Abstract It was shown in  be a propositional language of type (2, 2, 2, 2, 0, 0). Let Γ, Π be finite sequences of L-formulas and ϕ, ψ, ξ be L-formulas. The sequent calculus LJ * \c is defined by the following axioms and rules

Romà J Adillon

Ventura Verdú

CiteSeerX

Bulletin of the Section of Logic
Volume 31/3 (2002), pp. 125–134
Romà J. Adillon
Ventura Verdú
ON A SUBSTRUCTURAL GENTZEN SYSTEM, ITS
EQUIVALENT VARIETY SEMANTICS AND ITS
EXTERNAL DEDUCTIVE SYSTEM
Abstract
It was shown in [1] that the Gentzen system GLJ∗\c, the deductive system IPC∗\c
and the equational system
〈
L, |=RL
〉
associated with the variety of residuated
lattices are equivalent in the sense of [8] and [9]. In this paper we show that if
we delete the rules for the implication connective → from the sequent calculus
LJ∗\c, then the Gentzen system obtained from this sequent calculus, denoted
by G∧,∨,¯, is algebraizable and the variety of bounded commutative integral
l-monoids is its equivalent algebraic semantics. As a consequence of this result
we obtain that the contraction rule is not derivable in G∧,∨,¯, so we can say
that G∧,∨,¯ is a substructural Gentzen system. It is also shown that there is no
deductive system equivalent to G∧,∨,¯ or to
〈
L, |=BCILM
〉
and that the variety
of BCILM is an algebraic semantics for the external deductive system S0G∧,∨,¯
associated with G∧,∨,¯, with defining equation p ≈ 1. Finally we prove that this
deductive system S0G∧,∨,¯ is not protoalgebraic.
Let us recall the definition of the sequent calculus LJ∗\c.
Definition 1. ([1, definition 2], cf. [7]) Let L = {∧,∨,¯,→, 0, 1} be a
propositional language of type (2, 2, 2, 2, 0, 0). Let Γ, Π be finite sequences
of L-formulas and ϕ, ψ, ξ be L-formulas. The sequent calculus LJ∗\c is
defined by the following axioms and rules:
126 Romà J. Adillon and Ventura Verdú
ϕ ⇒ ϕ (Axiom 1) 0 ⇒ ϕ (Axiom 2) ∅ ⇒ 1 (Axiom 3)
Γ, ϕ, ψ, Π ⇒ ξ
Γ, ψ, ϕ, Π ⇒ ξ (i ⇒)
Γ ⇒ ξ
Γ, ϕ ⇒ ξ (w ⇒)
Γ ⇒ ϕ ϕ, Π ⇒ ξ
Γ,Π ⇒ ξ (cut)
Γ ⇒ ϕ ψ, Π ⇒ ξ
ϕ → ψ, Γ, Π ⇒ ξ (→⇒)
Γ, ϕ ⇒ ψ
Γ ⇒ ϕ → ψ (⇒→)
Γ, ϕ ⇒ ξ Γ, ψ ⇒ ξ
Γ, ϕ ∨ ψ ⇒ ξ (∨ ⇒)
Γ ⇒ ψ
Γ ⇒ ϕ ∨ ψ (⇒ ∨1)
Γ ⇒ ϕ
Γ ⇒ ϕ ∨ ψ (⇒ ∨2)
Γ, ψ ⇒ ξ
Γ, ϕ ∧ ψ ⇒ ξ (∧1 ⇒)
Γ, ϕ ⇒ ξ
Γ, ϕ ∧ ψ ⇒ ξ (∧2 ⇒)
Γ ⇒ ϕ Γ ⇒ ψ
Γ ⇒ ϕ ∧ ψ (⇒ ∧)
Γ, ϕ, ψ ⇒ ξ
Γ, ϕ¯ ψ ⇒ ξ (¯ ⇒)
Γ ⇒ ϕ Π ⇒ ψ
Γ, Π ⇒ ϕ¯ ψ (⇒ ¯) .
Definition 2. Let LJ∗{∧,∨,¯,0,1}\c be the sequent calculus obtained by
deleting the rules for the implication connective → from LJ∗\c and let
G∧,∨,¯ =
〈
L,`LJ∗{∧,∨,¯,0,1}\c
〉
be the Gentzen system of type (ω, {1}) de-
termined by the sequent calculus LJ∗{∧,∨,¯,0,1}\c.
