Weak Subintuitionistic Logics
Fatemeh Shirmohammadzadeh Maleki,
Department of Mathematics,
Shahid Beheshti University, G. C., Evin, Tehran, Iran.
E-mail: [email protected]
Dick de Jongh,
Institute for Logic, Language and Computation,
University van Amsterdam. E-mail: [email protected]
Abstract
A systemWFof subintuitionistic logic is introduced, weaker than Corsi’s basic subintuitionistic sys-temF. A derivation system with and without hypotheses is given in line with the authors’ derivation system forF. A neighborhood semantics is introduced with a somewhat more complex definition than the neighborhood semantics for non-normal modal logics. Completeness is proved forWFwith respect to this neighborhood semantics, and similarly for some logics between WF and F which characterize nice frame classes. The study by the authors of the conservativity ofIPCoverFwith respect to some classes of implications is extended toWF, and shows clearly the difference in strength between the two logics. Study of translations of these weak subintuitionistic logics into non-normal modal logics turned out to be hard because of the difference between their respective neighborhood structures and leaves us with some open problems.
Keywords: subintuitionistic logic, intuitionistic logic, conservativity, neighborhood semantics, trans-lation.
1
Introduction
Subintuitionistic logics were studied by Corsi in 1987 [2], who introduced a basic systemFand by Restall in 1994 [9], who defined a similar systemSJ, both with Kripke models in which no assumption of preservation is made and also not of reflexivity and transitivity. Fcannot prove formulas likeA→(B→A). Corsi showed thatFcan be translated into the modal logicK just asIPC intoS4.
A much studied extension of F, Basic logicBPC, was introduced by Visser in 1981 [11] and shown in 2015 by K. Sano and M. Ma [10] to be translatable into the modal logicWK4.
Neighborhood structures are the standard semantic tool used to study non-normal modal logic. In a neighborhood model for modal logic, each state is associated with a collection of subsets of the universe and a modal formulaϕis true at a statewif the set of all states in whichϕis true is a neighborhood ofw.
M. Moniri and F. Shirmohammadzadeh Maleki in 2015 [8] presented a neighborhood semantics forIPCandBPC. In this paper we will introduce a systemWF, weaker than F, for which we define a neighborhood semantics. The neighborhood semantics needs to be more complex than the neighborhood semantics of modal logic.
The structure of this paper is as follows. In Section 2 we introduce the logicWF, it is created by replacing some axioms ofFby rules. The derivation system forWF. with and without hypotheses, is modeled on the one for Fof [5]. We prove a strong completeness theorem for WF. In Section 3 we prove strong completeness theorems
1
for the logics between WF and F, formed by combinations of five axioms and rules. In Section 4, we show that WFhas the finite model property. In Section 5 we study conservativity ofIPCoverWFwith respect to the simple implications introduced in [5]. There is a clear difference in strength compared to the systemF:IPC was proved to be conservative overFwith respect to the much more complex basic implications [5]. In Section 6 we investigate the relation betweenWFand modal logic. The logicWFis clearly related to the non-normal modal logicEN. But because of the difference of the models we were able to prove only `WF A ⇒ `EN A. The other direction remains
an open problem. A similar situation arises between the basic monotonic modal logic Mand our system WFIRIL.
2
Soundness and Completeness
In this section we will introduce the logicWF, a logic strictly weaker thanFand prove soundness and completeness ofWF.
Definition 2.1
A pair F =hW, g, NBiis called a Neighborhood Frameof subintuitionistic logic ifW is a non-empty set andNBis a neighborhood function fromW intoP((P(W))2) such that
1.∀w∈W, ∀X, Y ∈P(W), (X⊆Y ⇒ (X, Y)∈NB(w));
2.N B(g) ={(X, Y)∈(P(W))2| X⊆Y}.
Here gis called omniscient(i.e. has the property 2).
We use the existence of omniscient worlds in the proofs of soundness and of charac-terization of properties of frames.
Definition 2.2
A Neighborhood Model of subintuitionistic logic is a tuple M = hW, g, NB, Vi, where hW, g, NBiis a neighborhood frame of subintuitionistic logic andV :P →2W a valuation function on the set of propositional variablesP.
Definition 2.3
(Truth in a Neighborhood Model) Let M = hW, g, NB, Vi be a model and
w ∈ W. Truth of a propositional formula in a world w is defined inductively as follows.
