rem on KC
KC is the intermediate logic axiomatized by ¬ϕ∨ ¬¬ϕ. KC is complete with respect to finite rooted frames with unique top points. It is known that
KC proves exactly the same negation-free formulas asIPC. That is for any negation-free formula ϕ, KC ` ϕ iff IPC ` ϕ. Jankov proved in [19] that
KCis the strongest intermediate logic that has this property. In this section, we give a frame theoretic alternative proof of Jankov’s Theorem. The basic idea of the proof comes from Theorem 3.3.1 in the previous section and the next theorem. For more details on the next theorem, one may refer to [8], [9].
Theorem 3.4.1. If L is an intermediate logic strictly extending IPC, i.e. IPC⊂L⊆CPC, then for some n ∈ω, L`ψw for some w in U(n).
Proof. Suppose χ is a formula satisfying
L`χ and IPC0χ.
Then there exists a finite rooted frame Fsuch that F6|=χ. Introduce a new propositional variable pw for every point w in W, and define a valuation V
by letting V(pw) = R(w). Put n = |F|. By Theorem 3.2.3, there exists a
generated submodel U(n)w such that U(n)w is a p-morphic image of F. By
the construction, we know that different points ofhF, Vihave different colors, thus hF, Vi ∼=U(n)w.
Consider the de Jongh formula ψw. Suppose L 0 ψw. Then there exists
a descriptive frame G of L such that G 6|= ψw. By Theorem 3.3.1, F is a
p-morphic image of a generated subframe of G. Thus, by Theorem 2.3.9, F is an L model. Since L ` χ, we have that F |= χ, which leads to a contradiction.
Now, we define formulasϕ0
w andψw0 , which are negation-free modifications
of de Jongh formulas. They do a similar job for KC-frames as de Jongh formulas do for all frames. First, we introduce some terminologies.
For any finite set X of formulas with |X|>1, let ∆X =^{ϕ↔ψ :ϕ, ψ∈X}.
For the case that |X|= 1 or 0, we stipulate ∆X =>.
LetU(n)w0 =hW, R, Vi be a generated submodel with a largest element
t of U(n) such that
• t |=p1∧ · · · ∧pn;
• colV(w)6=colV(v) for all w, v ∈W such thatw6=v.
Let r be a new propositional variable.
Definition 3.4.2. We inductively define the formulas ϕ0
w and ψ0w for every
w∈W. Ifd(w) = 1, ϕ0 w =p1∧ · · · ∧pn, ψ0 w =ϕ0w →r.
Ifd(w) = 2, letq be the propositional letter in notprop(w) with the least index. Define ϕ0 w = ^ prop(w)∧∆notprop(w)∧((q →r)→q)2, ψ0 w =ϕ0w →q.
Ifd(w)>2 andw≺ {w1,· · · , wm}, then let
ϕ0 w := ^ prop(w)∧(_newprop(w)∨ m _ i=1 ψ0 wi → m _ i=1 ϕ0 wi) and ψ0 w :=ϕ0w → m _ i=1 ϕ0 wi.
We will prove for theϕ0
w and ψw0 formulas a lemma (Lemma 3.4.8) which
is an analogy of Theorem 3.3.1 for the ϕw and ψw formulas.
2Note that in the definition, it does not matter which q∈ notprop(w) is chosen. For
Lemma 3.4.3. Let V be a valuation on a frame F =hW, R,Pi and V0 is a valuation on F defined by V0(p) = ½ V(p), p6=r; ∅, p=r.
Then for any formula ϕ,
hF, Vi |=ϕ iff hF, V0i |=ϕ[⊥/r]3.
Proof. We prove the lemma by induction on ϕ that for any w∈W,
hF, Vi, w |=ϕiff hF, V0i, w |=ϕ[⊥/r].
ϕ=⊥. Thenw=t, hF, Vi, w 6|=⊥ and hF, V0i, w 6|=r.
The induction steps that ϕ= ψ ∧χ, ψ∨χ and ϕ = ψ → χ are proved easily.
Lemma 3.4.4. LetU(n)w0 be a model described above with a largest element
t. Then for anyw∈W, we have that
(i) `IPC ϕw[⊥/r]↔ϕ0w;
(ii) `IPC ψw[⊥/r]↔ψw0 .