Let us recall the definition of the variety of bounded commutative
integral l-monoids.
Definition 3. (cf. [2]) Let A = 〈A,∧,∨,¯, 0, 1〉 be an algebra of type
(2, 2, 2, 0, 0), A is a bounded commutative integral l-monoid if the following
conditions are satisfied:
1. 〈A,∧,∨, 0, 1〉 is a bounded lattice where 1 and 0 are the maximum
and minimum elements, respectively;
2. 〈A,¯, 1〉 is a commutative monoid, i.e., for every x, y, z ∈ A,
On a Substructural Gentzen System, its Equivalent Variety Semantics... 127
(a) (x¯ y)¯ z = x¯ (y ¯ z),
(b) x¯ y = y ¯ x,
(c) 1¯ x = x;
3. For all x, y, z ∈ A,
(x ∨ y)¯ z = (x¯ z) ∨ (y ¯ z).
The class of bounded commutative integral l-monoids will be denoted
by BCILM.
It is easy to see that BCILM satisfies the property of monotonicity of
¯, i.e., for every x, y, z ∈ A, if x ≤ y implies (x¯ z) ≤ (y ¯ z).
Notice that if we add the residuation property
x ≤ y → z iff x¯ y ≤ z
to the definition of BCILM, then we obtain a residuated lattice.
The following lemma will be used to show that G∧,∨,¯ is algebraizable.
Lemma 4. ([9, Lemma 2.5]) Let G be a Gentzen system of type (α, β), let
K be a quasivariety.
If there is a translation τ from G to K and a translation ρ from K to
G such that:
1. Γ ⇒ ∆ a`G ρτ (Γ ⇒ ∆) for all Γ ⇒ ∆ ∈ Seq(α,β)L ,
2. (ϕ,ψ)=||=K τρ(ϕ, ψ) for all (ϕ,ψ) ∈ Fm2L,
3. for every A ∈ K, the set R = {(X, Y ) ∈ Am ×An : m ∈ α, n ∈ β,
τA(X, Y ) ⊆ ∆A
}
is a G-filter,
4. for all T ∈ ThG, θT =
{
(ϕ,ψ) ∈ Fm2L : ρ(ϕ,ψ) ⊆ T
} ∈ ConKFmL,
then G is algebraizable with equivalent algebraic semantics K.
Theorem 5. G∧,∨,¯ is algebraizable with its equivalent algebraic seman-
tics, the variety BCILM, and with the translations τ from G∧,∨,¯ to BCILM
and ρ from BCILM to G∧,∨,¯ defined in the following way:
τ(p0, . . . , pm−1 ⇒ q0) =



( ⊙
i<m
pi
)
∧ q0 ≈
⊙
i<m
pi, if m ≥ 1
1 ≈ q0, if m = 0
ρ(p0 ≈ p1) = {p0 ⇒ p1; p1 ⇒ p0} .
128 Romà J. Adillon and Ventura Verdú
In order to simplify the notation we use
⊙
i<m
pi as an abbreviation for p0 ¯
(p1 ¯ (. . .¯ (pm−2 ¯ pm−1) . . .)).
Proof: To prove this result we will use Lemma 4.
1. Let us prove condition (i) : Γ ⇒ ψ a`LJ∗{∧,∨,¯,0,1}\c ρτ (Γ ⇒ ψ).
If Γ = ∅ we have ρτ (∅ ⇒ ψ) = {1 ⇒ ψ, ψ ⇒ 1} then by using the
axiom 3, the rules (w ⇒) and (cut) we obtain the condition (i).
If Γ = γ0, . . . , γn−1,
and we will denote the formula γ0 ¯ (γ1 ¯ (. . .¯ (γm−2 ¯ γm−1) . . .))
as
⊙
Γ, we have
ρτ (Γ ⇒ ψ) =
{⊙
Γ ⇒
(⊙
Γ
)
∧ ψ,
(⊙
Γ
)
∧ ψ ⇒
⊙
Γ
}
.
Note that the sequent (
⊙
Γ) ∧ ψ ⇒ ⊙Γ is derivable in G∧,∨,¯ by
using the rule (∧2 ⇒).
