionistic proposi-
tional logic. More precisely, we explore whether a projective substitution of a formula
is equivalent to its uniform post-interpolant, assuming the substitution leaves the vari-
ables of the interpolant unchanged. We show that in classical logic, this holds for all
formulas. Although such a nice property is missing in intuitionistic logic, we provide
Kripke semantical characterisation for propositions with this property.
As a main application of this, we show that the unification type of some extensions
of intuitionistic logic are finitary. In the end, we study admissibility for intuitionistic
logic, relative to some sets of formulae.
The first author of this paper recently considered a particular case of this relativised
admissibility and found it useful in characterising the provability logic of Heyting
Arithmetic.
Keywords: Unification, Projectivity, Admissible Rules, Intuitionistic Logic,
Uniform Interpolation.
1 Introduction
The notion of projectivity for propositional logics, first introduced by S. Ghi-
lardi [6,7,8]. Roughly speaking, a formula Ais pro jective if there is a substi-
tution θwhich unifies A, i.e. ⊢θ(A), and A⊢θ(B)↔Bfor every formula B.
The importance of projective formulas is mainly due to the following observa-
tion: any projective formula, has a single unifier which is more general than
all other unifiers. Then, by approximating formulas with pro jective ones, we
may fully describe all unifiers of a given formula [7,8,4,1]. Having this strong
tool in hand, we may then characterise admissible rules 3[10,11,12,5,9,18] for
a logic. The situation for classical logic is simple: Every satisfiable formula is
1Email: mojtahedy@gmail.com. This work is partially funded by FWO grant G0F8421N
and BOF grant BOF.STG.2022.0042.01.
2Email: Konstantinos.Papafilippou@uGent.be. Funded by the FWO-FWF Lead Agency
grant G030620N (FWO)/I4513N (FWF) and by the SNSF–FWO Lead Agency Grant
200021L 196176/G0E2121N.
3An inference rule A/B is admissible if every unifier of Ais also a unifier of B.
2 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
also projective. Hence no approximation is required and also no non-derivable
admissible rule exists (A/B is admissible iff A→Bis derivable).
Additionally, we may consider propositional language with two sorts of
atomics: variables var and parameters (constants) par. Variables are intended
to be substituted, while parameters are considered as constants and are never
substituted. Again one may wonder about the previous questions of unification
[4] and admissibility [14,13] for this two-sorted language. But this time, even
for classical logic, not all satisfiable formulas are projective. For example the
formula A:= p∧xfor atomic variable xand parameter pis satisfiable, while
it is not unifiable. This will rule out Abeing pro jective only because of it
not being unifiable. To mend this, we define the notion of E-projectivity by
replacing condition ⊢θ(A) by ⊢θ(A)↔Ein above definition. This time in
our example, the formula Ais p-projective. In this paper, we first observe that
Acan only be E-projective for a unique E(Lemma 2.1), which is annotated
as A∗. Then we show (Theorem 3.5) that all A’s are A∗-projective in classical
logic. For intuitionistic logic, we find a Kripke semantical characterisation for
those A’s that are A∗-pro jective (Theorem 4.2).
On the other hand, we have the well-studied notion of the uniform post-
interpolant. Given a formula A, the uniform post-interpolant of Awith respect
to par, is the strongest formula Bin the variable-free language such that A→B
holds. Of course, it might be the case that such a strongest formula does not
exist at all. Fortunately, several logics indeed have uniform post-interpolants,
namely intuitionistic logic [20] and all locally tabular (finite) logics like clas-
sical logic. Interestingly, it turns out (Theorem 2.2) that A∗is equal to the
uniform post-interpolant of A. As an application, we show that extensions of
intuitionistic logic with variable-free formulas have a finitary unification type.
More precisely, in the mentioned logics, we show that every unifiable formula
has a finite complete set of unifiers, i.e. a finite set of unifiers such that every
unifier is less general than at least one of them. In the end, we observe that ad-
missibility relative to the set of all variable-free formulas is trivial: A∼par B4
iff ⊢A→Bfor the intuitionistic logic.
Motivations for the study of relativised projectivity and admissibility come
from their applications in the study of the provability logic of HA 5[15]. More
precisely, the first author, studied [16] projectivity and admissibility relative to
the set NNIL 6and used it in a crucial way [15] to axiomatize the provability
logic of HA. Interestingly, [3,2] have also found relative unification beneficiary
for their study. Nevertheless, they consider the non-parametric language. The
main aim of this paper is to look at this relativised notion of projectivity and
admissibility, in some more general viewpoint. In this direction, Theorem 5.6
generalises the previous result in [16] from NNIL to all finite sets of formulas
4A∼
par Bis defined as follows: for every variable-free Eand substitution θsuch that
⊢E→θ(A) we have ⊢E→θ(B).
5Heyting Arithmetic
6NNIL is the set of formulas with No Nested Implications on the Left [21].
