A Cut-free Sequent Calculus for
Bi-Intuitionistic Logic
Linda Buisman
1
and Rajeev Gor´e
12
1
The Australian National University
Canberra ACT 0200, Australia
2
Logic and Computation Programme
Canberra Research Laboratory, NICTA
?
, Australia
{Linda.Buisman|Rajeev.Gore}@anu.edu.au
Abstract. Bi-intuitionistic logic is the extension of intuitionistic logic
with a connective dual to implication. Bi-intuitionistic logic was intro-
duced by Rauszer as a Hilbert calculus with algebraic and Kripke se-
mantics. But her subsequent “cut-free” sequent calculus for BiInt has
recently been shown by Uustalu to fail cut-elimination. We present a new
cut-free sequent calculus for BiInt, and prove it sound and complete with
respect to its Kripke semantics. Ensuring completeness is complicated by
the interaction between implication and its dual, similarly to future and
past modalities in tense logic. Our calculus handles this interaction using
extended sequents which pass information from premises to conclusions
using variables instantiated at the leaves of failed derivation trees. Our
simple termination argument allows our calculus to be used for auto-
mated deduction, although this is not its main purpose.
1 Introduction
Propositional intuitionistic logic (Int) has connectives →, ∧, ∨ and ¬, with
¬ϕ definable as ¬ϕ := ϕ →⊥. Propositional dual intuitionistic logic (DualInt)
has connectives −< , ∧, ∨ and ∼ , with ∼ ϕ definable as ∼ ϕ := >−< ϕ. Bi-
intuitionistic logic (BiInt) or subtractive logic or Heyting-Brouwer logic is the
union of Int and DualInt. It is a conservative extension of both and was first
studied by Rauszer [11, 12].
Rauszer’s Kripke semantics for BiInt involve a reflexive and transitive bi-
nary relation R, and its converse R
−1
, similar to the normal tense logic Kt.S4.
Specifically, a world w makes ϕ → ψ true if every R-successor v that makes
ϕ true also makes ψ true, and a world w makes ϕ−< ψ true if there exists an
R-predecessor v where ϕ holds but ψ does not. Thus, ϕ−<ψ (“ϕ excludes ψ”)
is a natural dual to ϕ → ψ (“ϕ implies ψ”).
While there are many cut-free sequent systems for Int (e.g., [15, 6, 5]) and
DualInt (e.g., [16, 4]), the case for BiInt is less satisfactory. Rauszer presented
?
National ICT Australia is funded by the Australian Government’s Dept of Commu-
nications, Information Technology and the Arts and the Australian Research Council
through Backing Australia’s Ability and the ICT Centre of Excellence program.
a sequent calculus for BiInt in [11] and “proved” it cut-free, but Uustalu [17]
has recently shown that the BiInt-valid formula p → (q ∨ (r → ((p−< q) ∧ r))
cannot be derived in Rauszer’s calculus without the cut rule. Uustalu’s example
also shows that Crolard’s sequent calculus [3] for BiInt is not cut-free. Uustalu’s
example fails in these calculi because certain sequent rules are restricted to
singleton succedents or antecedents in their conclusions, and these fail to capture
the interaction between → and −< . Uustalu and Pinto have apparently given
a cut-free sequent-calculus for BiInt [19, 18] using labelled formulae which use
the Kripke semantics directly in the rules. But we have been unable to examine
their rules or proofs, as only the abstract of their work has been published.
We present a new purely syntactic cut-free sequent calculus for BiInt. We
avoid Rauszer’s and Crolard’s restrictions on the antecedents and succedents for
certain rules by basing our rules on Dragalin’s GHPC [5] which allows multiple
formulae on both sides of sequents. To maintain intuitionistic soundness, we re-
strict the premise of the implication-right rule to a singleton in the succedent.
Dually, the premise of our exclusion-left rule is restricted to a singleton in the
antecedent. But using Dragalin’s calculus and its dual does not give us BiInt
completeness. We therefore follow Schwendimann [13], and use sequents which
pass relevant information from premises to conclusions using variables instan-
tiated at the leaves of failed derivation trees. We then recompute parts of our
derivation trees using the new information, similarly to the restart technique of
[9]. Our calculus thus uses a purely syntactic addition to traditional sequents,
rather than resorting to a semantic mechanism such as labels. Our termination
argument also relies on two new rules from
´
Svejdar [14].