So we have to show that Γ ⇒ ψ a`LJ∗{∧,∨,¯,0,1}\c
⊙
Γ ⇒ (⊙ Γ) ∧ ψ:
`)
⊙
Γ ⇒ ⊙ Γ Γ ⇒ ψ⊙ Γ ⇒ ψ (¯ ⇒)
n−1
⊙
Γ ⇒ (⊙Γ) ∧ ψ (⇒ ∧)
a)
γn−2 ⇒ γn−2 γn−1 ⇒ γn−1
γn−2, γn−1 ⇒ γn−2 ¯ γn−1 (⇒ ¯)
...
Γ ⇒ ⊙Γ(⇒ ¯)
⊙
Γ ⇒ (⊙Γ) ∧ ψ ψ ⇒ ψ(⊙ Γ) ∧ ψ ⇒ ψ (∧1 ⇒)⊙
Γ ⇒ ψ (cut)
Γ ⇒ ψ (cut)
1. Condition (ii) : ϕ ≈ ψ=||=BCILM {ϕ ∧ ψ ≈ ϕ, ψ ∧ ϕ ≈ ψ} is trivial.
2. Let us prove condition (iii) that is, for all A ∈ BCILM, the set
R =
{
(X, a) ∈ Am ×A : m ∈ ω, τA(X, a) ⊆ ∆A
}
is a G∧,∨,¯-filter.
On a Substructural Gentzen System, its Equivalent Variety Semantics... 129
Notice that
R =
{
(X, a) ∈ Am ×A : m ∈ ω\ {0} ,
⊙
X ≤ a
}
∪
∪{(∅, a) ∈ {∅} ×A : 1 ≤ a} .
This set contains all the pairs (a, a), (0, a) and (∅, 1) with a ∈ A,
so R contains the interpretation of the axioms. Now we will show
that R is closed under the interpretation of the inference rules of
LJ∗{∧,∨,¯,0,1}\c.
Let us check, for example, the following:
• Right introduction rule for ¯ :
Γ ⇒ ϕ Π ⇒ ψ
Γ, Π ⇒ ϕ¯ ψ .
If Γ = ∅ and Π = ∅, we have to show that if (∅, a) ∈ R and
(∅, b) ∈ R, then (∅, a ¯ b) ∈ R. We have that 1 ≤ a, 1 ≤ b and
by using the fact that 1 is the maximum element, we obtain that
a = b = 1, by applying the property (2)-(c) we have 1 = a ¯ b,
so (∅, a¯ b) ∈ R.
If Γ = ∅ and Π 6= ∅ (analogous for Γ 6= ∅ and Π = ∅), we have to
show that if (∅, a) ∈ R and (Y, b) ∈ R, then (Y, a¯ b) ∈ R. We
have that 1 ≤ a and ⊙ Y ≤ b and by using the fact that 1 is
the maximum element, we obtain that a = 1, then by applying
the property (2)-(c) we have b = a ¯ b, we obtain ⊙ Y ≤ b, so
(Y, a¯ b) ∈ R.
If Γ 6= ∅ and Π 6= ∅, we have to show that if (X, a) ∈ R and
(Y, b) ∈ R, then ((X, Y ), a ¯ b) ∈ R. We have that ⊙X ≤ a
and
⊙
Y ≤ b by applying the monotonicity of ¯ we obtain
(
⊙
X) ¯ (⊙ Y ) ≤ a ¯ b, and by the associative property of ¯
we have ((X,Y ), a¯ b) ∈ R.
• Left introduction rule for ∧ :
Γ, ψ ⇒ ξ
Γ, ϕ ∧ ψ ⇒ ξ .
If Γ = ∅, we have to show that if (b, c) ∈ R, then (a ∧ b, c) ∈ R.
We have that b ≤ c, and by using the property a ∧ b ≤ b we
obtain a ∧ b ≤ c, so that (a ∧ b, c) ∈ R.
130 Romà J. Adillon and Ventura Verdú
If Γ 6= ∅, we have to show that if ((X, b) , c) ∈ R, then
((X, a ∧ b) , c) ∈ R. We have that (⊙ X)¯ b ≤ c, and by using
the property a ∧ b ≤ b and the monotonicity of ¯ we obtain
(
⊙
X)¯ (a∧ b) ≤ (⊙ X)¯ b; so that (⊙ X)¯ (a∧ b) ≤ c, that
is ((X, a ∧ b), c) ∈ R.
