Download the simple substitution property of the intermediate propositional logics

Bulletin of the Section of Logic
Volume 18/3 (1989), pp. 94–99
reedition 2006 [original edition, pp. 94–99]
Katsumi Sasaki
THE SIMPLE SUBSTITUTION PROPERTY OF THE
INTERMEDIATE PROPOSITIONAL LOGICS
We study about the intermediate propositional logics obtained from
the intuitionistic propositional logics by adding some axioms.
We define the simple substitution property for a set of additional axioms. Hitherto have been known only three intermediate logics that have
the additional axioms with the property, and in this paper, we add one
more example, that is, we give new axioms with the property to the logic
Sω . Further, we decide with one among the axioms composed of one propositional variable has the property.
For the intuitionistic logic, we use the system LJ presented by Gentzen.
We use, possibly with suffixes, lower case Latin letters for propositional
variables, and upper case Latin letters for formulas. For logical connectives, we use the symbols ¬, ∧, ∨ and ⊃ as usual, and we assume the order
of priority in this order. Formulas are also defined as usual. And further,
we assume the association to the left, concerning each one of the logical
connectives. We denote the set of formulas as W F F .
Now, we define the simple substitution property.
Let L be a logic defined as LJ + B1 + B2 + . . . + Bm , where Bi ’s
are axioms added to LJ. Let Πn (Bi ) be the conjunction of all the formulas obtained from Bi by substituting the propositional variables of Bi by
a1 , a2 , . . . , an in all possible ways.
Definition 1. Let A be a formula constructed with the propositional
variables a1 , a2 , . . . , an . If there holds
A ∈ L if and only if Πn (B1 ) ∧ Πn (B2 ) ∧ . . . ∧ Πn (Bm ) ⊃ A ∈ LJ,
The Simple Substitution property of the Intermediate propositional Logics
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then we say that the set of axioms {B1 , B2 , . . . , Bm } for L has the simple
substitution property.
We remark here that, if a set of axioms has the simple substitution
property, we can prove or disprove a formula in the logic by using the
decision procedure for LJ. So the simple substitution property provides a
decision procedure by an axiomatic way.
In Hosoi [1], it is proved that each one of the sets of axioms {a ∨ ¬a},
{¬a ∨ ¬¬a} and {¬a ∨ ¬¬a, a ∨ (a ⊃ b) ∨ ¬b} has the simple substitution property. The method used there for proving the simple substitution
property for these sets of axioms is also useful in our investigations.
Here, for Gödel’s Sω , we give a set of axioms which has the simple
substitution property.
The logic Sω has been axiomatized as
Sω = LJ + Z(a, b), where Z(a, b) = (a ⊃ b) ∨ (b ⊃ a),
but the set, of axioms {Z(a, b)} does not have the simple substitution
property, since Z(a, a) ⊃ ¬a ∨ ¬¬a 6∈ LJ while ¬a ∨ ¬¬a ∈ Sω .
Here, we introduce a new axiom:
Z 0 (a, b) = (a ⊃ b) ∨ (a ⊃ b ⊃ a).
We also use the following axiom:
Q(a) = ¬a ∨ ¬¬a.
Since Z(a, a ⊃ b) ⊃ Z 0 (a, b) ∈ LJ and Z 0 (a, b) ⊃ Z(a, b) ∈ LJ, we
have the following theorem.
Theorem 2. Sω = LJ + Z 0 (a, b).
The next theorem is well known and we used it already above.
Theorem 3. Q(a) ∈ Sω .
Next, we prove the following theorem.
Theorem 4. The set of axioms {Q(a), Z 0 (a, b)} for Sω has the simple
substitution property.
Proof. The following formula are provable in LJ.
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Katsumi Sasaki
Q(B) ⊃ Q(¬B),
Q(B) ∧ Q(C) ⊃ Q(B ∧ C),
Q(B) ∧ Q(C) ⊃ (B ∨ C),
Q(B) ∧ Q(C) ⊃ Q(B ⊃ C),
Q(B) ⊃ Z 0 (¬B, C),
Q(B) ∧ Q(C) ⊃ Z 0 (B, ¬C),
Z 0 (B, D) ∧ Z 0 (C, D) ⊃ Z 0 (B ∧ C, D),
Z 0 (B, C) ∧ Z 0 (B, D) ⊃ Z 0 (B, C ∧ D),
Z 0 (B, D) ∧ Z 0 (C, D) ⊃ Z 0 (B ∨ C, D),
Z 0 (B, C) ∧ Z 0 (B, D) ∧ Z 0 (C, D) ⊃ Z 0 (B, C ∨ D),
Z 0 (B, C) ∧ Z 0 (C, D) ⊃ Z 0 (B ⊃ C, D),
Z 0 (B, D) ∧ Z 0 (C, D) ⊃ Z 0 (B, C ⊃ D).
By the method of [1], we can conclude the proof.
Now, we investigate all the axioms composed of only one variable
whether they have the simple substitution property. The axioms with one
variable have been studied minutely in Nishimura [3]. He defined a sequence
of basic formulas {Ni (a) | i = 0, 1, . . . , ∞} as follows. (In this paper, we
write Ni (a) for his Pi , since Pi ’s are used for different axioms in the field
of intermediate logics.)