1. M, wp ⇔ w∈V(p);
2. M, wA∧B ⇔ M, wAandM, wB;
3. M, wA∨B ⇔ M, wAor M, wB;
4. M, wA→B ⇔ ((A)M,(B)M)∈NB(w);
5. M, w1⊥,
where (A)M denotes the truth set ofA. If X ⊆W is such that X = (A)M then we
Definition 2.4
A formula A istrue in a modelM=hW, g, NB, Vi,MAif for allw∈W, M, wA
and if all models force A, we writeAand call Avalid. A formula A isvalid on a frameF =hW, g, NBi,F AifAis true in every model based on that frame.
In the following definition of neighborhood models we use the more standard neigh-borhood function.
Definition 2.5
AnN-Neighborhood Frame is a tripleF=hW, g, Ni. N is a neighborhood
func-tion from W into 22W, g ∈ W, and for each w ∈ W we have W ∈ N(w) and
N(g) ={W}(gis calledomniscient). AnN-Neighborhood Modelis a quadruple
M =hW, g, N, Viwith V :P →2W a valuation function on the set of propositional
variablesP.
Definition 2.6
(Truth in an N-Neighborhood Model) Let M = hW, N, Vi be a model and
w ∈ W. Truth of a propositional formula in a world w is defined inductively as in Definition 2.3 with the following clause for →:
M, wA→B ⇔ {v |vA ⇒ vB} ∈N(w).
It maybe easier to think of{v| vA ⇒ vB}as the set ((A)M)c∪(B)M. Here we denoteW −(A)M by ((A)M)c.
Indeed, we opted forN-neighborhood frames first but were not able to prove com-pleteness forWFwith respect to these frames.
Lemma 2.7
For everyN-neighborhood modelMN =hW, g, N, Vi, there is a pointwise equivalent
neighborhood model MNB=hW, g, NB, Vi.
Proof.The proof is straightforward by considering, for eachw∈W,
NB(w) ={(X, Y)|(X)c∪Y ∈N(w)}.
In the other direction the connection is not clear.
Definition 2.8
WFis the logic given by the following axioms and rules,
1.A→A∨B 9. A→A
2.B→A∨B 10. A AB→B
3.A∧B→A 11. AA∧BB
4.A∧B→B 12. B→AA
5. A→AB→BA∧C→C 13. A→AB→BC→C
6. A→AC∨B→B→CC 14. (AA→↔CB)↔C(B↔→DD)
7.A∧(B∨C)→(A∧B)∨(A∧C) 8.⊥→A
The rules are to be applied in such a way that, if the formulas above the line are theorems of WF, then the formula below the line is a theorem as well. We return to the rules when we discuss deduction from hypotheses. We write`for`WF.
(A→B)∧(A→C)→(A→B∧C)
(A→B)∧(C→B)→(A∨C→B)
(A→B)∧(B →C)→(A→C)
Typical for intuitionistic logic is that often axioms and their corresponding rules are different in strength. That comes out nicely here.
First we will show thatWFhas the disjunction property.
Definition 2.9 ([6])
We define|A by induction on A, as follows
1. Not|p,
2.|A∧B iff |Aand|B,
3.|A∨B iff |Aor|B,
4.|A→B iff `A→B and (if|Athen|B).
Theorem 2.10
|A ⇔ `A.
Proof.The proof is a trivial modification of the standard one forIPC.
Theorem 2.11
If`A∨B then`Aor`B.
Proof.Assume`A∨B, by Theorem 2.10,|A∨B. So|Aor|B. Again by Theorem
2.10,`Aor `B.
Theorem 2.12
The formulaA→B is valid if and only if for all models M,
(A)M ⊆(B)M.
Proof.⇒: LetA→B, so for allM and for allw∈M, ((A)M,(B)M)∈NB(w).
So, for all M, ((A)M,(B)M)∈NB(g). Hence, by Definition 2.1(2), (A)M ⊆(B)M.
⇐: We should prove that for all M and for all w ∈M, ((A)M,(B)M)∈ NB(w).
By assumption (A)M ⊆(B)M, so by definition of neighborhood frames for allM and
for allw∈M, ((A)M,(B)M)∈NB(w). That is,A→B.
Corollary 2.13
A↔B if and only if for all modelsM, (A)M = (B)M.
Proof.Obvious.
Lemma 2.14
(a) IfAandA→B, then B.
(b) If A→B andA→C, then A→B∧C.
(c) IfA→C andB→C, thenA∨B→C.
(d) If A→B andB→C, thenA→C.
Proof.We only prove (a) and (e). The other cases are similar.