• Left introduction rule for ∨ :
Γ, ϕ ⇒ ξ Γ, ψ ⇒ ξ
Γ, ϕ ∨ ψ ⇒ ξ .
If Γ = ∅, we have to show that if (a, c) ∈ R and (b, c) ∈ R, then
(a ∨ b, c) ∈ R. We have that a ≤ c and b ≤ c, and by using
the fact that ∨ is the supremum, we obtain a ∨ b ≤ c, so that
(a ∨ b, c) ∈ R.
If Γ 6= ∅, we have to show that if ((X, a), c) ∈ R and ((X, b), c) ∈
R, then ((X, a ∨ b), c) ∈ R. We have that (⊙ X) ¯ a ≤ c and
(
⊙
X) ¯ b ≤ c, and by using the distributive property (3) we
have (
⊙
X)¯ (a∨ b) = ((⊙X)¯a)∨ ((⊙ X)¯ b) and by using
the fact that ∨ is the supremum, we obtain (⊙ X)¯ (a∨ b) ≤ c,
so ((X, a ∨ b), c) ∈ R.
The other rules are left to the reader.
3. Let us prove condition (iv), that is for all T ∈ ThG∧,∨,¯,
θT =
{
(ϕ, ψ) ∈ Fm2{∧,∨,¯,0,1} : ρ(ϕ, ψ) ⊆ T
}
∈ ConBCILMFm{∧,∨,¯,0,1}.
It is easy to show that θT is a congruence.
Now let ϕ ≈ ψ be one of the equations which define the BCILM. We
have to show that ρ(ϕ,ψ) ⊆ T for all T ∈ ThG∧,∨,¯. It suffices to
show that ∅ `LJ∗{∧,∨,¯,0,1}\c ϕ ⇒ ψ and ∅ `LJ∗{∧,∨,¯,0,1}\c ψ ⇒ ϕ.
Let us check, for example, the following:
• x ∧ x = x :
We have to show that ∅ `LJ∗{∧,∨,¯,0,1}\c ϕ ∧ ϕ ⇒ ϕ
and ∅ `LJ∗{∧,∨,¯,0,1}\c ϕ ⇒ ϕ ∧ ϕ. By using the rules (∧1 ⇒)
and (⇒ ∧), respectively, the result follows.
• x¯ y = y ¯ x :
We have to show that ∅ `LJ∗{∧,∨,¯,0,1}\c ϕ ¯ ψ ⇒ ψ ¯ ϕ. It is
straightforward by using the rules (⇒ ¯), (i ⇒) and (¯ ⇒).
On a Substructural Gentzen System, its Equivalent Variety Semantics... 131
• 1¯ x = x :
We have to show that ∅ `LJ∗{∧,∨,¯,0,1}\c 1 ¯ ϕ ⇒ ϕ
and ∅ `LJ∗{∧,∨,¯,0,1}\c ϕ ⇒ 1¯ ϕ. Consider the following proofs:
ϕ ⇒ ϕ
1, ϕ ⇒ ϕ (w ⇒)
1¯ ϕ ⇒ ϕ (¯ ⇒)
∅ ⇒ 1 ϕ ⇒ ϕ
ϕ ⇒ 1¯ ϕ (⇒ ¯)
• (x ∨ y)¯ z = (x¯ z) ∨ (y ¯ z):
We have to show that ∅ `LJ∗{∧,∨,¯,0,1}\c (ϕ ∨ ψ)¯ ξ ⇒ (ϕ¯ ξ) ∨
(ψ ¯ ξ). Consider the following proof:
ϕ ⇒ ϕ ξ ⇒ ξ
ϕ, ξ ⇒ ϕ¯ ξ (⇒¯)
ϕ, ξ ⇒ (ϕ¯ ξ) ∨ (ψ ¯ ξ) (⇒ ∨2)
ψ ⇒ ψ ξ ⇒ ξ
ψ, ξ ⇒ ψ ¯ ξ (⇒¯)
ψ, ξ ⇒ (ϕ¯ ξ) ∨ (ψ ¯ ξ) (⇒ ∨1)
ϕ ∨ ψ, ξ ⇒ (ϕ¯ ξ) ∨ (ψ ¯ ξ)
(ϕ ∨ ψ)¯ ξ ⇒ (ϕ¯ ξ) ∨ (ψ ¯ ξ) (¯ ⇒)
(∨ ⇒)
Now we have to prove that ∅ `LJ∗{∧,∨,¯,0,1}\c (ϕ¯ ξ)∨ (ψ¯ ξ) ⇒
