The Non-reflexive Counterpart of Grz
Tadeusz Litak
[email protected]
Birkbeck College / London Knowledge Lab
23–29 Emerald Street
London WC1N 3QS
On this special occassion, I would like to thank professor Ono for all the unforgettable years
spent in the Ishikawa prefecture, for being counted among the LSoOL’s and for everything I
was lucky to bring from Japan, in particular my PhD and my wife.
Domo arigatou gozaimasu. Kokoro kara kansha itashimasu!
Abstract
The paper studies the weak Grzegorczyk logic wGrz. In particular, we discuss
the relationship between Grz, GL and wGrz as an interesting example of the relationship between an extension of T, its irreflexive counterpart and its non-reflexive
counterpart.
Our main object of interest is the weak Grzegorczyk logic or the acyclic logic wGrz.
It is a proper subsystem of both the Löb logic GL and the Grzegorczyk logic Grz. The
corresponding Kripke frames are transitive and Noetherian (containing neither nontrivial cycles nor infinite ascending chains), but neither reflexive nor irreflexive in general. In our terminology, wGrz is the non-reflexive counterpart of Grz and GL is the
irreflexive counterpart of Grz; in some cases, both notions coincide, but not here. We
will see that wGrz:
• is not the intersection of GL and Grz (Theorem 3.1), in fact, it seems much
weaker a system than GL ∩ Grz (Theorem 3.2);
• just like Grz and GL, contains K4 even though the transitivity axiom does not
need to be explicitly postulated (Theorem 2.2);
• has the finite model property (Theorem 2.5, proved before in [1] and in [13]) and
the finite model property of Grz is derivable from the fmp of wGrz;
• is closed under the Gabbay irreflexivity rule (Theorem 2.7). Therefore, the irreflexive counterpart of a given logic cannot always be obtained by closing the
non-reflexive counterpart under the Gabbay rule;
• is topologically complete under the d-interpretation of the modal diamond. However, in marked contrast to both Grz and GL, despite having fmp, wGrz is
not complete with respect to finite topological spaces. Neither with respect to
Alexandroff or metric spaces (Remark 2.8);
1
• has no canonical, strongly complete or even completion admitting extension of
infinite depth, which implies analogous results for NExtGrz and NExtGL (Theorem 3.3, generalizing non-canonicity of wGrz proved in [1]). This holds despite
the fact that wGrz
• has uncountably many pre-finite extensions (Theorem 3.2), in contrast both to
Grz and GL. Therefore, NExtwGrz seems significantly more complicated than
NExtGrz and NExtGL.
In the author’s opinion, these observations show that wGrz is a natural setting to
study properties common to both and Grz and GL. In addition, they allow to pose some
interesting questions concerning the role the assumption of reflexivity (or irreflexivity)
plays in the semantics for a given logic. For this reason, the note begins with Section 1
systematizing relationships between an extension of T and its non-reflexive/irreflexive
counterparts.
The original version of this note was written when the author was not aware of preexisting work on the subject except for the recent paper by Dawar and Otto [8], which
does not mention wGrz, but uses heavily the corresponding class of finite frames.
Nevertheless, K4 ⊕ Grz2 (see Table 1 below)—the same logic as wGrz, as it turns
out—has attracted the attention of the research community. Its finite model property was established first by Amerbauer [1]. The result has been improved upon by
Gabelaia [13], who has showed also that K4 ⊕ Grz2 is topologically complete under
d-interpretation. A very interesting reference on NExtwGrz is [10] extending the famous Blok-Esakia Theorem by showing that this lattice is isomorphic to the lattice of
extensions of the modalized Heyting calculus (mHC). That paper established also that
K4 ⊕ Grz2 = Grz+ (in the notation of the present paper). This made clear that GL
is not the weakest logic into which Grz can be embedded via (·)+ .
As we will see, when wGrz is axiomatized as in the present paper, Esakia’s result
follows immediately from a general observation (clause 12 of Lemma 1.1 below). This
is one of examples showing that it is profitable to discuss the relationship between Grz,
GL and wGrz in the broad setting introduced in Section 1.