If we were interested only in decision procedures, we could obtain a decision
procedure for BiInt by embedding it into the tense logic Kt.S4 [20], and using
tableaux for description logics with inverse roles [9]. However, an embedding into
Kt.S4 provides no proof-theoretic insights into BiInt itself. Moreover, the restart
technique of Horrocks et al. [9] involves non-deterministic expansion of disjunc-
tions, which is complicated by inverse roles. Their actual implementation avoids
this non-determinism by keeping a global view of the whole counter-model under
construction. In contrast, we handle this non-determinism by syntactically en-
coding it using variables and extended formulae, neither of which have a semantic
content. Our purely syntactic approach is preferable for proof-theoretic reasons,
since models are never explicitly involved in the proof system: see Remark 2.
In Section 2, we define the syntax and semantics of BiInt. In Section 3, we in-
troduce our sequent calculus GBiInt and give an example derivation of Uustalu’s
interaction formula. We prove the soundness and completeness of GBiInt in Sec-
tions 4 and 5 respectively. A version with full proofs can be found in [2].
2 Syntax and Semantics of BiInt
The formulae Fml of BiInt are built from a denumerable set of Atoms and the
constants > and ⊥ using the connectives ∧, ∨, →, −<, ¬, and ∼. The length of
a formula χ is just the number of symbols it contains. We use classical first-order
w ϕ ∨ ψ if w ϕ or w ψ
w ϕ ∧ ψ if w ϕ & w ψ
w ¬ϕ if ∀u ∈ W.[wRu ⇒ (u 2 ϕ)]
w ϕ → ψ if ∀u ∈ W.[wRu ⇒ (u 2 ϕ or u ψ)]
w ∼ ϕ if ∃u ∈ W.[uRw & u 2 ϕ]
w ϕ −<ψ if ∃u ∈ W.[uRw & u ϕ & u 2 ψ]
Fig. 1. BiInt semantics
logic when reasoning about BiInt at the meta-level. A BiInt frame is a pair
hW, Ri, where W is a non-empty set of worlds and R ⊆ W × W is a binary
reflexive transitive relation. A BiInt model is a triple M = hW, R, ϑi, where
hW, Ri is a BiInt frame and the truth valuation ϑ is a function W × Atoms →
{true, false} which obeys: ∀w ∈ W.ϑ(w, >) = true; ∀w ∈ W.ϑ(w, ⊥) = false; and
which obeys persistence, also known as truth monotonicity:
∀u, w ∈ W.∀p ∈ Atoms.(ϑ(w, p) = true & wRu) ⇒ (ϑ(u, p) = true).
Given a model M = hW, R, ϑi, a world w ∈ W and an atom p ∈ Atoms, we
write w p if ϑ(w, p) = true. We pronounce as “forces”, and we pronounce
2 as “rejects”. The forcing of compound formulae is defined in Fig. 1. Since ¬
and ∼ can be derived from → and −< respectively, we restrict our attention
to →, −<, ∧, ∨. We obtain persistence for compound formulae by induction on
their length, and then reverse persistence for compound formulae follows from
persistence because the truth valuation is binary:
∀M = hW, R, ϑi.∀u, w ∈ W.∀ϕ ∈ Fml.(w ϕ & wRu ⇒ u ϕ)
∀M = hW, R, ϑi.∀u, w ∈ W.∀ϕ ∈ Fml.(w 2 ϕ & uRw ⇒ u 2 ϕ).
We write to mean the empty set. Given a formula ϕ and two sets of formulae
∆ and Γ , we write ∆, Γ for ∆ ∪ Γ and we write ∆, ϕ for ∆ ∪ {ϕ}. Given a
model M = hW, R, ϑi and a world w ∈ W, we write w Γ (w forces Γ ) if
∀ϕ ∈ Γ.w ϕ, and we write w =| ∆ (w rejects ∆) if ∀ϕ ∈ ∆.w 2 ϕ. We
deliberately use “=|” for rejection of sets to emphasize that every member of the
set is rejected, instead of “2”, which could be seen as “some member is rejected”.