(a) Let A → B. By Theorem 2.12, for all models M, (A)M ⊆ (B)M. By
assumption, for all modelsM, (A)M =W. So also, (B)M =W. Hence,B.
(e) By Corollary 2.13, it is sufficient to show that for all modelsM, (A→C)M =
(B→D)M. By assumption and Corollary 2.13, we have (A)M = (B)M and (C)M =
(D)M.Therefore,{v| ((A)M,(C)M)∈NB(v)}={v |((B)M,(D)M)∈NB(v)}.That
is, (A→C)M = (B →D)M.
Observation 2.15
The formula (p → q)∧(p → r) → (p → q∧r) is not valid in the class of all neighborhood frames.
Proof.Consider the neighborhood frameF =hW, g, NBiwith
W ={g, w},NB(w) ={(W,{w}),(W,{g})} ∪{(X, Y)∈(P(W))2 |X ⊆Y}.
Also consider the valuationV(p) =W, V(r) ={g},V(q) ={w}. We have,
(W,{w})∈NB(w) ⇒ wp→q,
(W,{g})∈NB(w) ⇒ wp→r,
(W,∅)∈/ NB(w) ⇒ w1p→q∧r.
So, ((p→q)∧(p→r))M *(p→q∧r)M and therefore,
(((p→q)∧(p→r))M,(p→q∧r)M)∈/ NB(g).
That is,g1(p→q)∧(p→r)→(p→q∧r).
Observation 2.16
The formula (p → q)∧(r → q) → (p∨r → q) is not valid in the class of all neighborhood frames.
Proof.Consider the neighborhood frameF =hW, g, NBiwith
W ={w, g},NB(w) ={({g},{w})} ∪{(X, Y)∈(P(W))2 |X ⊆Y}.
Also consider the valuationV(p) =V(q) ={w},V(r) ={g}. We have,
({w},{w})∈NB(w) ⇒ wp→q,
({g}),{w})∈NB(w) ⇒ wr→q,
(W,{w}))∈/ NB(w) ⇒ w1p∨r→q.
So ((p→q)∧(r→q))M *(p∨r→q)M and therefore,
(((p→q)∧(r→q))M,(p∨r→q)M)∈/NB(g).
That is,g1(p→q)∧(r→q)→(p∨r→q).
Observation 2.17
The formula (p→q)∧(q→r)→(p→r) is not valid on the class of all neighborhood frames.
Proof.Consider the neighborhood frameF =hW, g, NBiwith
Also consider the valuationV(p) ={w},V(r) =∅andV(q) ={g}. We have,
({w},{g})∈NB(w) ⇒ wp→q,
({g},∅)∈NB(w) ⇒ wq→r,
({w},∅)∈/NB(w) ⇒ w1p→r.
So, ((p→q)∧(q→r))M *(p→r)M and therefore,
(((p→q)∧(q→r))M,(p→r)M)∈/NB(g).
That is,g1(p→q)∧(q→r)→(p→r).
For the strong completeness theorem we will show that if Σ0A then there exists a state ∆ in the canonical model such that ∆ Σ and ∆ 1 A. For this purpose we need to have some definitions and propositions. Actually the definitions of theory and of derivation from hypotheses are identical to the ones forFin [5].
Definition 2.18
A set of sentences ∆ is atheoryif and only if
1.A, B∈∆ ⇒ A∧B ∈∆,
2.`A→B ⇒ (ifA∈∆, thenB∈∆),
3. If`A ⇒ A∈∆.
Definition 2.19
(a) We define Γ`A if there is a derivation of A from Γ and theorems of WFusing the rules AA∧BB and A AB→B where in the latter case the restriction is thatA→Bhas to be provable inWF.
(b) We define ΓA iff for all M,w∈M, ifM, wΓ then M, wA.
Proposition 2.20
∆ is a theory⇐⇒ ∆`A if and only ifA∈∆.
Proof.⇒: The proof from right to left is immediate. The other direction is by
induction on the length of the derivation. If Ais theorem of WF, then by definition of theoryA∈∆.
If ∆ ` A and ∆ ` B, then, by induction hypothesis A ∈ ∆ and B ∈ ∆. So by definition of theory A∧B∈∆.
If`A→B and ∆`A, then, by induction hypothesisA∈∆ and by definition of theoremB ∈∆.
⇐: This direction is easy. We only check case 2 of the definition of theory. Let
` A → B and A ∈ ∆. Then by assumption ∆ ` A, and so ∆ ` B. Again by assumptionB ∈∆.