Acknowledgements. The bulk of this work has been done while the author was supported by the Japan Science and Technology Agency Jinzai Yosei program, the first
version of the paper has been completed thanks to the support by the Netherlands Organization for Scientific Research (NWO) under grant number 680–50–0613 and at
present, the author is supported by the EPSRC grant EP/F002262/1. The author is
grateful to Guram Bezhanishvili for his criticism and detailed information about the
bibliography of the subject, which lead to a substantial revision of the paper, to the
anonymous referee for his comments and to the editors for their patience and assistance.
1
Preliminaries
We are assuming familiarity with basic notions and tools of modal logic, such as modal
algebras, Kripke frames or general frames (see [6]). For arbitrary W , ∆W := {hx, xi |
2
Table 1: Logics discussed in this paper
T
=K
K4
=K
wGrz = K
Grz2 = K
Grz = K
Triv = K
⊕
⊕
⊕
⊕
⊕
⊕
2p → p
2p → 22p
2+ (2(p → 2p) → p) → p
2(2(p → 2p) → p) → 2p
2(2(p → 2p) → p) → p
p ↔ 2p
wK4 = K ⊕ 2+ p → 22p
S4 = K4 ⊕ T
GL = K ⊕ 2(2p → p) → 2p
A∗ = GL ⊕ 22p → 2(2+ q → p) ∨ 2(2+ p → q)
Ver = K ⊕ 2⊥
Table 2: Semantic operators
R
K(ref lexive)
K(arbitrary)
Λ⊇T
Λ
(·)◦
R ∪ ∆W
−
{F◦ | F ∈ K}
−
Log((Mod(Λ))◦ )
(·)•
R − ∆W
{F• | F ∈ K}
Log((Mod(Λ))• )
(·)◦
−
{F | F◦ ∈ K}
−
Log((Mod(Λ))◦ )
−
x ∈ W }. Let α+ be the result of replacing every 2β with 2+ β := β ∧ 2β, i.e.,
⊥+ := ⊥, p+ := p, (α → β)+ := α+ → β+ , (2α)+ := 2+ α+ . We will focus
on extensions of K defined in Table 1; we will also use the abbreviation For for the
inconsistent logic K ⊕ ⊥. It is known that both Grz and GL extend K4 [5, 6].
Nevertheless, when the axiom Grz is replaced by Grz2 , i.e., a variant where the
consequent is prefixed by the box operator, the resulting logic does not contain K4.
For the relationship between wGrz and K4, see Theorem 2.2 below. Given a set of
formulas Γ, ModΓ will denote the class of frames where Γ is valid and given a class of
frames K, LogK will denote the set of formulas valid in K.
Tables 2 and 3 define a number of operations on logics, frames and classes of
frames. F• , F◦ , Λ• , Λ◦ are called, respectively, irreflexivization of F, reflexivization of
F, the irreflexive counterpart of Λ, the non-reflexive counterpart of Λ. We could have
made all the operations totally defined, but it seemed optimal to leave some cells empty
for greater clarity (to avoid, e.g., the non-reflexive counterpart of K being empty, when
K consists of irreflexive frames). Elements of [Λ]+ are called + -counterparts of Λ.
Lemma 1.1
1. For an arbitrary class K of reflexive frames, K ∩ K • = ∅ and
•
K ∪ K ⊆ K◦ . Hence, Log(K◦ ) ⊆ Log(K • ) ∩ Log(K).
2. If Λ ⊇ T is a complete logic, i.e., Λ = Log(Mod(Λ)), then Λ◦ ⊆ Λ ∩ Λ• .
3. For every β, β+ ↔ β ∈ T.
Table 3: Syntactic operators
Λ⊇T
Λ
(·)+
−
T ⊕ {α | α+ ∈ Λ}
(·)+
K ⊕ {α+ | α ∈ Λ}
−
3
[·]+
{Λ0 | α ∈ Λ iff α+ ∈ Λ0 }
−
4. For every Λ ⊇ T, NExtT ∩ [Λ]+ = {Λ}; that is, Λ is its own + -counterpart and
the only element of [Λ]+ extending T.
5. Every element of [Λ]+ is below an element of [Λ]+ ↑—the class of maximal
elements of [Λ]+ .
6. For every Λ ⊇ T, Λ+ is the smallest element of [Λ]+ . Also, Λ = T(Λ+ )+ .
7. For every consistent Λ ⊇ T, T 6⊆ Λ+ and hence Λ+ ( Λ.
8. For every model hF, V i, every x ∈ W and every formula α, hF◦ , V i , x α iff
hF, V i , x α+ .