Proof. We prove the lemma by induction on d(w).
d(w) = 1. Then w=t, ϕw[⊥/r] = ϕ0w and ψw[⊥/r] =ψ0w, so the lemma
holds trivially. d(w) = 2. Then we have `ϕw ↔ ^ prop(w)∧(_notprop(w)∨ ¬(p1∧ · · · ∧pn)→p1∧ · · · ∧pn) `ϕw ↔ ^
prop(w)∧(_notprop(w)∨ ¬^notprop(w)→^notprop(w))
`ϕw ↔
^
prop(w)∧(_notprop(w)→^notprop(w))
∧(¬^notprop(w)→^notprop(w)) `ϕw ↔ ^ prop(w)∧ ^ s∈notprop(w) (s→^notprop(w)) ∧(¬^notprop(w)→^notprop(w)) `ϕw ↔ ^ prop(w)∧∆notprop(w)∧(¬q→q).
3We writeϕ[p/ψ] for the formula obtained by substituting all occurrences ofpinϕby
Thus, `ϕw[⊥/r]↔ϕ0w and
`ψw[⊥/r]↔(ϕw[⊥/r]→ϕt[⊥/r])
`ψw[⊥/r]↔(ϕ0w →ϕ0t)
`ψw[⊥/r]↔ψ0w.
The induction step thatd(w)>2 is proved easily by applying the induc- tion hypothesis.
For anywinU(n)w0 described above, by Theorem 3.2.6,U(n)w0 6|=ψw for eachw∈R(w0). Thus, by the Lemma 3.4.3 and Lemma 3.4.4, the underlying
frame of U(n)w0 falsifies ψw0 . Hence 6`IPC ψ0w for eachψ0w, wherew∈R(w0).
We will use this fact later in the proof of Theorem 3.4.9.
For anyw, v with wRv in then-universal model U(n), by Theorem 3.2.6 and Theorem 3.2.4, it is easy to prove that `IPC ϕv → ϕw. The next
lemma shows that the ϕw and ϕv formulas have the same property. Note
that Theorem 3.2.4 is not applicable for the ϕ0
w and ψw0 formulas. So here
we prove the next theorem directly from the construction of the ϕ0
w and ψw0
formulas.
Lemma 3.4.5. Let U(n)w0 =hW, R, Vi be a model described above and let
w, v be two points in W with wRv. Then we have that `IPC ϕ0v →ϕ0w.
Proof. For any finite rooted model M = hW0, R0, V0i with the root r and
some point x∈R0(r), suppose M, x|=ϕ0
v. We show that M, x|=ϕ0 w. (3.19) Ifd(v) = 1, then M, x|=p1∧ · · · ∧pn. (3.20) Clearly, M, x|=^prop(w). (3.21) We show (3.19) by induction on d(w).
d(w) =d(v) + 1 = 2. Then clearly (3.20) implies that M, x |= q, which implies that
M, x|= (q→r)→q.
d(w)> d(v) + 1. For any wi ∈ Sw, since d(wi)< d(w), by the induction hypothesis, M, x|=ϕ0 wi, thusM, x|= W wi∈Swϕ 0 wi and M, x|=_newprop(w)∨ _ wi∈Sw ψ0 wi → _ wi∈Sw ϕ0 wi. (3.22)
Hence, together with (3.21), (3.19) is obtained.
If d(v) = 2, then clearly (3.21) holds. We show (3.19) by induction on
d(w). d(w) =d(v) + 1. Then v is an immediate successor of w, and (3.22) follows from M, x|=ϕ0
v. Thus, together with (3.21), (3.19) is obtained.
d(w)> d(v) + 1. For any wi ∈ Sw, since d(wi)< d(w), by the induction
hypothesis, we have that M, x |= ϕ0
wi and so (3.22) holds. Thus, together
with (3.21), (3.19) is obtained.
Ifd(v)>2, then clearlyM, x|=ϕ0
v implies (3.21). By a similar argument
as above, we can show that (3.22) holds, thus, (3.19) is obtained.
A similar result to the next lemma for the de Jongh formulaϕw can also
be obtained by a similar argument.
Lemma 3.4.6. Let M=hW0, R0, V0i be any model and U(n)
w0 =hW, R, Vi
be a model described above. Put V00=V0 ¹ {p
1,· · · , pn}. For any point w in
U(n)w0 and any point x in M, if M, x|=ϕ0
w, M, x6|=ϕ0w1,· · · ,M, x6|=ϕ
0
wm, (3.23)
where w≺ {w1,· · · , wm}, then colV00(x) = colV(w).