(ϕ ∨ ψ)¯ ξ. Consider the following proof:
ϕ ⇒ ϕ
ϕ ⇒ ϕ ∨ ψ (⇒ ∨2)
ψ ⇒ ψ
ψ ⇒ ϕ ∨ ψ (⇒ ∨1)ξ ⇒ ξ
ϕ, ξ ⇒ (ϕ ∨ ψ)¯ ξ (⇒ ¯)
ξ ⇒ ξ
ψ, ξ ⇒ (ϕ ∨ ψ)¯ ξ (⇒ ¯)
ϕ¯ ξ ⇒ (ϕ ∨ ψ)¯ ξ (¯ ⇒) ψ ¯ ξ ⇒ (ϕ ∨ ψ)¯ ξ (¯ ⇒)
(ϕ¯ ξ) ∨ (ψ ¯ ξ) ⇒ (ϕ ∨ ψ)¯ ξ (∨ ⇒)
2
As a consequence of this theorem, we have that the equational theory
of BCILM is decidable.
Theorem 6. The contraction rule is not derivable in G∧,∨,¯.
Proof: Assume that the contraction rule is derivable, that is:
Γ, ϕ, ϕ, Π ⇒ ξ `LJ∗{∧,∨,¯,0,1}\c Γ, ϕ, Π ⇒ ξ
then by the theorem 5 we have that
(⊙
Γ
)
¯ϕ¯ϕ¯
(⊙
Π
)
≤ ξ|=BCILM
(⊙
Γ
)
¯ϕ¯
(⊙
Π
)
≤ ξ (1)
132 Romà J. Adillon and Ventura Verdú
Consider the 〈∧,∨,¯, 0, 1〉-reduct of the MV-algebra of three-elements,
that is A =
〈{
0, 12 , 1
}
,∧,∨,¯, 0, 1〉, where 0 < 12 < 1 and x¯ y = ¬(x →
¬y). As 12 ¯ 12 = ¬( 12 → ¬ 12 ) = ¬(12 → 12 ) = ¬1 = 0, we have that there
is an interpertation of the antecedent of (1) that is true (0 ≤ 0) but the
interpretation of the consequent of (1) is false. So the contraction rule is
not derivable in G∧,∨,¯. 2
Theorem 7. The variety BCILM is not the equivalent algebraic semantics
for any deductive system.
Proof: In [6, proposition 2.1] it is proved that for the four-element dis-
tributive lattice A = 〈{0, a, b, 1} ,∧,∨〉, with 0 < b < a < 1, the Leibniz
operator ΩA cannot be an isomorphism between the filters of an arbi-
trary deductive system and the congruences of the algebra A. Consider
A′ = 〈{0, a, b, 1} ,∧,∨,¯, 0, 1〉, where ¯ = ∧, then A′ ∈ BCILM. It is
easy to check that the proof of [6, proposition 2.1] applies in our case to
show that BCILM is not the equivalent algebraic semantics of any deductive
system. 2
Now we will consider the external deductive system associated with
G∧,∨,¯.
Definition 8. Let L = {∧,∨,¯, 0, 1}. Let S0G∧,∨,¯ =
〈
FmL,`S0G∧,∨,¯
〉
be the external deductive system associated with G∧,∨,¯; that is, `S0G∧,∨,¯
is the finitary and structural consequence relation on the set FmL defined
by:
Γ `S0G∧,∨,¯ ϕ iff {∅ ⇒ γ : γ ∈ Γ} `LJ∗L\c ∅ ⇒ ϕ, for all Γ ∪ {ϕ} ⊆ FmL.
Theorem 9. S0G∧,∨,¯ and G∧,∨,¯ are not equivalent.