9. Mod◦ (Λ) = Mod(Λ+ ), hence Λ◦ = Log(Mod(Λ+ )), i.e., Λ◦ is the smallest
complete extension of Λ+ . If T ⊆ Λ is complete, i.e., Λ = Log(Mod(Λ)), then
Λ◦ , Λ• ∈ [Λ]+ .
10. If Λ ⊇ T and [Λ]+ contains at least one complete logic Γ—for example, if
Λ+ = Λ◦ —then Λ is complete too. Moreover, if K is any class of frames s.t.
Γ = Log(K), then Λ is complete with respect to K ◦ ; e.g., the finite model
property of Λ+ implies finite model property of Λ.
11. For any complete Λ, Λ+ = Λ◦ .
12. If Λ = T ⊕ Γ, i.e., Γ axiomatizes Λ over T, then Λ+ = K ⊕ {γ+ | γ ∈ Γ}.
Proof: Proofs are given only for some clauses.
Clause 4 is a consequence of clause 3.
Clause 5: a routine application of The Kuratowski-Zorn Lemma, as [Λ]+ is nonempty by clause 4 and is easily seen to be closed under sums of chains.
Clause 7. By Makinson’s Theorem, Λ ⊆ Triv. Hence, Λ+ ⊆ Triv+ = K ⊕ p →
2p. But this logic is a sublogic of Ver and T 6⊆ Ver.
Clause 8. A generalization of Lemma 4.1 in [14]. The only inductive clause to be
proved is the one for α = 2β. hF◦ , V i , x 2β iff x ∈ 2R◦ V (β) = V (β) ∩ 2R V (β)
iff hF, V i , x α+ . This also yields clause 9.
Clause 10. α 6∈ Λ iff α+ 6∈ Γ iff (by the completeness assumption) for some
F ∈ K, F 2 α+ iff (by clause 8) F◦ 2 α. As F◦ ∈ K ◦ ⊆ ModΛ, we get the result.
This also gives the clause 11.
Clause 12. We need to show that for arbitrary α ∈ Λ, α+ ∈ K ⊕ Γ0 , where
0
Γ := {γ+ | γ ∈ Γ}. It boils down to showing that α+ holds in any model M = hF, V i
satisfying all substitution instances of Γ0 , as every logic is complete with respect to a
class of models (not frames!). Assume that α+ fails at some point of M. Then clause
8 implies α fails at the same point in M◦ . It means there is a substitution instance β
of some γ ∈ Γ which fails at M◦ . But then β+ is a substitution instance of γ+ which
fails in M.
a
4
As we see, it is enough to prove completeness of Λ+ to get a completeness result
for Λ itself. For the converse, see Problem 1 below.
Below, we group some facts concerning the logics we are interested in which are
either known or easily derivable from known ones.
Lemma 1.2
1. K = T+ = T◦ = T• [6].
2. wK4 = S4+ = S4◦ = S4• [9].
3. GL = Grz• and GL ∈ [Grz]+ [14].
4. NExtGL ∩ [Grz]+ ↑= {A∗ } and thus Grz• 6∈ [Grz]+ ↑ [18].
5. For every Λ ∈ NExtGrz, [Λ]+ does not have a greatest element.
Proof: Clause 5 follows from clause 3 and the fact that Grz ⊕ GL = For, as Grz 6⊆
Ver and GL 6⊆ Triv.
a
1.1
Neighborhood frames and topological semantics
We could have attempted developing the whole theory sketched above in terms of
neighborhood frames. Let us recall the definition first.
Definition 1.3 (Neighborhood frames) A (normal) neighborhood frame is a structure
F := hW, f : W 7→ P(P(W ))i, where for every x, f (x) is a non-empty family of sets
s.t. for every X, Y ∈ P(W ), X ∩ Y ∈ f (x) iff X, Y ∈ f (x). The (dual) operator
associated with f is 2f X := {x ∈ W | X ∈ f (x)}.
The closure condition on f (x) corresponds to normality of the modal operator. As
all the logics we are interested in are normal, we drop the word. Chellas’ monograph [7]
takes neighborhood frames as the basic modal semantics. As stated therein (and rather
easy to compute), T and K4 correspond, respectively, to the following conditions:
T
(t) for every x, x belongs to f (x),
(iv) for every x and every X ∈ f (x), 2f X belongs to f (x).