Theorem 2.21
(Weak Deduction Theorem)A`B if and only if`A→B.
Proof.⇒: By induction on the length of the proof.
IfB is an axiom. Then`B, so by rule 11,`A→B. IfA`Ais covered`A→A.
IfA`B andA`C. By induction hypothesis`A→B and`A→C, so by rule 5,
If A ` B and ` B → C. Then by induction hypothesis ` A → B, so by rule 12,
`A→C.
⇐: Let ` A → B and Γ = {A} then by Definition 2.19, we have Γ ` B, that is
A`B.
Corollary 2.22
(a)A1, ..., An`B if and only if`A1∧...∧An→B.
(b) ∆`B if and only if, for someA1, ..., An ∈∆,`A1∧...∧An →B.
Proof.The proof is easy.
Definition 2.23
A set of sentences ∆ is primeif and only if, ifA∨B∈∆, thenA∈∆ orB ∈∆.
Theorem 2.24
If Σ0D then there is a prime theory ∆ such that ∆⊇Σ, D /∈∆.
Proof.By assumption and by definition of provability we conclude that D /∈ Σ.
Enumerate all formulas, with infinitely many repetitions: B0, B1, ...and define,
∆0= Σ,
∆n+1= ∆n∪ {Bn} if ∆n, Bn 0D,
∆n+1= ∆n otherwise.
Take ∆ to be the union of all ∆n. Clearly, ∆0D.We must show that ∆ is a prime
theory. Assume that F ∈ ∆, G∈ ∆ and F ∧G /∈ ∆. Let F = Bi, G =Bj and
F ∧G = Bn with j ≥ i ≥ n. So, ∆n, F ∧G ` D and hence ∆n, F, G ` D. But
F, G∈∆j, so ∆n, F, G0D, a contradiction.
Now if ` A→ B and A ∈∆. Asuume B /∈∆. Let B =Bn, then, ∆n, Bn ` D
and so ∆n, A` D, because from ` A → B and A we can derive B. But this is a
contradiction, sinceA∈∆.
Assume that F ∨G ∈ ∆, and F /∈ ∆, G /∈ ∆. Let F = Bn and G = Bk, so
∆n, F ` D and ∆k, G ` D. Then by Corollary 2.22, there exist ¯B1, ...,B¯m ∈ ∆n
such that`B¯1∧...∧B¯m∧F →D and also there existB
0
1, ..., B
0
m0 ∈∆k such that
` B10 ∧...∧Bm0 0 ∧G→D. W.l.o.g. take n≥k, then ¯B1, ...,B¯m, B
0
1, ..., B
0
m0 ∈∆n.
Thus by some steps and using some rules ofWFwe will have,
`( ¯B1∧...∧B¯m)∧(B
0
1∧...∧B
0
m0)∧(F∨G)→D (2.1)
Again by Corollary 2.22 and (2.1) we have ∆n, F ∨G`D. But that cannot be true
sinceF∨G∈∆.
Assuming that ` F, we want to show that F ∈∆. Let F =Bn and F /∈ ∆, so
∆n, F `D. From ∆nwe can deriveF, so ∆n`D. But this is a contradiction, hence
F ∈∆. So ∆ is a prime theory.
Since ∆ is a theory and ∆0D,D /∈∆.
Definition 2.25
Let WWF be the set of all prime theories. Given a formula A, we define the setJAK
as follows,
Lemma 2.26
Let C and D are formulas. Then
(a)JC∧DK=JCK∩JDK.
(b)JC∨DK=JCK∪JDK.
(c)JCK⊆JDK iff `C→D.
(d)JCK=JDK iff `C↔D.
Proof.The proofs are easy. We only prove (c).
(c) Let0C →D. Then by the Weak Deduction Theorem C0 D. Let Σ ={C}, then by Theorem 2.24, there exist a prime theory Γ such that, C ∈ Γ and D /∈ Γ. That is JCK*JDK.
Now let`C→D. Assume Γ∈WWF, C ∈Γ. Then by definition of theoryD∈Γ.
SoJCK⊆JDK.
When constructing a canonical model the states of the world will be prime theories, i.e. elements ofWWF. Consider the functionNBWF:WWF→P((P(WWF))2) such that
for each Γ∈WWF,
NBWF(Γ) ={(JAK,JBK)|A→B∈Γ}.
In the completeness proof we need to be sure that, if (JAK,JBK) ∈ NBWF(Γ), then
A→B∈Γ. This does not follow directly from the definition. It only follows directly thatC, Dexist such thatC→D∈Γ, andJCK=JAKandJDK=JBK. In the following lemma we obtain what is needed to make the argument go through.