Mojtahedi and Papafilippou 3
Γ. However, it appears that there is no straightforward generalisation for the
characterisation of admissibility relative to NNIL.
2 Definitions and basic facts
Language
In the sequel, we assume that par and var are two sets of parameters and
variables, respectively. Unless otherwise stated, we assume that both sets var
and par are infinite. Then the propositional language L, is the set of all Boolean
combinations of atomic propositions atom := var ∪par. Boolean connectives are
⊥,∧,∨and →. Note that all connectives are binary, except for falsity ⊥which
is nullary, i.e. an operator without argument. Then ⊤,¬Aand A↔Bare
shorthand for ⊥ → ⊥,A→ ⊥ and (A→B)∧(B→A), respectively. Sets of
formulas are indicated by Γ, ∆, Π, Xand Y. We use x, y, . . . and p, q, . . . and
a, b, . . . and A, B, C, D, E, F, . . . as meta-variables for variables, parameters,
and formulas, respectively. Optionally, we may also use subscripts for them.
Let L(X) indicate all Boolean combinations of propositions in X. Hence, for
example, L(var ∪par) is equal to L. Moreover, Γpar indicates the set Γ ∩ L(par).
We use Γ ⊢iAto indicate the intuitionistic derivability of Afrom the set Γ. We
also use Γ ⊢cAfor the classical derivability. Whenever we state a definition
for both intuitionistic and classical cases, we simply use ⊢without superscript.
Apar-extension of a logic is an extension of that logic by adding some E∈
L(par) to its set of axioms. Note that this additional axiom is not considered
as a schema.
Kripke models
A Kripke model is a tuple K= (W, 4, V ), whose frame (W, 4), is a reflexive
transitive ordering. We write w≺uto w4uand w6=u. Moreover, we use
<and ≻as inverses of the relations 4and ≺, respectively. A root of K, is the
minimum element in (W, 4) if it exists, and we say that Kis rooted if it has
a root. We say that Kis finite if Wis so. In this paper, it is assumed that
all Kripke models are finite and rooted. We use the notation K, w Afor the
validity / forcing of Aat the node win the model K.KAis defined as
∀w∈WK, w A. Given w∈W, we define Kwas the restriction of Kto all
nodes that are accessible from w. Furthermore, K(w) indicates the valuation
of Kat w, i.e. the set of all atomics asuch that w V a.
Substitutions
Substitutions are functions θ:L −→ L with the following properties: (they are
indicated by lowercase Greek letters θ, τ, λ, γ , . . .)
•θcommutes with connectives. This means that for binary connective ◦we
have θ(A◦B) = θ(A)◦θ(B) and for ⊥we have θ(⊥) = ⊥.
•θ(p) = pfor every p∈par.
Given two substitutions θand γ, we define θγ as the composition of θand γ:
θγ(a) := θ(γ(a)). Given a substitution θand the Kripke model K= (W, 4, V ),
define θ∗(K) as a Kripke model with the same frame of K, and with valuation
4 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
V′as follows: w V ′aiff K, w θ(a). Then it can be easily observed that
K, w θ(A) iff θ∗(K), w A, for every w∈Wand A∈ L. If no confusion is
likely, 7we may simply use θ(K) instead of θ∗(K).
Given two substitutions θand γ, we say that θis more general than γ(γ
less general than θ), annotated as γ≤θ, if there is a substitution λsuch that
⊢γ(x)↔λθ(x).
Ralative projectivity
An A-identity is some θsuch that
∀a∈atom A⊢θ(a)↔a. (1)
Let Ebe a proposition in the parametric language L(par). An E-fier of A
is some θsuch that ⊢θ(A)↔E. An E-fier for some E∈Γpar := Γ ∩ L(par)
is also called a Γ-fier. Note that here Γ is not necessarily a subset of L(par).
⊤-fier and L-fier are also called unifier and parametrifier, respectively. We say
that Ais E-projective (Γ-projective), if there is some θwhich is A-identity and
E-fier (Γ-fier) of A. In this case we say that θis a projective E-fier (Γ-fier) of
Aand Eis the projection of A. As we will see in Lemma 2.1, the projection is
unique, modulo provable equivalence. So we use the notation A∗for its unique
projection, if it exists. Then L-pro jectivity is also called par-pro jectivity. The
well-known notion of projectivity, as introduced by S. Ghilardi [6,7], coincided
with ⊤-projectivity in this general setting.
Given A∈ L and a set Θ of substitutions, we say that Θ is a complete set
of unifiers of A, if the following holds: (1) every θ∈Θ is a unifier of A, (2)
every unifier of Ais less general than some θ∈Θ. We say that the unification
type of a logic is:
•unitary, if every unifiable formula of the logic, has a singleton complete
set of unifiers,
•finitary, if every unifiable formula of the logic, has a finite complete set of
unifiers.
Note that in our definition, if the unification type of a logic is unitary, it is also
finitary.
