Bulletin of the Section of Logic
Volume 18/2 (1989), pp. 63–71
reedition 2005 [original edition, pp. 63–71]
Nobu-Yuki Suzuki
INTERMEDIATE LOGICS CHARACTERIZED BY
A CLASS OF ALGEBRAIC FRAMES WITH
INFINITE INDIVIDUAL DOMAIN
From semantical point of view, an essential nature of predicate logics
depends on the fact that individual domains are infinite – not potentially
but actually infinite. Ono [2] proved that a class C of algebraic frames
characterizes an intermediate predicate logic (an IPL) if and only if there
exists at least one algebraic frame (P, U ) in C whose individual domain U
is infinite. Hence, a class of algebraic frames each of which has an infinite
individual domain always characterizes an IPL. Our starting point is the
following question: What kind of an IPL does such a class characterize?
We say such an IPL is ω + -complete.
In Section 1, we will define a syntactical property which we call the
pseudo-relevance property (PRP, for short), which can be regarded as a
weak version of Craig’s interpolation property. We will show that if L is
an algebraically complete IPL with the above PRP, then L is ω + -complete.
Next, we will show that the ω + -completeness of an IPL L implies PRP of
L, provided that K(≡ ¬¬∀x(p(x) ∨ ¬p(x))) is provable in L. By making
use of this result, we will show that completeness does not always imply
ω + -completeness in Section 2.
In the following, we assume the readers’ familiarity with Ono [2].
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Nobu-Yuki Suzuki
1. We will repeat some terminology and notation. We refer readers
to Ono [2] for details.
As usual, we fix a first-order language L, which contains neither constants nor function symbols (see Ono [2]). For each non-empty set U , we
denote by L[U ] the language obtained from L by adding the name u of
each u ∈ U as new constants. We sometimes identity L[U ] with the set of
all sentences of L[U ]. A pair (P, U ) of a non-degenerate pseudo-Boolean
algebra P and a non-empty set U is said to be an algebraic frame if P is
λ-complete, where λ is the cardinality of U . An assignment f of an algebraic frame (P, U ) is a mapping of the set of all atomic sentences in L[U ]
to P = <P, ∪, ∩, →, 0, 1 >. We extend it to a mapping of L[U ] to P as
follows:
(A1)
f (A ∨ B)
(A2)
f (A ∧ B)
(A3) f (A ⊃ B)
(A4)
f (¬A)
(A5) f (∀xA(x))
(A6) f (∃xA(x))
=
=
=
=
=
=
f (A) ∪ f (B),
f (A) ∩ f (B),
f (A) → f (B),
fT(A) → 0,
Su∈U f (A(u)),
u∈U f (A(u)).
A formula A (of L) is said to be valid in an algebraic frame (P, U )
if f (A) = 1 holds for every assignment f of (P, U ), where A is the universal closure of A. The set of all formulas of L that are valid in an algebraic frame (P, U ) is denoted by L+ (P, U ). As is well-known, L+ (P, U )
contains all formulas provable in the intuitionistic predicate logic H, and
is closed under modus ponens, the rule of generalization and the rule of
substitution. An intermediate predicate logic (an IPL) L is said to be (algebraically)
complete if there exists a class C of algebraic frames such that
T
L = (P,U )∈C L+ (P, U ).
Definition 1.1. An algebraic frame (P, U ) is said to be an ω + -frame if
+
U is infinite. Let L be an IPL. L is said
T to be ω +-complete if there exists
+
a class C of ω -frames such that L = (P,U )∈C L (P, U ).
It is obvious that ω + -completeness implies completeness. In Section 2,
we will prove that ω + -completeness does not follow from completeness.
Next, we will introduce a syntactical property which plays an important
rôle in this paper.
Intermediate Logics Characterized by a Class of Algebraic Frames with...
65
Definition 1.2. Let L be an IPL. L is said to have the pseudo-relevance
property (PRP) if L satisfies the following condition (*):
(*) For every formula A and B (in L) which contain no predicate
variable in common, if A ⊃ B is provable in L then either ¬A or B is
provable in L.
We remark here that PRP can be regarded as a weak version of Craig’s
interpolation property (see e.g., Ono [3]). We can define PRP analogously
for intermediate propositional logics. However, it is well-known that every
intermediate propositional logic has PRP (see Komori [1]). On the other
hand, PRP is not trivial for IPL’s.
Theorem 1.3. For every intermediate propositional logic J other than
the classical one, there exist at least countably infinite predicate extensions
of J each of which does not have PRP.