Definition 5. The basic formulas are defined inductively as follows.
Nω (a) = a ⊃ a, N0 (a) = a ∧ ¬a,
N1 (a) = a, N2 (a) = ¬a,
N2n+3 (a) = N2n+1 (a) ∨ N2n+2 (a) (n = 0, 1, 2, . . .),
N2n+4 (a) = N2n+3 (a) ⊃ N2n+1 (a) (n = 0, 1, 2, . . .).
We list some of Nishimura’s results without proofs.
Theorem 6. Any one of the formulas with the variable a is equivalent to
one and the only one of the basic formulas in LJ.
Theorem 7.
other in LJ.
Any pair of the basic formulas are not equivalent to each
We remark that Ni (a) ∈ LJ if and only if i = ∞.
Theorem 8. As logics,
LJ + N0 (a) = LJ + N1 (a) = LJ + N2 (a) = LJ + N4 (a) (= W F F ),
LJ + N3 (a) = LJ + N6 (a) (= LK),
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97
LJ + N5 (a) = LJ + N8 (a) (= LQ),
where, in parentheses, we show the usual names for the logics.
It is known that the sets of axioms {Nω (a)}, {N0 (a)}, {N3 (a)} and
{N5 (a)} have the simple substitution property.
Further, it can be easily proved that N1 (a), N2 (a), N4 (a), N6 (a) and
N8 (a) do not have the simple substitution property.
By Nishimura’s figure 2 in [3], we know that
LQ ⊇
\ LJ + Ni (a) ⊇
\ LJ if and only if i = 7 or 9 ≤ i < ∞.
Next, we prove the following Lemma.
Lemma 9. For i = 7 or 9 ≤ i < ∞, any one of the sets of axioms {Ni (a)}
does not have the simple substitution property.
Proof. In Maksimova [2], it is proved that the logics having Craig’s
interpolation theorem are only seven:
LJ, LK, LQ, Sω , LJ + a ∨ (a ⊃ b ∨ ¬b),
LJ + a ∨ (a ⊃ b ∨ ¬b) + Z(a, b) and
LJ + a ∨ (a ⊃ b ∨ ¬b) + Z(a, b) ∨ (a ⊃ ¬b) ∧ (¬b ⊃ a).
Among these logics, only LK, LQ and LJ are axiomatized by Ni ’s. So
we remark that, if i = 7 or 9 ≤ i < ∞, then Craig’s interpolation theorem
never holds for LJ + Ni (a).
Now, we suppose that, for some i, where i = 7 or 9 ≤ i < ∞, the set
of axioms {Ni (a)} has the simple substitution property. And suppose that,
for some formulas B(b, c) and C(c, d),
B(b, c) ⊃ C(c, d) ∈ LJ + Ni (a).
Using the simple substitution property of {Ni (a)},
(Ni (b) ∧ Ni (c) ∧ Ni (d)) ⊃ (B(b, c) ⊃ C(c, d)) ∈ LJ.
Hence,
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Katsumi Sasaki
(Ni (b) ∧ Ni (c) ⊃ B(b, c)) ⊃ (Ni (c) ∧ Ni (d) ⊃ C(c, d)) ∈ LJ.
Using the interpolation property of LJ, we obtain that there exists a formula D(c) such that,
(Ni (b) ∧ Ni (c) ⊃ B(b, c)) ⊃ D(c) ∈ LJ
and
D(c) ⊃ (Ni (c) ∧ Ni (d) ⊃ C(c, d)) ∈ LJ.
On the other hand,
Ni (x) ∈ LJ + Ni (a) (x ∈ {b, c, d}).
Hence,
B(b, c) ⊃ D(c) ∈ LJ + Ni (a)
and
D(c) ⊃ C(c, d) ∈ LJ + Ni (a).
Hence, LJ + Ni (a) has the interpolation property, but this is contradictory to Maksimova’s result.
So, we obtain the Lemma.
Corollary 10. The sets of axioms with one variable which have the simple substitution property are only {Nω (a)}, {N0 (a)}, {N3 (a)} and {N5 (a)}.
Acknowledgement. The author would like to express his hearty thanks
to Prof. Tsutomu Hosoi and Prof. Hiroakira Ono. Prof. Hosoi gave the
problems as in this paper and guided him during these researches. The
name “the simple substitution property” is given by Prof. Hosoi. Prof.
Ono showed the elegant proof for the Lemma 9 using Maksimova’s result to
the author. Previously, the author proved this lemma by a direct method.
References
[1] T. Hosoi, Pseudo two-valued evaluation method for intermediate
logics, Studia Logica 45 (1986), pp. 3–8.
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[2] L. L. Maksimova, Craig’s theorem in superintuitionistic logics and
amalgamated varieties of pseudo-Boolean algebras, Algebra i Logika 16
(1977), pp. 643–681.
[3] I. Nishimura, On formulas of one variable in intuitionistic propositional calculus, The Journal of Symbolic Logic 25 (1960), pp. 327–331.
Department of Information Sciences
Science University of Tokyo
Noda City, CHIBA 278, JAPAN