All the above definitions of pro jectivity rely on a background logic in the
context. More precisely, we have two different notions of Γ-projectivity: one
for classical logic and one for intuitionistic logic. To simplify the notation, we
always hide the dependency on the logic. This will not cause much confusion,
since it is quite clear from the context which logic we are talking about. To be
more precise, in this section, we provide all statements for both classical and
intuitionistic logic. Later in section 3, classical logic is our background logic,
whereas for the rest of the paper, the background logic is intuitionistic logic.
Lemma 2.1 (Uniqueness of projections) Projection is unique, modulo
provable equivalence.
7Confusions are due to this fact that (θγ)∗=γ∗θ∗.
Mojtahedi and Papafilippou 5
Proof. Let ⊢θi(A)↔Biand A⊢θi(x)↔xand Bi∈ L(par) for i= 1,2
and every x∈var. Then A⊢θ1(A)↔Aand thus ⊢A→B1. Therefore, we
have ⊢θ2(A→B1) and since θ2is identity over the language L(par), we may
deduce ⊢θ2(A)→B1, and thus ⊢B2→B1. By a similar argument we may
prove ⊢B1→B2, and thus ⊢B1↔B2.✷
Uniform post-interpolant and upward approximations
Given a formula A, we say that Bis its uniform post-interpolant with respect
to par, if we have (1) B∈ L(par), (2) ⊢A→Band (3) for every C∈ L(par)
s.t. ⊢A→Cwe have ⊢B→C. It can be easily proved that such a Bis unique
(modulo provable equivalence), if it exists. Hence, we use the notation ⌈A⌉for
the uniform post-interpolant of Awith respect to par. It is a well-known fact
that the uniform post-interpolant for classical and intuitionistic logic exists. In
the following theorem, we will show that the unique projection is equal to the
uniform post-interpolant with respect to the set par of atomics.
Theorem 2.2 For every par-projective A, we have A∗=⌈A⌉.
Proof. Let A⊢θ(x)↔xfor every x∈var and ⊢θ(A)↔A∗and A∗∈
L(par). We will check that all 3 required conditions for it to be a uniform
post-interpolant hold. (1) Trivially, we have A∗∈ L(par). (2) By the argument
provided in the proof of Lemma 2.1, we have ⊢A→A∗. (3) Let C∈ L(par)
s.t. ⊢A→C. Hence ⊢θ(A)→θ(C) and since C∈ L(par) we get θ(C) = C.
Thus, ⊢A∗→C.✷
Given A∈ L, one might wonder if par-projectivity of Acould be defined
using the standard notion of projectivity. We will prove that par-projectivity of
Ais equivalent to projectivity of A↔ ⌈A⌉, for both classical and intuitionistic
logic. One direction is easy and stated in the following lemma. However, for
the other direction, we provide separate proofs for intuitionistic and classical
logic. For intuitionistic logic, we take advantage of the semantic characterisa-
tion of par-projectivity (see Corollary 4.3). For classical logic, Lemma 3.3 and
Lemma 3.4 imply that A↔ ⌈A⌉is projective for every A!
Lemma 2.3 Every projective unifier of A↔ ⌈A⌉is also a projective paramet-
rifier of A. Therefore, projectivity of A↔ ⌈A⌉implies par-projectivity of A.
Proof. Let θbe a projective unifier of A↔ ⌈A⌉. Hence ⊢θ(A↔ ⌈A⌉)
and ⊢θ(A)↔ ⌈A⌉. This shows that θis a parametrifier of A. Also, for
every variable xwe have A↔ ⌈A⌉ ⊢ θ(x)↔x. Since ⊢A→ ⌈A⌉, we have
⌈A⌉ → A⊢θ(x)↔x. Hence, a fortiori, we have A⊢θ(x)↔x. Thus, θis a
projective parametrifier of A.✷
Theorem 2.4 Every projective unifier of A↔ ⌈A⌉is a most general ⌈A⌉-fier
of A.
Proof. Let θbe a pro jective unifier of A↔ ⌈A⌉. It is obvious that θis also
a⌈A⌉-fier of A. To show that it is a most general ⌈A⌉-fier of A, let γbe some
⌈A⌉-fier of A. Hence γis also a unifier of A↔ ⌈A⌉. Since projective unifiers
are most general unifiers, we may infer that γis less general than θ.✷
6 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
3L-Projectivity for Classical Logic
In this section we will prove that in Classical Logic, every A∈ L is par-
projective. First, let us see some definitions.
We say that an atomic xis positive (negative) in Aif for all interpretations I
and Jsuch that I(x)≤J(x) and I(a) = J(a) elsewhere, we have I(A)≤J(A)
(I(A)≥J(A)). We use the order ≤on truth-falsity values so that falsity is
less than truth.
Lemma 3.1 A variable xis positive in Aiff ⊢cA→A[x:⊤].