Proof. Let an intermediate propositional logic J other than the classical
one be fixed. Define formulas Yn (n ≥ 1), Zn (n ≥ 1), and N Rn (n ≥ 1) by
Ym
≡ p1 (x);
m
V
≡ ( ¬pi (x)) ∧ pm (x) (m ≥ 2),
Zn
≡
Y1
i=1
n
V
∃xYi , and
i=1
N Rn
≡ Zn ⊃ (q ∨ ¬q),
where each pi (i = 1, 2, . . .) is a monadic predicate variable and q is a
propositional variable. Since each N Rn (n ≥ 2) is provable in the classical
predicate logic C, we see that J∗ +N Rn (n ≥ 2) is an IPL, where J∗ +N Rn
is the logic obtained from the minimum predicate extension J∗ of J by
adding N Rn as a new axiom. Since the associated propositional formula
α(Zn ) of each Zn (n ≥ 2) is logically equivalent to “absurdum” in H,
each α(N Rn ) is provable in J. Hence, for each n (n ≥ 2), J∗ + N Rn is
a predicate extension of J. Since J∗ + N Rn is strictly weaker than C,
neither ¬Zn nor q ∨ ¬q is provable in J∗ + N Rn , and hence J∗ + N Rn
fails to have PRP. It remains to prove that J∗ + N Rm 6= J∗ + N Rn if
m 6= n. Let S2 be the three-valued linear pseudo-Boolean algebra. Then,
if n > m ≥ 2, an algebraic frame (S2 , {1, 2, . . . , m}) validates N Rn but
not N Rm . Hence, J∗ + N Rn ⊂ L+ (S2 , {1, 2, . . . , m}) 6∈ N Rm . Therefore,
J∗ + N Rm 6= J∗ + N Rn .
q.e.d.
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Nobu-Yuki Suzuki
Since there are uncountably many intermediate propositional logics,
Theorem 1.3 provides us with uncountably many ILP’s without PRP. For
complete IPL’s, PRP is a sufficient condition to be ω + -complete. That is,
Theorem 1.4.
complete.
Let L be a complete IPL. If L has PRP, then L is ω + -
Proof. Assume that L is complete with respect to a class C of algebraic frames. Let D be the T
class of ω + -frames in C. We will show
that if L has PRP, then L = (P,U )∈D L+ (P, U ). Recall that if L =
T
L+ (P, U ), then L is ω + -complete. Suppose that L is a proper
(P,U )∈D T
subset of (P,U )∈D L+ (P, U ). Then there exists a sentence A such that
(1) A is not provable in L.
(2) A is valid in every ω + -frame in C.
Define sentences F1 , F2 and F in by
F1
F2
F in
≡ ∀xr(x, x) ∧ ∀x∀y(r(x, y) ∨ r(y, x)),
≡ ∀x∀y∀z(r(x, y) ∧ r(y, z) ⊃ r(x, z)), and
≡ F1 ∧ F2 ⊃ ¬∀x∃y¬r(y, x),
where r is binary predicate variable. Ono [2, Theorem 2.6] proved that
F in is valid in every algebraic frames whose individual domain is finite.
Hence, A ∨ F in is valid in every algebraic frame in C. It follows from this
that A ∨ F in is provable in L. Since A ∨ F in ⊃ (¬F in ⊃ A) is provable in
H, ¬F in ⊃ A is provable in L. Without loss of generality, we can assume
that ¬F in and A contain no predicate variable in common. Recall that
¬¬F in is not provable in C. So, ¬¬F in ≡ ¬(¬F in) is not provable in L.
From this and (1), it follows that L does not have PRP.
q.e.d.
The converse of Theorem 1.4 does not always hold (see Section 2).
However, the converse holds under some additional conditions. Let K be
the formula ¬¬∀x(p(x) ∨ ¬p(x)), where p is a monadic predicate variable.
Note that K is provable in C.
Theorem 1.5. Let L be an IPL in which K is provable. If L is ω + complete, then L has PRP.
In the rest of this Section, we will prove Theorem 1.5. Our proof proceeds
similarly to the case of intermediate propositional logics in Komori [1, Theorem 2.1]. He made use of Glivenko’s theorem. The condition that, “K
Intermediate Logics Characterized by a Class of Algebraic Frames with...
67
is provable” has a closed relationship to Glivenko’s theorem in IPL’s as
follows.
Fact 1.6. (Umezawa [5]) Let L be an IPL. The following two conditions
on L are equivalent:
(i) K is provable in L,
(ii) For every formula A, ¬A is provable in C if and only if ¬A is
provable in L.
Lemma 1.7. Let a formula A ≡ A(x1 , . . . , xn ) contains no free variables
other than x1 , . . . , xn . If ¬A is not provable in C, then, for every ω + -frame
(P, U ), there exist an assignment f of (P, U ) and elements u1 , . . . , un ∈ U
such that f (A(u1 , . . . , un )) = 1P where 1P is the greatest element of P .