Lemma 2.27
If NBWF : WWF → P((P(WWF))2) is a function such that for each Γ ∈ WWF,
NBWF(Γ)={(JAK,JBK)|A→B∈Γ}. Then (JAK,JBK)∈NBWF(Γ) impliesA→B∈Γ.
Proof.Assume (JAK,JBK)∈NBWF(Γ). Then for someC,D,JAK=JCK,JBK=JDK,
C→D∈Γ. By Lemma 2.26, we have `A↔C and`B↔D. Then by rule 14 we will have`(A→B)↔(C→D). By assumption,C→D∈Γ. Hence, by definition of prime theory we conclude that, A→B∈Γ.
Now we want to define the canonical model forWF.
Definition 2.28
TheCanonical modelMWF=hW
WF, g, NBWF, ViofWFis defined by:
1.gis the set of theorems of WF,
2. For each Γ∈W and all formulasA andB,
NBWF(Γ)={(JAK,JBK)|A→B ∈Γ},
3. Ifp∈At, thenV(p) =JpK={Γ |Γ∈WWF and p∈Γ}.
By Theorem 2.11, WFhas the disjunction property and therefore is a prime theory. It is easy to see that in the canonical modelWFis omniscient. Also for all Γ∈WWF,
ifJAK⊆JBK, then (JAK,JBK)∈NBWF(Γ). Since ifJAK⊆JBK, then by Lemma 2.26,
Theorem 2.29
(Truth Lemma)For any formulaE, ifMWFis the canonical model ofWF, then
(E)MWF =JEK.
Proof.By induction on E. The atomic case holds by the definition of canonical
model.
(E:=A∧B) Let Γ∈WWFand ΓA∧B then ΓAand ΓB. By the induction
hypothesisA∈Γ and B∈Γ. Γ is a theory soA∧B∈Γ.
Now letA∧B∈Γ. We have `A∧B→Aand`A∧B→B, hence by definition of theory we conclude that A∈ Γ andB ∈Γ. By induction hypothesis Γ A and ΓB so ΓA∧B.
(E:=A∨B) Let Γ∈WWFand ΓA∨B.Then ΓAor ΓB. By the induction
hypothesis A ∈Γ or B ∈Γ. We have ` A→ A∨B and `B →A∨B, so by the definition of theory we conclude that A∨B ∈Γ.
Now letA∨B∈Γ. Γ is a prime soA∈Γ orB ∈Γ. By induction hypothesis we conclude that ΓAor ΓB. That is ΓA∨B.
(E:=A→B) Let Γ∈WWF, then,
ΓA→B ⇐⇒ ((A)MWF,(B)MWF)∈NBWF(Γ)
(by induction hypothesis) ⇐⇒ (JAK,JBK)∈NBWF(Γ)
(by Lemma 2.27) ⇐⇒ A→B∈Γ.
Theorem 2.30
The logicWFis sound and strongly complete with respect to the class of neighborhood frames.
Proof.Soundness is straightforward (in fact already shown in earlier lemmas).
Let Σ0A, then by Theorem 2.24, there is a prime theory ∆⊇Σ such thatA /∈∆. So, in the canonical model we haveMWF,∆Σ, MWF,∆1A. That is, Σ1A.
3
Completeness for Logics between
WF
and
F
In this section we consider some formulas which characterize special classes of frames. We form the logics axiomatized by some of these formulas and prove their complete-ness.
Definition 3.1
For every neighborhood frame F = hW, g, NBi, we list some relevant properties as follows. Here X, Y, Zare definable subsets ofP(W).
1. F is closed under intersection if and only if for allw∈W, if (X, Y)∈NB(w), (X, Z)∈NB(w) then (X, Y ∩Z)∈NB(w).
2. F is closed underunionif and only if for all w∈W, if (X, Y)∈NB(w), (Z, Y)∈
NB(w) then (X∪Z, Y)∈NB(w).
3. F satisfiestransitivity if and only if for allw∈W, if (X, Y)∈NB(w), (Y, Z)∈
NB(w) then (X, Z)∈NB(w).
4. F is closed underupsetif and only if for allw∈W, if (X, Y)∈NB(w) andY ⊆Z
then (X, Z)∈NB(w).
5. F is closed under downset if and only if for allw ∈W, if (X, Y)∈NB(w) and
Lemma 3.2
The formula (p → q)∧(p → r) → (p → q∧r) characterizes the class of frames
F =hW, g, NBisatisfying closure under intersection.