Proof. Left-to-right: Let xbe positive in Aand I|=Aseeking to show that
I|=A[x:⊤]. Let Jbe the interpretation that is always equal to Iexcept for
J(x) := ⊤. Then obviously I(x)≤J(x) and thus I(A)≤J(A). Since I|=A
we get J|=Aand thus I|=A[x:⊤].
Right-to-left: Let ⊢cA→A[x:⊤], I(x)< J (x) (note that this implies
J|=x) and I(a) = J(a) elsewhere. Assume I|=Ato show J|=A. Since
I|=A→A[x:⊤] we get I|=A[x:⊤], which is just J|=A.✷
Given A∈ L and some x∈var, define the substitution θA
xas follows:
θA
x(y) := (y:y6=x
A∧y:y=x
Lemma 3.2 Given A, the variable xis positive in θA
x(A). Furthermore, if y
is positive in A, then it is also positive in θA
x(A).
Proof. Let I6|=xand J|=xand I(a) = J(a) for every atomic a6=x.
Moreover, assume that I|=θA
x(A), seeking to show J|=θA
x(A). If I(A) =
J(A), then obviously I(θA
x(A)) = J(θA
x(A)) and we are done. Hence, we have
the following cases:
•I|=Aand J6|=A. Then we have I(θA
x(x)) = J(θA
x(x)) and thus
I(θA
x(A)) = J(θA
x(A)).
•I6|=Aand J|=A. Then we have J(θA
x(x)) = J(x) and thus J(θA
x(A)) =
J(A). This means that J|=θA
x(A).
At the end, we must show that if yis positive in A, then it is also positive in
θA
x(A). Let I6|=yand J|=yand I(a) = J(a) for every atomic a6=y. Moreover,
assume that I|=θA
x(A), seeking to show J|=θA
x(A). If I(A) = J(A), then
obviously I(θA
x(A)) = J(θA
x(A)) and we are done. So we have the following
cases:
•I|=Aand J6|=A. This case contradicts the positiveness of yin A.
•I6|=Aand J|=A. Then we have J(θA
x(x)) = J(x) and thus J(θA
x(A)) =
J(A). This means that J|=θA
x(A).
✷
Lemma 3.3 The following items are equivalent:
(i) Ais unifiable.
Mojtahedi and Papafilippou 7
(ii) Ais projective.
(iii) ⌈A⌉=⊤.
Proof. (i)⇒(ii): Let ⊢τ(A) and define the substitution ǫτas follows:
ǫτ(x) := (A∧x)∨(¬A∧τ(x)) for every variable x.
It is obvious that ǫτis A-identity. Moreover one may easily prove that both
A⊢cǫτ(A) and ¬(A)⊢cǫτ(A) and thus ⊢cǫτ(A).
(ii)⇒(iii): This holds by Theorem 2.2.
(iii)⇒(i): We use induction on nA, the number of non-positive variables oc-
curring in A. First, observe that if nA= 0, then for the substitution τ
which replaces all variables with ⊤, by Lemma 3.1 we have ⊢A→τ(A) and
τ(A)∈ L(par). Then, since ⌈A⌉=⊤, we get ⊢cτ(A), and thus Ais unifiable.
As induction hypothesis, assume that every formula Bwith nB=nA−1 and
⌈B⌉=⊤is unifiable. Take some positive variable xin Aand let B:= θA
x(A).
Since A⊢θA
x(y)↔yfor every y, we get ⊢A→Band thus ⌈B⌉=⊤.
Moreover, Lemma 3.2 implies that nB=nA−1 and hence by the induction
hypothesis, there is a substitution γwhich unifies B. Therefore, γ◦θA
xunifies
A, as desired. ✷
Lemma 3.4 ⌈⌈A⌉ → A⌉=⊤.
Proof. Let E∈ L(par) be such that ⊢c(⌈A⌉ → A)→E. It suffices to show
that ⊢cE. By assumption, we have ⊢cA→E, and thus ⊢c⌈A⌉ → E. On the
other hand, ⊢c(⌈A⌉ → A)→Ealso implies ⊢c¬⌈A⌉ → E. Thus ⊢cE.✷
Theorem 3.5 All formulas are par-projective. Moreover, ⌈A⌉ → Afor every
Ais projective.
Proof. Given A∈ L, let B:= ⌈A⌉ → A. Lemma 3.4 implies that ⌈B⌉=⊤.
Hence Lemma 3.3 implies that Bis pro jective. Thus, Lemma 2.3 gives the
desired result. ✷
Recall from section 2 that a par-extension of a logic is an extension by an
L(par)-formula.
Corollary 3.6 The unification type of par-extensions of classical logic is uni-
tary.
Proof. Let A∈ L and E∈ L(par) and assume that B:= E→Ais unifiable,
seeking to find a single unifier of Bthat is more general than all of its unifiers.
Since Bis unifiable, there is some γsuch that ⊢cγ(B), and hence ⌈B⌉=⊤.
Then Lemma 3.3 implies that Bis pro jective. Let θbe its projective unifier.