Together, these two additional conditions on neighborhood frames yield the same class
of structures as the definition of topological spaces in terms of neighborhood bases
(cf. any handbook of set-theoretic topology). In other words, neighborhood frames for
S4 are nothing else than topological spaces, with 2 being the interior operator and 3
being the closure operator. The operation of irreflexivization from Table 2 could be
also defined in terms of neighborhood frames as f • (x) = {X | X ∪ {x} ∈ f (x)}. On
topological spaces, this would replace neighborhoods with punctured neighborhoods.
It means that whenever hW, f i is a S4-frame (i.e., a topological space), 3f • is the
derivative operator of the same space. wK4 is complete with respect to this derivative
operator semantics (see, e.g., [9]; this actually follows from clause 2 of Lemma 1.2
above). Both the c-interpretation of the modal operator 3 as the closure operator and
the d-interpretation as the derivative operator have been studied in considerable depth
5
by Esakia [9, 10], Shehtman [19] or Bezhanishvili et al. [3]; in fact, the idea of both
interpretations dates back to McKinsey and Tarski [17] or even to Kuratowski [16].
Nevertheless, as we will see in Remark 2.8, defining Λ• in terms of neighborhood
semantics would have some surprising consequences.
2
Axiomatization and completeness of wGrz
Lemma 2.1 The wGrz axiom is equivalent to the + -translation of the Grz axiom. In
other words, wGrz = Grz+ .
Proof: It is enough to observe that (p → 2p) ↔ (p → 2+ p) and (2(p → α) →
p) ↔ (2+ (p → α) → p) are tautologies of classical propositional logic.
a
Even without using clause 3 of Lemma 1.2, we can prove that wGrz ⊆ GL. A
syntactic derivation of Grz+ in GL can be found in [5, Chapter 12].
As was mentioned above, K4 ⊆ GL ∩ Grz. This has been proved independently
by de Jongh, Sambin and Kripke for GL and by Blok for Grz—see [2] or [5] for an
account. Blok’s derivation does not yield automatically that all logics in [Grz]+ are
extensions of K4—the fact that Grz ⊇ S4 implies only that Grz+ ⊇ wK4. But
transitivity does indeed follow from the weak Grzegorczyk axiom.
Theorem 2.2 K4 is a sublogic of wGrz.
Proof: A generalization of Theorem 5.48 in [6]. Let F := hW, R, P i be a general
frame for wGrz and let x ∈ W . Assume x 2 2p → 22p. Then there is y s.t. xRy,
y p ∧ ¬2p and x 2 β := p ∧ (2p → 22p). We will show that x 2+ (2(β →
2β) → β), a contradiction with the fact that F wGrz. First, as y β ∧ ¬2β,
we have that x 2 2(β → 2β) and hence x 2(β → 2β) → β. Second, assume
for some u s.t. xRu, u 2(β → 2β) and yet u 2 β. As u p because of being a
successor of x, it means u 2p ∧ ¬22p. This in turn means there is v s.t. uRv and
v p ∧ ¬2p. Replacing x and y above with u and v, we obtain a contradiction with
u 2(β → 2β).
a
Now let us consider Grz2 . This logic does not contain K4; the reader is invited
to find a counterexample. On the other hand, it is well-known that over T the axiom
of Grz2 is equivalent to what we called here the Grzegorczyk axiom, i.e., Grz =
T ⊕ Grz2 .
Theorem 2.3 Grz2 is a proper sublogic of wGrz.
Proof: wGrz is equivalent to (2(p → 2p) → p) → (2(2(p → 2p) → p) → p). By
using axioms of K, we obtain 2(2(p → 2p) → p) → 2(2(2(p → 2p) → p) → p).
This is where we have to use Theorem 2.2 to deduce
2(2(p → 2p) → p) → 22(2(p → 2p) → p) ∧ 2(2(2(p → 2p) → p) → p).
The rest of the proof is left to the reader. The proper containment follows from Theorem 2.2: wGrz contains K4, while Grz2 does not.
a
6
However, as proved by Gabelaia [13], wK4 ⊕ Grz2 does contain K4. Together
with the next result, it implies that over wK4, Grz2 and wGrz are equivalent.
Theorem 2.4 The weak Grzegorczyk axiom is derivable in K4 ⊕ Grz2 .