Proof. We prove the lemma by induction on d(w).
d(w) = 1. Then (3.23) means thatM, x|=p1∧ · · · ∧pn. Note thatw=t
also satisfies U(n)w0, t |=p1∧ · · · ∧pn. So colV00(x) =colV(w).
d(w) = 2. Then (3.23) implies that
M, x|=^prop(w), (3.24) M, x|= ∆notprop(w), (3.25)
M, x6|=p1∧ · · · ∧pn. (3.26)
First, from (3.24), it follows that
Next, it follows from (3.26) that there exists pi (1 ≤ i ≤ n) such that
pi 6∈prop(x), which by (3.27) implies that pi ∈notprop(w). Thus, by (3.25),
M, x6|=Wnotprop(w) and so
notprop(x)∩ {p1,· · ·, pn} ⊇notprop(w).
Together with (3.27), we obtain colV00(x) = colV(w).
d(w)>2. Then (3.23) implies (3.24) and
M, x6|=_newprop(w), (3.28) M, x6|=ψ0wi, (3.29) for all wi ∈Sw. From (3.24), we obtain (3.27). From (3.28), we obtain
notprop(x)∩ {p1,· · · , pn} ⊇newprop(w). (3.30)
It follows from (3.29) that for eachwi ∈Sw, there exists y∈R0(x) such that
y and wi satisfy (3.23). Since d(wi)< d(w), by the induction hypothesis, we
have that colV00(y) = colV(wi), which implies that
notprop(x)∩ {p1,· · · , pn} ⊇notprop(y)∩ {p1,· · ·, pn}=notprop(wi).
Together with (3.30), we obtain
notprop(x)∩ {p1,· · · , pn} ⊇newprop(w)∪
[
wi∈Sw
notprop(wi) =notprop(w).
The above and (3.27) proves colV00(x) =colV(w).
The next lemma is crucial in the proof of Lemma 3.4.8. For the ϕw
formulas, a similar lemma (Corollary 3.2.12) is obtained as a corollary of the results on the connection of H(n) and U(n). However, this method cannot be generalized to the ϕ0
w formulas. Here we prove the next lemma directly.
Lemma 3.4.7. Let M and U(n)w0 be models described above. For any point
w in U(n)w0 and any point x in M, if M, x|=ϕ
0
w, then there exists a unique
v ∈R(w) such that M, x|=ϕ0 v, M, x6|=ϕ0v1,· · · ,M, x6|=ϕ 0 vm, (3.31) where v ≺ {v1,· · · , vm}.
Proof. Suppose M, x|= ϕ0
w. If for all wi ∈ Sw, M, x 6|=ϕ0wi, then w satisfies
(3.31). Now suppose that for some wi0 ∈ Sw, M, x |= ϕ0wi0. We show that
there exists v ∈R(w) satisfying (3.31) by induction on d(w).
d(w) = 1. Then trivially v =w satisfies (3.31).
d(w) > 1. Since M, x |= ϕ0
wi0 and d(wi0) < d(w), by the induction hypothesis, there exists v ∈W, such that wi0Rv and v satisfies (3.31). And clearly, wRv.
Next, supposev0 ∈R(w) also satisfies (3.31). By Lemma 3.4.6,
colV(v0) = colV00(x) = colV(v),
which by the property of U(n)w0 means that v0 =v.
LetFbe a finite rooted frame with a largest element x0. For every point
xinF, we introduce a new propositional variable px and define a valuationV
onFby lettingV(px) =R(x). Letn=|F|. By Theorem 3.2.3, there exists a
generated submodel U(n)w of U(n) such thatU(n)w is a p-morphic image of
hF, Vi. Since different points inhF, Vihave different colors, the p-morphism is injective and so hF, Vi ∼= U(n)w. Note that U(n)w has a top point t and
t |=p1∧ · · · ∧pn.
The next lemma is a modification of the Jankov-de Jongh Theorem (The- orem 3.3.1) proved in the previous section. Both the statement of the lemma and the proof are generalized from those of Theorem 3.3.1.
Lemma 3.4.8. For every finite rooted frame F with a largest element, let
U(n)w be the model described above. Then for every descriptive frame G,
G6|=ψ0
w iff F is a p-morphic image of a generated subframe of G.
Proof. Let U(n)w = hW, R,P, Vi. Suppose F is a p-morphic image of a
generated subframe ofG. By Theorem 3.2.6, U(n)w 6|=ψw, thus F6|=ψw. By
Lemma 3.4.3 and Lemma 3.4.4, we know thatF6|=ψ0
w. By applying Theorem
2.3.9, G6|=ψ0
w is obtained.