We claim that θis also its most general unifier. Take any unifier τof B. Since
B⊢cθ(x)↔x, we get τ(B)⊢cτθ(x)↔τ(x) and thus ⊢cτ θ(x)↔τ(x). This
shows that τis less general than θ.✷
8 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
4Γ-Projectivity for Intuitionistic Logic
As we have shown in previous section, classical logic is well-behaving for par-
projectivity in the following sense: all formulas are classically par-projective
(Theorem 3.5). Nevertheless, such a nice behaviour does not hold in intuition-
istic logic. For example the formula A:= x∨ ¬x, is not par-projective by the
following argument. First, observe that ⌈A⌉=⊤. Hence, a parametrifier of
Ashould be a unifier. By the disjunction property for intuitionistic logic, A
has only two unifiers: those who replace the variable xby truth, and by falsity
respectively; namely θ1(x) := ⊤and θ2(x) := ⊥. This means that Adoes not
have a most general unifier. On the other hand, if Awere ⊤-projective, its
projective unifier should be a most general unifier, a contradiction with our
previous observation. Thus, Ais not par-projective.
Given that not all formulas are par-pro jective, the question arises on which
formulas are par-projective. In this section, we will characterise par-projectivity,
via Kripke semantics. It is a variant of [7] in which it is proven that pro jec-
tivity is equivalent to extendability for a given formula in the non-parametric
language. 8Let us first look at some definitions.
Given X⊆atom, we say that K1is a X-variant of K2, if (1) they share the
same frame, (2) the evaluations at every node other than the root are the same,
and (3) the evaluations at the root for all a∈Xare the same. An ∅-variant is
simply called a variant. We say that Ais weakly valid in K, notation K
−A,
if Ais valid on every node other than the root.
Definition 4.1 We say that Ais B-extendable, if ⊢iA→Band for every
Kripke model Kwith K
−Aand KB, there is some par-variant K′of K
such that K′A. Additionally, for a given set Γ of formulas, we say that Ais
Γ-extendable, if there is some B∈Γpar such that Ais B-extendable.
Given A∈ L and a set of variables X, we define θX
Aas follows:
θX
A(x) := (A→x:x∈X
A∧x:x6∈ X
Theorem 4.2 E-extendability and E-projectivity are equivalent notions for ev-
ery E∈ L(par).
Proof. Right-to-left: Let θbe a pro jective E-fier of A, i.e. A⊢θ(x)↔xfor
every x∈var and ⊢θ(A)↔E. We will show that Ais E-extendable. Let
K
−Awith KEand take K′:= θ(K). Then it is not difficult to observe
that K′is indeed a par-variant of Kand K′A.
Left-to-Right: Assume that Ais E-extendable. Hence ⊢iA→Eand for every
Kwith KEand K
−A, there is a par-variant K′of Ksuch that K′A.
We will construct a substitution θ, which is a projective E-fier of A. Our
8Roughly speaking, an extendable formula is a formula whose Kripke models could be
extended from below.
Mojtahedi and Papafilippou 9
construction is by iterated compositions of substitutions θX
A, and is the same
as the one first introduced by S. Ghilardi in [7].
Let X:= {x1, x2,...,xm}be the set of all variables occurring in A. Con-
sider an ordering of the powerset of X, namely P(X) = {X0, X1,...,Xk}such
that Xi⊆Xj⇒i≥j. Then define θA:= θX0
A◦θX1
A◦...◦θXk
Awhich is
an A-projection, since it is composition of A-projections. We claim that θAis
a projective E-fier of A. Since θAis a cumulative composition of A-identity
substitutions, and A-identity substitutions are closed under compositions, we
may infer that θAis also A-identity. So, it only remains to show that θAis
E-fier of A. By induction on the height nof the Kripke model K, we prove
KθA(A)↔E.
As induction hypothesis, let K
−θA(A)↔E. Given a node wof K, let
Kwbe the restriction of Kto all nodes that are accessible from w, including w
itself. Then, by the induction hypothesis, we have KwθA(A)↔Efor every
wother than the root. If K1E, since ⊢iθA(A)→E, we get KθA(A)↔E.
So, we may assume that KE. Therefore, by E-extendability of A, there is a
par-variant K′of Ksuch that K′A. Then Lemma 4.4 implies that KθA(A)
and so KθA(A)↔E.✷
Corollary 4.3 par-projectivity of Ais equivalent to projectivity of ⌈A⌉ → A.