Proof: 2(2(p → 2p) → p) → 22p holds in K4 ⊕ Grz2 . Together with the fact
that 22p → 2(p → 2p) belongs to K, it yields the result.
a
Thus, wK4 ⊕ Grz2 = K4 ⊕ Grz2 is the same system as wGrz. Observe that in
this way, we obtained an alternative proof of Esakia’s [10] result that K4 ⊕ Grz2 =
Grz+ . We turn to semantics. Recall that Mod(Grz) is the class of conversely wellfounded partial orders. It follows that (Mod(Grz))◦ is the class of transitive frames
which contain no proper cycles and no infinitely ascending chains—such frames are
called Noetherian.
Theorem 2.5 (Amerbauer [1]) wGrz = Grz◦ . Moreover, wGrz has the finite
model property.
Proof: We only need to prove the “moreover” part. wGrz is a subframe logic and all
such extensions of K4 have the finite model property, as proved by Fine [11]. The fmp
can be also proved directly in pretty much the same way as for Grz and GL, i.e., via a
selective filtration of the canonical model; such a proof can be found in [13].
a
Remark 2.6 By clause 10 of Lemma 1.1, this result implies the finite model property
of Grz itself. And together with clause 9 of Lemma 1.1, this in turn implies the second
statement of clause 3 of Lemma 1.2.
It seemed natural to conjecture that for any Λ ⊇ T, Λ• can be obtained from
Λ◦ by closing the set of theorems of the latter under Gabbay’s Irreflexivity Rule:
p ∧ 2¬p → β
, for p not appearing in β. This turned out false. In fact, GL is
β
not derivable from wGrz using the Gabbay rule even after the addition of the past
modality with corresponding axioms, despite the fact that the Gabbay rule allows for
some general “irreflexive completeness” results in tense logic [12, 20]. Recall that the
minimal tense extension of a modal logic Λ is obtained by extending the language with
the past modality and adding—together with the basic modal axioms for —the
axioms
p → 3p
p → 2p
Theorem 2.7 The closure of the minimal tense extension of wGrz under the Gabbay
rule is conservative over formulas in the basic modal language, i.e., involving no past
modality .
Proof: For every formula β in the basic modal language which does not belong to
wGrz, there exists a finite Noetherian transitive model M = hW, R, V i, where β
fails at some point y. Define M0 = hW 0 , R0 , V 0 i as follows: bulldoze all the reflexive
points of M (that is, replace them by infinite ascending chains), and take V 0 to be the
natural valuation inherited from V . The admissible subsets in the general tense frame
7
generated by all sets of of the form V 0 (φ) together with all singletons in W 0 are those
whose intersection with every infinite ascending chain obtained via bulldozing is either
finite or cofinite. Hence, the general tense frame obtained this way allows no valuation
refuting wGrz and admits the Gabbay rule, as all the points are irreflexive.
Now define S ⊆ W × W 0 as follows: xSx0 iff x0 is a copy of x, i.e., x itself if
x was irreflexive and otherwise an element of the ascending chain replacing x. S is a
bisimulation for 3, hence any point y 0 s.t. ySy 0 fails β in M0 .
a
Remark 2.8 The closure of wGrz under the Gabbay rule can be also inferred from
the result of Gabelaia [13]: under the d-interpretation of 3 as topological derivative,
wGrz is complete with respect to topological spaces. The Gabbay rule is easily seen
to be sound under this interpretation. Gabelaia’s result implies that defining Λ• in
terms of neighborhood frames would make GL distinct from Grz• . Moreover, an
asymmetry between Grz and S4 or T visible in Lemma 1.2 would disappear: for
Λ ∈ {Grz, S4, T}, Λ◦ would be equal to Λ• (observe, however, that Triv• would
be still equal to Ver and thus distinct from Triv◦ —i.e., we still would not be able
to deduce Λ◦ = Λ• even for a complete Λ). In short, given all the results we have
seen until now, nothing would distinguish GL among all elements of [Grz]+ : both
above and below GL there would be elements of this class complete with respect to
“irreflexive” topological semantics. It is rather surprising, given how natural it seems
to think of GL as “Grz minus reflexivity”. Let us also note here that despite its
fmp and topological completeness, wGrz is not complete with respect to any class
of finite spaces, Alexandroff spaces or metric spaces. This is a corollary of results of
Bezhanishvili et al. [4]: every such space is heredetarily irresolvable iff it is scattered,
i.e., a space for GL.