Proof. Since U is infinite, C is complete with respect to an ω + -frame
(2, U ), where 2 is the two-valued Boolean algebra, i.e., 2 = {0, 1}. Hence,
there exist an assignment g of (2, U ) and elements u1 , . . . , un ∈ U such
that
(3) g(A(u1 , . . . , un )) = 1 in 2.
Define a mapping f of the set of all atomic sentences in L[U ] to P by
1P if g(p(v1 , . . . , vm )) = 1,
f (p(v1 , . . . , vm )) =
0P otherwise,
for each m−ary predicate variable p and each vi ∈ U (i = 1, . . . , m), where
0P is the least element of P . Extending f as an assignment of (P, U ), we
have that for every sentence C of L[U ],
1P if g(C) = 1,
f (C) =
0P otherwise.
It follows from (3) that f (A(u1 , . . . , un )) = 1P .
Proof of Theorem 1.5. Suppose that ¬A and B contain no predicate
variable in common, and none of them is provable in L. Without loss of
generality, we can assume that A and B contain no free variables other
than x1 , . . . , xn . We write A(x1 , . . . , xn ) and B(x1 , . . . , xn ) for A and B
respectively. Then, there exist an ω + -frame (P, U ), an assignment f1 of
(P, U ) and elements u1 , . . . , un ∈ U such that
(4) L ⊂ L+ (P, U ), and
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Nobu-Yuki Suzuki
(5) f1 (B(u1 , . . . , un )) 6= 1P in P .
By Fact 1.6, ¬A(x1 , . . . , xn ) is not provable in C. Hence, by Lemma
1.7, there exist an assignment f2 of (P, U ) and elements v1 , . . . , vn ∈ U
such that
(6) f2 (A(v1 , . . . , vn )) = 1P in P .
Define mappings σ1 and σ2 of the set of atomic sentences in L[U × U ] to
L[U ] by
σ1 (p((a1 , b1 ), . . . , (am , bm ))) = p(a1 , . . . , am ), and
σ2 (p((a1 , b1 ), . . . , (am , bm ))) = p(b1 , . . . , bm ),
for every m-ary predicate variable p and every (ai , bi ) ∈ U × U (i =
1, . . . , m). By induction, σ1 and σ2 can be extended to mappings of L[U ×U ]
to L[U ] such that for every C ∈ L[U × U ],
(7) σ1 (C((a1 , b1 ), . . . , (am , bm ))) = C(a1 , . . . , am ), and
(8) σ2 (C((a1 , b1 ), . . . , (am , bm ))) = C(b1 , . . . , bm ).
Next, f is defined to be a mapping which sends each atomic sentence
p((a1 , b1 ), . . . , (am , bm )) of L[U × U ] for m-ary predicate variable p and
for (ai , bi ) ∈ U × U (i = 1, . . . , m) to the following element of P :
f1 (p(a1 , . . . , am )) if p occurs in B,
f (p((a1 , b1 ), . . . , am , bm ))) =
f2 (p(b1 , . . . , bm )) otherwise.
We extend f to an assignment of (P, U ) by induction. Then, we can prove
that for every C ∈ L[U × U ],
(9) if every predicate variable in C appears in B, f (C) = f1 (σ1 (C)),
(10) if every predicate variable in C appears in A, f (C) = f2 (σ2 (C)).
Thus, by (7), (9) and (5), we have that
f (B((u1 , v1 ), . . . , (un , vn ))) = f1 (B(u1 , . . . , un )) 6= 1P .
And, by (8), (10) and (6),
f (A((u1 , v1 ), . . . , (un , vn ))) = f2 (A(v1 , . . . , vn )) = 1P .
Therefore, f ((A ⊃ B)((u1 , v1 ), . . . , (un , vn ))) 6= 1P . It follows from this
that A ⊃ B is not valid in (P, U × U ). Since U is infinite, U × U can
be identified with U as an individual domain. Thus, we have A ⊃ B 6∈
L+ (P, U ). Hence, A ⊃ B is not provable in L by (4).
q.e.d.
Intermediate Logics Characterized by a Class of Algebraic Frames with...
69
Theorem 1.5 provides us with a powerful device for proving non-ω + completeness of a given IPL. It suffices to show that K is provable in it
and it does not have PRP.
Is it possible to eliminate the condition that K is provable in L?
The answer is negative (see Section 2). On the other hand, there exists
an ω + -complete IPL with PRP in which K is not provable, namely the
intuitionistic predicate logic H.
2. In this Section, we will compare completeness, ω + -completeness
and PRP.
Theorem 2.8. There exists a complete IPL which is not ω + -complete.
That is, ω + -completeness does not follow from completeness.