SupposeG6|=ψ0
w. Then there exists a modelN onG such thatN6|=ψw0 .
Consider the generated submodel N0 = N
V0(ϕ0
w) = hW
0, R0,P0, V0i of N.
SinceV0(ϕ0
w) is admissible, by Lemma 2.6.13,N0 is admissible. Define a map
f :W0 →W by takingf(x) =v iff
N0, x|=ϕ0
v, N0, x6|=ϕ0v1,· · · ,N
0, x6|=ϕ0
where v ≺ {v1,· · · , vk}.
Note that for every x ∈ N0, N0, x |= ϕ
w, thus by Lemma 3.4.7, there
exists a unique v ∈R(w) satisfying (3.32). So f is well-defined.
We show that f is a surjective frame p-morphism of hW0, R0,P0i onto hW, R,Pi. Suppose x, y ∈ N0 with xR0y, f(x) = v and f(y) = u. Since
N0, x|=ϕ0
v, we have that N0, y |=ϕ0v. By Lemma 3.4.7, there exists a unique
u0 ∈ R(v) such that u0 and y satisfy (3.32). So, since u and y also satisfy
(3.32), by the uniqueness, u0 =u and vRu.
Next, suppose x ∈ N0 and v, u ∈ W such that f(x) = v and vRu. We
show that
there exists y∈N0 such that f(y) = u and xR0y. (3.33)
Ifd(v) = 1, then u=v, so trivially xR0x and f(x) =v =u.
If d(v) = 2, then if u = v, we have that (3.33) trivially holds. Now suppose u=t. Since f(x) = v,v and xsatisfy (3.32), so
N0, x|=^prop(v)∧∆notprop(v)∧((q→r)→q). (3.34)
It then follows that N0, x|= (q →r)→q. Note that `IPC((q→r)→q)→ ¬¬q.
Thus, N0, x |= ¬¬q, which means that there exists y ∈ W0 such that xR0y
and N0, y |=q. Since
N0, y |=^prop(v)∧∆notprop(v),
we have that N0, y |=p
1∧ · · ·pn, i.e. f(y) = u.
Ifd(v) >2, then since x and v satisfy (3.32), by the definition of ϕ0
v, we
must have that
N0, x6|=ψ0
vi, (3.35)
for 1≤i≤k. We now show by induction on d(u) that (3.33) holds.
d(u) = d(v)−1. Then u is an immediate successor of v and u satisfies (3.35). There are two cases. Case 1: d(u) = 2. Then it follows from N0, x6|=
ψ0
u that there exists y ∈ R0(x) such that N0, y |= ϕ0u and N0, y 6|= q. The
latter implies that N, y 6|=p1∧ · · · ∧pn, i.e. N, y 6|=ϕ0t. Thus y and u satisfy
Case 2: d(u)> 2. Then by the definition of ψ0
u, N0, x6|=ψ0u implies that
there exists y∈R0(x) such that y and u satisfy (3.32), thus f(y) = u.
d(u)< d(v)−1. Then there exists an immediate successor vi0 of v such that vi0Ru. By the basic step of the induction, there exists y∈W
0 such that
xR0y and f(y) = v
i0. Sinced(vi0)< d(v), by the induction hypothesis, there exists z ∈W0 such thatyR0z and f(z) =u. And clearly, xR0z.
For any upset X ∈ P, we have that X = Sv∈XR(v). By applying Lemma 3.4.5, Lemma 3.4.7 and using a same argument as that in the proof of Theorem 3.3.1, we can show that for every v ∈X,
f−1(R(v)) =V0(ϕ0v). So, f−1(X) = f−1([ v∈X R(v)) = [ v∈X f−1(R(v)) = [ v∈X V0(ϕ v).
Since X is finite, we obtain f−1(X)∈ P0.
Lastly, we show that f is surjective. First, by a similar argument as above, we can show that for w ∈ W, there exists x ∈ W0 such that f(x) =
w. Next, for every v ∈ W, we have that wRv. So by (3.33), there exists
y∈R0(x)⊆W0, such thatf(y) =v.
Hence,f is a surjective frame p-morphism ofhW0, R0,P0iontohW, R,Pi.
Then sinceF∼=hW, R,Pi, Fis a p-morphic image of hW0, R0,P0i, which is a
generated subframe of G.