3
The lattice NExtwGrz
Theorem 3.1 wGrz is a proper sublogic of Grz ∩ GL.
Proof: Consider a Kripke frame consisting of a root point and two maximal, incomparable points. One of the maximal points is reflexive, the remaining ones are irreflexive.
To show it is not a frame for Grz ∩ GL, use Theorem 4.11 in [6].
a
Let us conclude with some observations on the lower regions of NExtwGrz, the
lattice of normal extensions of wGrz. The lattice in question is in many aspects more
complicated than both NExtGrz and NExtGL. To give one example: call a logic
prefinite if the logic is not finitely generated (i.e., generated by a single finite frame) but
all its proper sublogics are. It is known that NExtGrz contains exactly three prefinite
logics [6, Theorem 12.13] and that NExtGL contains countably many prefinite logics
[6, Theorem 12.16]. By contrast, we can show
Theorem 3.2 There are uncountably many prefinite logics in NExtwGrz.
Proof: Modify the proof of Theorem 13.25 in [6].
8
a
For
Triv
Ver = Triv•
A∗
[Grz]+↑
Grz
GL = Grz•
S4
wGrz = Grz+ = Grz◦
Grz2
K4
wK4 = S4+ = S4•
T
K = T+ = T•
Figure 1: The relationships between the logics discussed in this paper
Nevertheless, many results previously obtained for NExtGrz and NExtGL transfer
to NExtwGrz. It is well-known that neither Grz nor GL are canonical and Amerbauer
[1] proved the same result for K4 ⊕ Grz2 (that is, wGrz). In other words, the underlying Kripke frames of the canonical models for these systems are not Noetherian.
These results can be generalized both in strength and scope. That is, we can prove that
• Grz, GL do not have properties more general than canonicity (see [6, Section
6] for references). Examples of such properties include strong Kripke completeness (every consistent set of formulas can be satisfied in a Kripke frame for the
logic) and, still broader, admissibility of completions: every algebra in the corresponding variety can be embedded in a lattice-complete algebra from the same
variety.
• Every extension of GL or Grz of infinite depth fails to possess the above properties. Recall that a logic Λ is of finite depth if there is n ∈ ω s.t. no Kripke frame
for Λ contains a chain longer than n and of infinite depth otherwise.
wGrz provides a universal setting for such negative results.
Theorem 3.3 No Λ ⊇ wGrz of infinite depth admits completions; hence, no such Λ
is strongly Kripke complete.
Proof: Take any sequence of finite frames for Λ containing for every n ∈ N cofinitely
many frames of length greater than n. Let A be an ultraproduct of dual algebras of
those frames. A contains a family of distinct atoms {an }n∈N s.t. for every m < n,
am ≤ 3an and an ≤ 2¬am , The set of formulas equal to > under the valuation
9
V (pi ) = {ai } cannot be simultaneously satisfied in any lattice-complete algebra from
the variety corresponding to wGrz, in particular in any Kripke frame.
a
Forthcoming [15] provides an analogus result for multi-valued logics. The reader
is referred to Esakia [10] for more information on NExtwGrz.
4
Open problems
There are still some questions the author has not been able to answer.
Problem 1 Does the converse of clause 10 of Lemma 1.1 hold, i.e., does completeness
of Λ imply that Λ+ = Λ◦ ?
Problem 2 As Lemma 1.2 shows, Λ• does not have to belong to [Λ]+ ↑ even when
Λ• 6⊆ Λ. Is it true then that exactly one element of [Λ]+ ↑ lies above Λ• ? Observe that
if this conjecture is true, then clause 5 of Lemma 1.2 does not hold for some extensions
of T, e.g., S4 would be then the greatest element of [S4]+ .
Problem 3 Provide syntactic derivations of the K4 axiom in wGrz and in wK4 ⊕
Grz2 .
Problem 4 Find the unique Λ ⊇ T s.t. Grz2 ∈ [Λ]+ . In general, given a finite set of
axioms for a logic ∆, is there an algorithm to obtain an axiomatization for the unique
Λ ⊇ T s.t. ∆ ∈ [Λ]+ ? Will this axiomatization be finite too?
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