Proof. One direction has already been proved in Lemma 2.3. For the other
way around, assume that Ais par-projective and, hence, ⌈A⌉-projective. Then
by Theorem 4.2, Ais also ⌈A⌉-extendable. Again, by Theorem 4.2, it is enough
to show that ⌈A⌉ → Ais ⊤-extendable. So let K
−⌈A⌉ → A, seeking to find
some par-variant K′of Ksuch that K′⌈A⌉ → A. If K1⌈A⌉, take K′:= K
and we are done. Otherwise, by ⌈A⌉-extendability of A, there is some par-
variant K′of Ksuch that K′A, as desired. ✷
Question 1: With the above corollary, we have a nice characterisation of
projectivity of ⌈A⌉ → A. We may wonder about (characterisation of) its dual
notion, namely pro jectivity of A→ ⌊A⌋, in which ⌊A⌋indicates the uniform
pre-interpolant of A.
Lemma 4.4 Let K′θA(A)is a par-variant of Kand K
−θA(A). Then
KθA(A).
Proof. We first make the observation that for any model K 6Athen for any
set Yand any x∈Y∩var,θY
A(K)xiff x∈Y. Now for every i≤klet
θi
A=θX0
A◦...◦θXi
A, which are all A-identities. As such, if Kθi
A(A) for some
i≤k, then Kθj
A(A) for all i≤j≤k. Hence, it is sufficient to show that if
K
−θi
A(A) and K′θj
A(A) with i≤j≤k, then Kθr
A(A) for some r≤k.
We prove this by induction on the height nof K.
•n= 0. Let Ybe the valuation of θi
A(K′) at its root and Y∩var =Xjfor
some j. If θj−1
A(K)A, then this step is done. If not, then for every variable
x∈X,θj
A(K)xiff x∈Y, and so θj
A(K)A.
•Assume the induction claim for every model of height k < n and let Kbe a
10 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
model of height nsuch that K
−θi
A(A) with ibeing least with this property.
Let jbe least for which there exists a par-variant K′of Kwith K′θj
A(A),
then j≥ias K′θj
A(A) implies that K′
−θj
A(A) hence K
−θj
A(A) and
so i≤j.
Let Ybe the valuation of θj
A(K′) at its root and let rbe such that Y∩var =
Xr. If r < j, then since θj−1
A(K′)6A, we get for every variable x∈Xthat
θj
A(K′)xiff x∈Xj, a contradiction. Hence Y∩var =Xrfor some r≥j.
If j=ror θr−1
A(K)Athen θr
A(K)A. Otherwise, since θr−1
A(K)6Awe
once more have for every variable x∈Xthat θr
A(K)xiff x∈Y∩var and
so θr
A(K)A.
✷
5 Unification type of extensions of intuitionistic logic
Thanks to [7], we know that the unification type of intuitionistic logic is fini-
tary. In this section, we show in Theorem 5.7 that the unification type of
par-extensions of intuitionistic logic is also finitary.
Convention. All over this section, we assume that atom, the set of atomic
formulas is finite.
We say that two Kripke models are par-equivalent (K1≡par K2), if they share
the same frame and for every node wand every p∈par we have K1, w piff
K2, w p.
Given a class Mof Kripke models, PMindicates the disjoint union of
Kripke models in Mwith a fresh root below with empty valuation.
Definition 5.1 A class of Kripke models Kis called B-extendable, if: KB
and for every finite (including empty) M⊆K, if a variant Kof PMforces
B, then there is a par-variant of Kwhich belongs to K. We say that Kis
Γ-extendable, if there is some B∈Γpar such that Kis B-extendable. Then
⊤-extendability is also called extendability. It is not difficult to observe that A
is B-extendable (Definition 4.1) iff its class of models is so.
Let c→(A) indicate the maximum number of nested implications in A. More
precisely,
•c→(A) = 0 for A∈atom ∪ {⊥}.
•c→(A∧B) = c→(A∨B) = max{c→(A), c→(B)}.
•c→(A→B) := 1 + max{c→(A), c→(B)}.
In what follows, we will make use of the notions of bounded bisimilarity and
bounded sub-bisimilarity. We define these notions inductively for given models
Kand K′with respective roots w0and w′
0:
•K⇋0K′iff K(w0) = K′(w′
0).
•K⇋n+1 K′iff
·Forth: ∀w∈ K ∃ w′∈ K′Kw⇋nK′
w′and
·Back: ∀w′∈ K′∃w∈ K Kw⇋nK′
w′.
Mojtahedi and Papafilippou 11
•K ≤0K′iff K(w0)⊇ K′(w′
0).
•K ≤n+1 K′iff
·Forth: ∀w∈ K ∃ w′∈ K′Kw⇋nK′
w′.
Lemma 5.2 Let K ≤nK′and c→(A)≤n. If K′A, then KA.
Proof. See [7]. ✷
Then for a class of Kripke models K, we define
hKin:= {K :∃K′∈K(K ≤nK′)}.
We call a class Kof Kripke models stable, iff for every K ∈ Kand every node
wof K,Kw∈K. Furthermore, we say that Kis ≤n-downward closed, if
K′≤nK ∈ Kimplies K′∈K. Observe that if hKin=Kthen Kis stable
and ≤n-downward closed.