Proof.Let F be closed under intersection and M = hW, g, NB, Vi be any model
based on F. We have to prove for all w∈W,
(((p→q)∧(p→r))M,(p→q∧r)M)∈NB(w).
For this purpose it is sufficient to show that, ((p→q)∧(p→r))M ⊆(p→q∧r)M.
Letw∈W,wp→qandwp→rthen,
(V(p), V(q))∈NB(w) (3.1)
(V(p), V(r))∈NB(w) (3.2)
The frame is closed under intersection, so by (3.1) and (3.2), (V(p), V(q)∩V(r))∈
NB(w). So, w p → q∧r. Hence, by definition of neighborhood frames for all
w∈W,
(((p→q)∧(p→r))M,(p→q∧r)M)∈NB(w).
For the other direction, we use contraposition. Suppose that the class is not closed under intersection. Then there is a frame F and w∈F such that (X, Y)∈NB(w) and (X, Z)∈NB(w) but (X, Y ∩Z)∈/ NB(w). Consider the valuation V such that,
V(p) =X, V(r) =Z andV(q) =Y. Then,
(V(p), V(q))∈NB(w) ⇒ wp→q,
(V(p), V(r))∈NB(w) ⇒ wp→r,
(V(p), V(q∧r))∈/NB(w) ⇒ w1p→q∧r.
So ((p→q)∧(p→r))M *(p→q∧r)M.Then by definition of neighborhood frames
g1(p→q)∧(p→r)→(p→q∧r). ThereforeF 1(p→q)∧(p→r)→(p→q∧r).
Lemma 3.3
The formula (p → q)∧(r → q) → (p∨r → q) characterizes the class of frames
F =hW, g, NBisatisfying closure under union.
Proof.LetF be closed under union andM =hW, g, NB, Vibe any model based on
F. We have to prove for allw∈W,
(((p→q)∧(r→q))M,(p∨r→q)M)∈NB(w).
For this purpose it is sufficient to show that, ((p→q)∧(r→q))M ⊆(p∨r→q)M.
Letw∈W,wp→qandwr→qthen,
(V(p), V(q))∈NB(w) (3.3)
(V(r), V(q))∈NB(w) (3.4)
The frame is closed under union, so by (3.3) and (3.4), (V(p)∪V(r), V(q))∈NB(w). So,wp∨r→q. Hence by definition of neighborhood frames for allw∈W,
For the other direction, we use contraposition. Suppose that the class is not closed under union. Then there is a frame F and w ∈ F such that (X, Y)∈ NB(w) and (Z, Y) ∈ NB(w) but (X ∪Z, Y) ∈/ NB(w). Consider the valuation V such that,
V(p) =X, V(r) =Z andV(q) =Y. Then,
(V(p), V(q))∈NB(w) ⇒ wp→q,
(V(r), V(q))∈NB(w) ⇒ wr→q,
(V(p∨r), V(q))∈/NB(w) ⇒ w1p∨r→q.
So ((p→q)∧(r→q))M *(p∨r→q)M).Then by definition of neighborhood frames
g1(p→q)∧(r→q)→(p∨r→q). ThereforeF 1(p→q)∧(r→q)→(p∨r→q).
It is very notable that, if we use N-neighborhoods instead of neighborhoods, the formulas (p→ q)∧(p →r)→ (p→ q∧r) and (p →q)∧(r → q) →(p∨r → q) characterize the same class of frames. This is clearly undesirable and strong evidence that our definition of neighborhood is the right one.
Lemma 3.4
The formula (p → q)∧(q → r) → (p → r) characterizes the class of frames F =
hW, g, NBisatisfying transitivity.
Proof.LetF satisfy transitivity andM =hW, g, NB, Vibe any model based onF.
We have to prove for all w∈W,
(((p→q)∧(q→r))M,(p→r)M)∈NB(w).
For this purpose it is sufficient to show that, ((p→q)∧(q→r))M ⊆(p→r)M.Let
w∈W,wp→qandwq→rthen,
(V(p), V(q))∈NB(w) (3.5)
(V(q), V(r))∈NB(w) (3.6)
The frame satisfies transitivity, so by (3.5) and (3.6), (V(p), V(r)) ∈ NB(w). So,
wp→r. Hence, by definition of neighborhood frames for allw∈W,
(((p→q)∧(q→r))M,(p→r)M)∈NB(w).