Proof: This theorem is a consequence of the fact that G∧,∨,¯ is algebraiza-
ble with its equivalent algebraic semantics, the variety BCILM, (theorem 5)
and the fact that BCILM is not the equivalent algebraic semantics for any
deductive system (theorem 7). 2
Theorem 10. The variety BCILM is an algebraic semantics of the de-
ductive system S0G∧,∨,¯ with defining equation ϕ ≈ 1, that is, for every
Γ ∪ {ϕ} ⊆ Fm{∧,∨,¯,0,1},
On a Substructural Gentzen System, its Equivalent Variety Semantics... 133
Γ `S0G∧,∨,¯ ϕ iff {1 ≈ ϕi : ϕi ∈ Γ} |=BCILM 1 ≈ ϕ.
Proof: By the definition of S0G∧,∨,¯ and theorem 5, we have that
Γ `S0G∧,∨,¯ ϕ iff {∅ ⇒ ϕi : ϕi ∈ Γ} `LJ∗{∧,∨,¯,0,1}\c ∅ ⇒ ϕ iff
{1 ≈ ϕi : i ∈ I} |=BCILM 1 ≈ ϕ. 2
Theorem 11. S0G∧,∨,¯ is not protoalgebraic.
Proof: To show this result we will check that the Leibniz operator is not
order-preserving on the S0G∧,∨,¯-filters of the algebra A = 〈{0, a, 1},∧,∨,¯,
0, 1〉 of type (2, 2, 2, 0, 0), where 0 < a < 1, ∧ and ∨ are infimum and
supremum, respectively, and ∧ = ¯.
First we prove that the S0G∧,∨,¯-filters of A are exactly the lattice
filters of A; that is, the S0G∧,∨,¯-filters of A are: {1}, {1, a} and {1, a, 0}.
Indeed,
• Suppose that F is a lattice filter of A. We have to show that F is a
S0G∧,∨,¯-filter; that is, if Γ `S0G∧,∨,¯ ϕ, h ∈ Hom(FmL,A) and h(Γ) ⊆
F , then h(ϕ) ∈ F . Consider the Gentzen system GLJ∗{∧,∨,¯,0,1} which
is obtained by adding the contraction rule to G∧,∨,¯. It is easy to
see that the external deductive system S0LJ∗{∧,∨,¯,0,1} associated with
GLJ∗{∧,∨,¯,0,1} coincides with the deductive system S obtained from
GLJ∗{∧,∨,¯,0,1} by means of Γ `S ϕ iff ∅ `LJ∗{∧,∨,¯,0,1} Γ ⇒ ϕ. This
deductive system S coincides with the fragment without implication
of the Intuitionistic Propositional Calculus. It is also known that the
filters of any distributive lattice A are the S-filters. Let us suppose
that Γ `S0G∧,∨,¯ ϕ, then Γ `S ϕ and as F is a S-filter we obtain that
h(ϕ) ∈ F .
• Suppose that F is a S0G∧,∨,¯-filter. We have to see that F is a lattice
filter of A. We have that {ϕ,ψ} `S0G∧,∨,¯ ϕ¯ ψ, since
{∅ ⇒ ϕ, ∅ ⇒ ψ} `LJ∗{∧,∨,¯,0,1}\c ∅ ⇒ ϕ¯ ψ and, therefore, if α, β ∈ A
and α ∈ F , β ∈ F , we have that α¯ β ∈ F .
We have that ϕ `S0G∧,∨,¯ ϕ ∨ ψ, since ∅ ⇒ ϕ `LJ∗L\c ∅ ⇒ ϕ ∨ ψ and,
therefore, if α, β ∈ A and α ≤ β and α ∈ F , we have that α∨ β ∈ F ,
that is, β ∈ F. So, F is a lattice filter of A.
134 Romà J. Adillon and Ventura Verdú
Finally let us see that the Leibniz operator is not order-preserving
on the S0G∧,∨,¯-filters of the algebra A. Consider the S
0
G∧,∨,¯-filters F1 =
{1} and F2 = {1, a} then we have that (0, a) ∈ ΩA ({1}), but (0, a) /∈
ΩA ({1, a}). So ΩA ({1}) 6⊆ ΩA ({1, a}). 2
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E. U. Estudis Empresarials Facultat de Matemàtiques
Universitat de Barcelona Universitat de Barcelona
Diagonal 696 Gran Via 585
08034 Barcelona, Spain 08007 Barcelona, Spain
adillon@emp.ub.es verdu@mat.ub.es


On a substructural Gentzen system, its equivalent variety semantics and its external deductive system

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On a substructural Gentzen system, its equivalent variety semantics and its external deductive system

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