Now we are ready to prove Jankov’s theorem on KC.
Theorem 3.4.9 (Jankov). If L is an intermediate logic such that L*KC,
then L`θ and IPC0θ for some negation-free formula θ.
Proof. Suppose χ is a formula satisfying
L`χ and KC0χ.
Then there exists a finite rootedKC-frameFwith a largest element such that F 6|= χ. For every point w in F, we introduce a new propositional variable
pw and define a valuation V on F by letting V(pw) = R(w). Let n = |F|.
By Theorem 3.2.3, there exists a generated submodel U(n)w of U(n) which
is a p-morphic image of hF, Vi. Note that U(n)w has a largest element t,
Consider the formulaψ0
w. Suppose L0ψ0w. Then there exists a descrip-
tive frame G of L such that G 6|= ψ0
w. By Lemma 3.4.8, F is a p-morphic
image of a generated subframe ofG. Thus, by Theorem 2.3.9,Fis anL-frame. Thus, since L`χ, we have that F|=χ, which leads to a contradiction.
Hence,L`ψ0
w. Note thatIPC0ψ0wandψw0 is negation-free, thusθ =ψw0
Chapter 4
Subframe logics and subframe
formulas
In this chapter, we summarize classic and recent results on subframe logics and subframe formulas. Subframe logics are intermediate logics that are characterized by a class of frames closed under subframes. The study of intuitionistic subframe logics was first inspired by the related results on modal subframe logics, where Fine [14] and Zakharyaschev [28] defined the subframe formulas and proved the finite model property of subframe logics. In [27], [29] (see also [6]) Zakharyaschev defined subframe formulas for intermediate logics, which are [∧,→]-formulas. It then follows from Zakharyaschev [29], [30] (see also [6]) that subframe logics are exactly those logics axiomatized by [∧,→]-formulas. N. Bezhanishvili proved in [3] that subframe logics can also be axiomatized by NNIL-formulas. G. Bezhanishvili and Ghilardi [2] gave an algebraic approach to subframe logics by using the tools of nuclei in topos theory and proved that a variety of Heyting algebras is nuclear iff it is a subframe variety. Also in [2], an alternative proof of the finite model property of subframe logics is given from the algebraic point of view.
In Section 4.1, we give definitions of subframes of intuitionistic general frames and subframe logics. In Section 4.2, we give the definition of sub- frame formulas in NNIL-form, which was done in [3]. In Section 4.3, we provide a frame-based proof of the property that subreductions preserve [∧,→]-formulas. In Section 4.4, we state the equivalent characterizations of subframe logics.
4.1
Subframe logics
In this section, we recall the important notions related to subframe logics. For more details on subframe logics, one may refer to [6].
Definition 4.1.1. Let F = hW, R,Pi and G = hV, S,Qi be intuitionistic general frames. A partial map f from W onto V is called a subreduction of F to Gif it satisfies the following
(R1’) For any w, v ∈dom(f), wRv implies f(w)Sf(v);
(R2’) f(w)Sv0 implies∃v ∈dom(f),wRv and f(v) = v0; (R3’) ∀X ∈ Q, f−1(X)↓∈ P.
Remark 4.1.2. If a subreduction is total, then (R3’) is equivalent to (R3). This means that any reduction is also a subreduction and any total subreduc- tion is also a reduction.
An intuitionistic general frame G = hV, S,Qi is called a subframe of an intuitionistic general frame F =hW, R,Pi, if hV, Si is a subframe of hW, Ri
and the inclusion map is a subreduction, i.e.,
U ∈ Q impliesR−1(U)∈ P.
Alternatively, we can define subframe in topological terminology. A Heyt- ing spaceY =hY, ν, Siis called asubframe of a Heyting spaceX =hX, τ, Ri
if hY, Si is a subframe of hX, Ri, hY, νi is a subspace ofhX, τi, and
U is a clopen ofY implies that R−1(U) is a clopen of X.
Also, a correspondence between subframe and nuclei can be found in [2].
Remark 4.1.3. If G is a subreduct of F, then G is a reduct (p-morphic
image) of a subframe of F.
There are many ways of to define a subframe logic. In subsequent sections, we will see that these characterizations are equivalent.
Definition 4.1.4. An intermediate logicL is called asubframe logic, if it is characterized by a class of frames that is closed under subframes (i.e. every subframe of an L-frame is also an L-frame).