For the other direction, we use contraposition. Suppose that the class does not satisfy transitivy. Then there is a frame F and w ∈ F such that (X, Y) ∈ NB(w) and (Y, Z)∈NB(w) but (X, Z)∈/ NB(w). Consider the valuation V such that, V(p) =
X, V(r) =Z andV(q) =Y. Then,
(V(p), V(q))∈NB(w) ⇒ wp→q,
(V(q), V(r))∈NB(w) ⇒ wq→r,
(V(p), V(r))∈/NB(w) ⇒ w1p→r.
So ((p→q)∧(q→r))M *(p→r)M).Then by the definition of neighborhood frames
Lemma 3.5
The rule (q→rp)→→(qp→r) characterizes the class of frames F = hW, g, NBi satisfying closure under downset.
Proof.Let for all M =hW, g, NB, Vion a frameF, which is closed under downset,
M p → q. We have to prove for all M on F, M (q → r) → (p → r). For this purpose we show that for all M on F, (q→r)M ⊆(p→r)M. Let w∈M and
w q → r then, (V(q), V(r)) ∈ NB(w). Also by assumption V(p) ⊆ V(q). The frame is closed under downset so, (V(p), V(r))∈ NB(w), thereforew p→ r. So (q→r)M ⊆(p→r)M, i.e. by Theorem 2.12, for allM onF,M (q→r)→(p→r). For the other direction, we use contraposition. Suppose that the class is not closed under downset. That is there is a frameF andw∈F such that (X, Y)∈NB(w),Z⊆ X but (Z, Y)∈/ NB(w). Consider the valuation V such that, V(p) =Z, V(r) =Y
and V(q) =X. So (V(q), V(r))∈NB(w), (V(p), V(r))∈/ NB(w) andV(p)⊆V(q), thenM, wq→r,M, w1p→randM p→q. Hence (q→r)M *(p→r)M so,
M 1(q→r)→(p→r).
Lemma 3.6
The rule (r→pp)→→q(r→q) characterizes the class of frames F = hW, g, NBi satisfying closure under upset.
Proof.Let for all M = hW, g, NB, Vi on a frame F, which is closed under upset,
M p → q. We have to prove for all M on F, M (r → p) → (r → q). For this purpose we show for all M on F, (r → p)M ⊆ (r → q)M. Let w ∈ M and
w r → p. Then, (V(r), V(p))∈NB(w). Also by assumptionV(p) ⊆V(q). The frame is closed under downset so, (V(r), V(q))∈ NB(w), thereforew r → q. So (r→p)M ⊆(r→q)M, i.e. by Theorem 2.12, for allM onF,M (r→p)→(r→q).
For the other direction, we use contraposition. Suppose that the class is not closed under upset. That is there is a frameF andw∈F such that (X, Y)∈NB(w),Y ⊆Z
but (X, Z)∈/ NB(w). Consider the valuation V such that,V(p) =Y, V(r) =X and
V(q) =Z. So (V(r), V(p))∈NB(w), (V(r), V(q))∈/ NB(w) andV(p)⊆V(q), then
M, w r → p, M, w 1 r→ q and M p→ q. Hence (r → p)M *(r →q)M so,
M 1(r→p)→(r→q).
In this section we will be interested in the following axiom schemas and rules.
(C) (A→B)∧(A→C)→(A→B∧C) (D) (A→B)∧(C→B)→(A∨C→B) (I) (A→B)∧(B →C)→(A→C) (IL) (C→AA)→→B(C→B)
(IR) (B→CA)→→B(A→C) Lemma 3.7
(a) IfWFC⊆L, then the canonical model of logicL is closed under intersection. (b) If WFD⊆L, then the canonical model of logicLis closed under union.
(c) IfWFI⊆L, then the canonical model of logic L satisfies transitivity. (d) If WFIL⊆L, then the canonical model of logicL is closed under upset.
Proof.We only prove (a) and (e). The other cases are similar.
(a) Suppose that in the canonical model of logicL, (X, Y)∈NB(Γ) and (X, Z)∈
NB(Γ). By definition of NB in the canonical model there exist formulas A,B and C such that (X, Y) = (JAK,JBK) and (X, Z) = (JAK,JCK), where A → B ∈ Γ and
A→C∈Γ. Hence (A→B)∧(A→C)∈Γ and so using (C),A→B∧C∈Γ. Hence (JAK,JB∧CK)∈NB(Γ). Therefore sinceJBK∩JCK=JB∧CK, (X, Y ∩Z)∈NB(w). SoNB is closed under intersection.