Lemma 5.3 Let Kbe a stable class of E-extendable Kripke models for some
E∈ L(par). Then for every n > c→(E),hKinis also stable and E-extendable.
Proof. By our above observation, we only need to show that hKinis E-
extendable. Since KEand c→(E)< n, we get from Lemma 5.2 that
hKinE. Let Mbe a finite subset of hKin; then due to the stability of K,
there is a minimal set M′⊆Ksuch that for every M ∈ Mand every w∈ M
there is a model M′∈M′such that Mw⇋n−1M′. W.l.o.g. M′will be finite
as there can only be at most finitely many non-⇋n−1Kripke models. Assume
that there is a variant Kof PMsuch that KEand let K′be a variant of
PM′with its root evaluated the same as K. We show that K⇋n−1K′.
•By our assumption for M′for every wother than the root, there is some
M′∈M′such that Kw⇋n−2M′and vice versa.
•The roots are 0-bisimilar to each other and, by the above, K⇋n−1K′.
Hence K′E, so there is a par-variant K′∈Kof K′. Consider the par-variant
Kof Kwhose root is evaluated the same as in K′. By omitting the backward
direction in the proof of K⇋n−1K′, one can easily show that K ≤nK′.✷
Corollary 5.4 Let Γbe finite. Then there is msuch that for every stable class
Kof Γ-extendable Kripke models and every n > m,hKinis also stable and
Γ-extendable.
Proof. Let m=max{c→(E) : E∈Γpar}and apply Lemma 5.3. ✷
Lemma 5.5 A class Kof Kripke models is equal to the class of all models of
some Awith c→(A)≤niff Kis ≤n-downward closed.
Proof. See [7]. ✷
Theorem 5.6 Given a finite Γand A, there is a finite set Πof Γ-projective
formulas with the following properties:
(i) ⊢iWΠ→A.
(ii) A∼ΓΠ. (see section 6 for the definition of ∼Γ.)
12 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
Proof. Let m0:= c→(A) and m1be the number mgiven from Corollary 5.4
and n:= max{m0, m1}. For a given E∈Γpar and substitution θwith ⊢iE→
θ(A), we find some Bθ
E∈ L such that:
(i) c→(Bθ
E)≤n.
(ii) Bθ
Eis E-projective.
(iii) ⊢iBθ
E→A.
(iv) ⊢iE→θ(Bθ
E).
Let us first see why this finishes the proof of this theorem. Define
Π := {Bθ
E:E∈Γpar &⊢iE→θ(A)}.
Since atom is finite, by (i) we have the finiteness of Π. Γ-projectivity of elements
of Π is guaranteed by (ii). Item (iii) implies ⊢iWΠ→A. Finally, by item (iv)
we have A∼ΓΠ.
So, let us go back to find Bθ
Ewith the listed properties. Define
K:= {θ(K) : KE}.
It is easy to observe that Kis stable and E-extendable. Then Lemma 5.3
implies that hKinis also stable and E-extendable. Lemma 5.5 implies that
there is some formula Bθ
Ewith c→(Bθ
E)≤nsuch that hKin={K :KBθ
E}.
Hence (i) is satisfied. Since hKinis extendable, Theorem 4.2 implies the
validity of item (ii). To check the validity of item (iii), we argue semantically.
Let KBθ
E. Therefore, K ≤nK′for some K′∈K. Since K′∈K, we
have K′=θ(K′′) for some K′′ E. Since K′′ Eand E∈ L(par), we also
have θ(K′′)E. On the other hand, since ⊢iE→θ(A), we have K′′ E→
θ(A). Hence K′′ θ(A) and thus θ(K′′ )A. This means K′Aand since
c→(A)≤nand K ≤nK′, Lemma 5.2 implies KA. Thus, ⊢iBθ
E→A. For
the item (iv), let KE. Then θ(K)∈K⊆ hKin. Hence θ(K)Bθ
Eand
thus Kθ(Bθ
E). ✷
Recall from section 2 that a par-extension of a logic is an extension by an
L(par)-formula.
Theorem 5.7 The unification type of par-extensions of intuitionistic logic is
finitary.
Proof. Let A∈ L and E∈ L(par) be given, seeking a finite complete set of
unifiers of Ain the intuitionistic logic extended by E. Take Γ := {E}and apply
Theorem 5.6 to obtain a finite set Π with the properties mentioned. Then it
can be easily verified that the set of projective E-fiers of elements of Π serves
as a finite complete set of unifiers, as desired. ✷
6 Parametric Admissibility: Relative to Γ
Given a logic L, the (multi-conclusion) admissibility relation for Lis defined as
follows:
A∼L∆ iff ∀θ(L⊢θ(A)⇒ ∃ B∈∆L⊢θ(B)).
Mojtahedi and Papafilippou 13
We simply write A∼LBfor A∼L{B}. Given E∈ L(par), we define ∼Eas
the admissibility relation for intuitionistic logic strengthened by an additional
axiom E. Note that since E∈ L(par), this logic is closed under substitutions,
and hence it is a logic indeed. The admissibility for intuitionistic logic ∼IPC
is annotated as ∼in the literature [10] and is known to be decidable [17].