Proof. Let L1 = L+ (2, ω) ∩ L+ (S2 , {0}), where ω is the first infinite
ordinal, which we will identify with the set {i; i < ω}. Clearly, L1 is a
complete IPL. It is a routine to check that K is valid both in (2, ω) and in
(S2 , {0}). Hence, K is provable in L1 . If L1 is ω + -complete, L1 must have
PRP by Theorem 1.5. So, it suffices to show that L1 does not have PRP.
Let p and q be a monadic predicate variable and a propositional variable
respectively. In (2, ω), q ∨ ¬q is valid, hence so is (∃xp(x) ∧ ∃x¬p(x)) ⊃
(q ∨ ¬q). In (S2 , {0}), ∃xp(x) ∧ ∃x¬p(x) is always assigned to the least
element of S2 . Hence (∃xp(x) ∧ ∃x¬p(x)) ⊃ (q ∨ ¬q) is valid in (S2 , {0}).
It follows from this that (∃xp(x) ∧ ∃x¬p(x)) ⊃ (q ∨ ¬q) is provable in L1 .
On the other hand, neither ¬(∃xp(x) ∧ ∃x¬p(x)) nor (q ∨ ¬q) is provable
in L1 .
q.e.d.
Theorem 2.9. There exists an ω + -complete IPL without PRP. That is,
PRP does not follow from ω + -completeness.
Proof. Define a set Q and an order on Q by
Q = {(x, y); x = 1, 2 and y = 1, 2, . . . , ω} ∪ {(0, 0)}
and
(x1 , y1 ) (x2 , y2 ) if and only if x2 ≤ x1 and y2 ≤ y1 ,
where ≤ is the natural order on ω∪{ω}. Then, Q together with coincides
with the complete pseudo-Boolean algebra illustrated in Figure 1. We write
Q for this algebra.
70
Nobu-Yuki Suzuki
◦ (0,0)
◦ (1,1)
@
@◦ (2,1)
(1,2) ◦
@
@◦ (3,1)
@
@◦
(2,2)
@
@
@
@
(3,2) ◦
·
@
@
·
· (ω, 1)
·
◦
·
·
(ω, 2) ◦
Figure 1
Clearly, L2 = L+ (Q, ω) is an ω + -complete IPL. We will show that L2
does not have PRP. Let Lin be the formula (q ⊃ r) ∨ (r ⊃ q), where q
and r are propositional variables. it can be easily verified that Lin is not
provable in L2 . Next, define an assignment f of (Q, ω) by
f (p(i)) = (i + 1, 1)
for each i ∈ ω. Then we have that f (¬¬K) = (ω, 1). hence, ¬¬K
is not provable in L2 . It remains to show that ¬K ⊃ Lin is valid in
(Q, ω). By an easy calculation, we have that (1, 1) f (Lin) for every
assignment f of (Q, ω). Verify that (ω, 1) f (p(i) ∨ ¬p(i)) holds for every assignment f of (Q, ω). Hence (ω, 1) f (∀x(p(x) ∨ ¬p(x))). Thus
f (¬K) = f (¬∀x(p(x) ∨ ¬p(x))) (ω, 1) → (ω, 2) = (1, 2), where → is the
relative pseudo-complementation of Q. Therefore f (¬K ⊃ Lin) = (0, 0)
for every assignment f of (Q, ω). Thus we have that ¬K ⊃ Lin is valid in
(Q, ω).
q.e.d.
Recall that K 6∈ L+ (Q, ω). It follows that in Theorem 1.5, we cannot
eliminate the condition “K is provable in L”. We illustrate the situation
simply in Figure 2. Note that each implication (→) was proved to be
proper.
Intermediate Logics Characterized by a Class of Algebraic Frames with...
71
ω + − complete + “K is provable”
↓
PRP + complete
↓
+
ω − complete
↓
complete
Figure 2
Acknowledgement. The author would like to express his thanks to
Professor Hiroakira Ono and Professor Mitio Takano for their comments
on the earlier versions of this article. He also wishes to thank Professor
Masazumi Hanazawa and Mr. Minoru Narui for discussions.
References
[1] Y. Komori, Logics without Craig’s interpolation property, Proceedings of the Japan Academy 54 (1978), pp. 46–48.
[2] H. Ono, A study of intermediate predicate logics, Publications
of Research Institute for Mathematical Sciences, Kyoto University
8 (1972), pp. 619–649.
[3] H. Ono, Some problems in intermediate predicate logics, Reports
on Mathematical Logic 21 (1987), pp. 55–67.
[4] H. Rasiowa, An algebraic approach to non-classical logics, Studies in Logic and the Foundation of Mathematics 78, North-Holland
Publishing Company, Amsterdam-London, 1974.
[5] T. Umezawa, On some properties of intermediate logics, Proceedings of the Japan Academy 35 (1959), pp. 575–577.
Mathematical Institute
Tôhoku University
Sendai 980, Japan