Γ
BiInt
∆ means: ∀M = hW, R, ϑi.∀w ∈ W.(w Γ ⇒ ∃ϕ ∈ ∆.w ϕ)
Γ 6
BiInt
∆ means: ∃M = hW, R, ϑi.∃w ∈ W.(w Γ & w =| ∆).
Thus Γ 6
BiInt
∆ means that Γ
BiInt
∆ is falsifiable. As usual, our sequent calculus
has a semantic reading which assumes that there exists an initial world w
0
in a
BiInt-model M where w
0
Γ and w
0
=| ∆. We then systematically apply the
sequent rules using backward proof-search to either construct M successfully,
giving us Γ 6
BiInt
∆, or conclude that M cannot exist, giving us Γ
BiInt
∆.
3 Our Sequent Calculus GBiInt
We now present GBiInt, a Gentzen-style sequent calculus for BiInt. The se-
quents have a non-traditional component in the form of variables that are instan-
tiated at the leaves of the derivation tree, and passed back to lower sequents from
premises to conclusion. Note that variables are not names for Kripke worlds.
We extend our syntax for presenting some of our sequent rules. The extended
BiInt formulae are defined as: if ϕ is a BiInt formula, then ϕ is an extended
BiInt formula, and if S/P is a set {{ϕ
0
0
, · · · , ϕ
n
0
}, · · · , {ϕ
0
m
, · · · , ϕ
k
m
}} of sets of
BiInt formulae, then
W
S and
V
P are extended BiInt formulae with intended
semantics
W
S ≡ (ϕ
0
0
∧ · · · ∧ ϕ
n
0
) ∨ · · · ∨ (ϕ
0
m
∧ · · · ∧ ϕ
k
m
)
V
P ≡ (ϕ
0
0
∨ · · · ∨ ϕ
n
0
) ∧ · · · ∧ (ϕ
0
m
∨ · · · ∨ ϕ
k
m
).
From now on, we implicitly treat extended BiInt formulae as their BiInt
equivalents. Given a BiInt model M = hW, R, ϑi, and a world w ∈ W, the
following semantics follows directly from their definition:
w
_
S if ∃Γ ∈ S.w Γ and w =|
^
P if ∃∆ ∈ P.w =| ∆.
We can now extend the definition of forcing and rejecting to extended BiInt
formulae in the obvious way. If Γ and ∆ are sets of extended BiInt formulae,
and ϕ is an extended BiInt formula, then w Γ if ∀ϕ ∈ Γ.w ϕ, and w =|
∆ if ∀ϕ ∈ ∆.w 2 ϕ.
A GBiInt sequent is an expression
S
P
Γ ` ∆, where the left hand side
(LHS) Γ is a set of extended BiInt formulae; the right hand side (RHS) ∆ is
a set of extended BiInt formulae; and the variables S and P are each a set of
sets of formulae. We sometimes write just Γ ` ∆, ignoring the variable values
for readability, but only when the values of the variables are not important to
the discussion. In terms of the counter-model under construction, we say that
a sequent
S
P
Γ ` ∆ is falsifiable [at w
0
in M] iff there exists a BiInt model
M = hW, R, ϑi and ∃w
0
∈ W such that w
0
Γ and w
0
=| ∆. Thus, a sequent
Γ ` ∆ is not falsifiable iff Γ
BiInt
∆. We say the variable conditions of a sequent
γ =
S
P
Γ ` ∆ hold iff γ is falsifiable at w
0
in some model M = hW, R, ϑi and
the following Successor/Predecessor conditions hold:
S-condition: ∃Σ ∈ S.∀w ∈ W.w
0
Rw ⇒ w Σ
P-condition: ∃Π ∈ P.∀w ∈ W.wRw
0
⇒ w =| Π.
A sequent rule is an expression of one of the two forms below
γ
1
· · · γ
n
(name)
γ
side conditions
γ
0
· · · γ
n
(name)
γ
side conditions
where n ≥ 0, and each γ
i
is a sequent. The rule has a name, a conclusion
γ, optional premise(s) γ
1
, · · · , γ
n
, optional side conditions, and universal
branching as indicated by a solid line or existential branching as indicated by a
dashed line (explained shortly).