One may easily observe that A∼EBiff (E→A)∼(E→B) for every
E∈ L(par). Thus, by characterising the admissibility relation ∼, we also have
a characterisation of ∼E.
Then we define the Γ-admissibility relation ∼Γ, as the intersection of all
∼E’s with E∈Γpar = Γ ∩par. In other words:
A∼ΓBiff ∀E∈Γpar A∼EB.
This time, the characterisation of ∼Γis not that simple. For instance, the
characterisation for Γ = NNIL, the set of No Nested Implications on the Left,
is the main result of [16]. This result has been used in an essential way in the
proof for characterisation of intuitionistic provability logic [15]. In this paper,
we will dig further and characterise one more salient case: Γ = L, the full
language. It turns out that A∼LBiff ⊢iA→B(Corollary 6.4).
There is yet another binary relation on L, called preservativity [19] and
annotated as A|≈ΓB, which is defined as follows:
A|≈ΓBiff ∀E∈Γ (⊢iE→A=⇒ ⊢iE→B).
Note the difference between the role of Ein ∼Γand |≈Γ. In the first, we quantify
Eover the set Γpar, a subset of Γ, whereas in the second, we quantify it over Γ
itself. Since both definitions are universal over Γ, the following trivially holds:
Remark 6.1 If Γ ⊆∆ then ∼Γ⊇∼∆and |≈Γ⊇ |≈∆.
In the following lemma, we show that Γ-admissibility implies b
Γ-
preservativity, where b
Γ is the set of Γ-projective formulas.
Lemma 6.2 ∼Γ⊆ |≈b
Γ.
Proof. Let E∈b
Γ such that ⊢iE→A, seeking to show ⊢iE→B. Since
E∈b
Γ, there is a substitution θthat projects Eto EΓ∈ L(par). Therefore, by
⊢iE→A, we have ⊢iEΓ→θ(A). Hence, by A∼ΓBwe have ⊢iEΓ→θ(B).
Since ⊢iE→EΓ, we have ⊢iE→θ(B), and thus by the E-projectivity of θ,
we have ⊢iE→B.✷
Apar-substitution is a substitution θsuch that θ(x)∈par for every x∈var.
Then we say that Γ is closed under par-substitutions if for every A∈Γ and
every par-substitution θwe have θ(A)∈Γ.
Lemma 6.3 Given Γclosed under par-substitutions and A∈Γ, the following
are equivalent:
(i) ⊢iA→B.
14 Projectivity meets Uniform Post-Interpolant:Classical and Intuitionistic Logic
(ii) A∼ΓB.
(iii) A|≈b
ΓB.
Proof. The proof of 1 →2 is obvious. 2 →3 holds by Lemma 6.2. We reason
for 3 →1 as follows. Let 0iA→B. Also, assume that X:= {x1,...,xn}in-
cludes all variables occurring in A, and {p1,...,pn}is a set of fresh parameters,
i.e. parameters not appeared in Aand B. Finally, assume that
E:= A∧
n
^
i=1
xi↔piand θ(x) := (pi:x∈Xand x=xifor 1 ≤i≤n
x:x6∈ X
It can be easily verified that
•Eis Γ-projective.
•⊢iE→A.
•0iE→B.
This implies that A6|≈b
ΓB, as desired. ✷
Note that in the proof of the above lemma, we are taking advantage of the
infiniteness of par.
As a result, if we take Γ as the full language L, we have the following:
Corollary 6.4 The following are equivalent:
(i) ⊢iA→B.
(ii) A∼LB.
(iii) A|≈ b
LB.
Given that A∼LBis equivalent to ⊢iA→B, asking for par-projective
approximations of a proposition A(Theorem 5.6), simplifies to the following
question.
Question 2: Given A∈ L, is it possible to find a finite set ΠAof L-pro jective
propositions with
⊢iA↔_ΠA.
By the arguments in this section, we already have a characterisation / decidabil-
ity of ∼Γfor finite Γ and for Γ = L. Moreover, [16] provides a characterisation
for the case Γ = NNIL, the set of No Nested Implications on the Left.9
Question 3: Axiomatize or provide decision algorithm for ∼Γin the following
cases for Γ:
(i) Weakly extendable formulae: We say that Ais weakly extendable if every
Kwith K
−Ahas a variant K′such that K′A.
9Since its appearance in the literature [21], NNIL has shown itself to be of great importance
in the study of intuitionistic logic.
Mojtahedi and Papafilippou 15
(ii) Prime formulae: A formula Ais prime if ⊢iA→(B∨C) implies ⊢iA→B
or ⊢iA→C. Note that every extendable formula is prime, but not
necessarily vice versa. 10
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10 For example the formula ¬p→(q∨r) is prime, while it is not extendable.