Our traditional rules (Fig. 2) are based on Dragalin’s GHPC [5] for Int be-
cause we require multiple formulae in the succedents and antecedents of sequents
for completeness; we have added symmetric rules for the DualInt connective −<.
The main difference is that our (→
L
) rule and the symmetric (−<
R
) carry their
(Id)
S:=
P:=
˛
˛
˛
˛
Γ, ϕ ` ∆, ϕ
(⊥
L
)
S:=
P:=
˛
˛
˛
˛
Γ, ⊥` ∆
(>
R
)
S:=
P:=
˛
˛
˛
˛
Γ ` ∆, >
S
1
P
1
˛
˛
˛
˛
˛
˛
Γ, ϕ ∧ ψ, ϕ, ψ ` ∆
(∧
L
)
S:=S
1
P:=P
1
˛
˛
˛
˛
˛
˛
Γ, ϕ ∧ ψ ` ∆
S
1
P
1
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ ∧ ψ, ϕ
S
2
P
2
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ ∧ ψ, ψ
(∧
R
)
S:=S
1
∪S
2
P:=P
1
∪P
2
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ ∧ ψ
S
1
P
1
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ ∨ ψ, ϕ, ψ
(∨
R
)
S:=S
1
P:=P
1
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ ∨ ψ
S
1
P
1
˛
˛
˛
˛
˛
˛
Γ, ϕ ∨ ψ, ϕ ` ∆
S
2
P
2
˛
˛
˛
˛
˛
˛
Γ, ϕ ∨ ψ, ψ ` ∆
(∨
L
)
S:=S
1
∪S
2
P:=P
1
∪P
2
˛
˛
˛
˛
˛
˛
Γ, ϕ ∨ ψ ` ∆
S
1
P
1
˛
˛
˛
˛
˛
˛
Γ, ϕ → ψ ` ϕ, ∆
S
2
P
2
˛
˛
˛
˛
˛
˛
Γ, ϕ → ψ, ψ ` ∆
(→
L
)
S:=S
1
∪S
2
P:=P
1
∪P
2
˛
˛
˛
˛
˛
˛
Γ, ϕ → ψ ` ∆
S
1
P
1
˛
˛
˛
˛
˛
˛
Γ, ψ ` ∆, ϕ −< ψ
S
2
P
2
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ −< ψ, ϕ
( −<
R
)
S:=S
1
∪S
2
P:=P
1
∪P
2
˛
˛
˛
˛
˛
˛
Γ ` ∆, ϕ −< ψ
For every rule with premises π
i
and conlusion γ, apply the rule only if:
∀π
i
.(LHS
π
i
6⊆ LHS
γ
or RHS
π
i
6⊆ RHS
γ
)
Fig. 2. GBiInt rules - traditional
principal formula and all side formulae into the premises. Our rules for ∧ and ∨
also carry their principal formula into their premises to assist with termination.
Note that there are other approaches to a terminating sequent calculus for Int,
e.g., Dyckhoff’s contraction-free calculi [6], or history methods by Heuerding
et al. [8] and Howe [10]. These methods are less suitable when the interaction
between Int and DualInt formulae needs to be considered, since they erase po-
tentially relevant formulae too soon during backward proof search. Moreover, we
found it easier to prove semantic completeness with our loop-checking method
than with history-based methods since both [8] and [10] prove completeness using
syntactic transformations of derivations. Consequently, while GBiInt is sound
and complete for the Int (and DualInt) fragment of BiInt, it is unlikely to be
as efficient on the fragment as these specific calculi.
Our rules for → on the right and −< on the left (Fig. 3) are non-traditional.
The (→
R
) and (−<
L
) rules have two premises instead of one, and they are con-
nected by existential branching as indicated by the dotted horizontal line.
Existential branching means that the conclusion is derivable if some premise is
derivable; thus it is dual to the conventional universal branching, where the con-
clusion is derivable if all premises are derivable. We chose existential branching
rather than two separate non-invertible rules so the left premise can communicate
information via variables to the right premise. This inter-premise communica-
tion and the use of variables is crucial to proving interaction formulae of BiInt,
and it gives our calculus an operational reading.
When applying an existential branching rule during backward proof search,
we first create the left premise. If the left premise is non-derivable, then it returns