(e) Suppose that in the the canonical model ofWFIR, (X, Y)∈NB(Γ) andZ ⊆X.
Then there exist formulas A,B and C, such that (X, Y) = (JAK,JBK), Z = JCK
and JCK ⊆ JAK. By definition of neighborhood in the canonical model and using Lemma 2.26, we conclude that A → B ∈ Γ and` C → A, and so using (IR) and
modus ponens, C →B ∈Γ. Hence (JCK,JBK) = (Z, Y)∈NB(Γ). Therefore NB is closed under downset.
Theorem 3.8
If Γ ⊆ {C,D,I,IL,IR}, thenWFΓ is sound and strongly complete with respect to
the class of neighborhood frames with all the properties defined by the schemes in Γ.
Proof.Immediate by Lemma 3.7.
The logic F is the smallest set of formulas closed under instances of WF, C, D
and I. Theorem 3.8 shows that Fis sound and complete with respect neighborhood models closed under intersection and union, and satisfying transitivity. A rooted subintuitionistic Kripke model for F is a quadruple M = hW, g, R, Vi with R lack-ing the properties of reflexivity, transitivity and preservation of rooted intuitionistic Kripke models [5]. The following theorem gives another proof of completeness for Fwith respect neighborhood models closed under intersection, union and satisfying transitivity, using the completeness of Fwith respect to Kripke models ([2, 9, 5]).
Theorem 3.9
For every rooted subintuitionistic Kripke model Mk =hW, g, R, Vi, there is a
point-wise equivalent neighborhood model Mn =hW, g, NB, Vi, closed under intersection
and union, and satisfying transitivity.
Proof.LetMk=hW, g, R, Vibe a rooted subintuitionistic kipke model andw∈W.
For each w∈W, we define R(w) ={u∈W |wRu} and
NB(w) ={(X, Y)|R(w)⊆(W−X)∪(Y)}.
We show thatMn =hW, g, NB, Viis a neighborhood model closed under intersection
and union, and satisfying transitivity. We know thatR(g) =W, so
NB(g) ={(X, Y)|W ⊆(W −X)∪(Y)}={(X, Y)|X ⊆Y}.
It is easy to show that with this definition NB is closed under intersection, union and satisfies transitivity. We only show that NB is closed under intersection. Let (X, Y) and (X, Z) be inNB(w). We want to show that (X, Y ∩Z)∈NB(w). So by definition of NB,
R(w)⊆(W−X)∪(Y), (3.7)
By (3.7) and (3.8),R(w)⊆(W−X)∪(Y∩Z). So, by definition ofNB, (X, Y∩Z)∈
NB(w). Now we prove that Mk and Mn are pointwise equivalent. The proof is by
induction on the complexity of formulas. We only consider the implication case. Let
Mk, w A→B. We want to prove that Mn, w A→B that is ((A)Mn,(B)Mn)∈
N(w). For this purpose it is sufficient to show thatR(w)⊆(W−(A)Mn)∪(B)Mn.
Let v ∈ R(w) and Mn, v A. Then wRv and by induction hypothesis Mk, v A.
Then, by assumption Mk, v B. So, by induction hypothesis, Mn, v B. That is
v∈(B)Mn and soR(w)⊆(W−(A)Mn)∪(B)Mn. Hence ((A)Mn,(B)Mn)∈N(w).
Now assume Mn, w A → B. We want to prove that Mk, w A → B. Let
wRv, Mk, v A. Then, by induction hypothesis, Mn, v A. Also, by assumption,
v ∈ R(w) and R(w) ⊆ (W −(A)M)∪(B)M. So Mn, v B. Again by induction
hypothesisMk, vB. That is Mk, wA→B.
4
Finite model property
In this section we will show that WFhas the finite model property. We are going to prove this result by means of finite theories in so-called adequate sets.
Definition 4.1
A set of formulas Σ isadequateif for all formulasA, B, ifA∨B∈Σ then so areA
andB, ifA∧B∈Σ then so are AandB, and ifA→B∈Σ then so areAandB.
Lemma 4.2
Let Σ be a set of formulas and letDbe a formula such that Σ∪ {D} ⊆Φ and Σ0D. Then there is a prime theory ∆⊆Φ such that
1. Σ⊆∆,
2.A, B∈∆ andA∧B∈Φ ⇒ A∧B ∈∆,
3. IfA, B∈Φ and`A→B ⇒ (ifA∈∆, thenB ∈∆), 4. IfA∈Φ and`A ⇒ A∈∆,
5. IfA∨B ∈∆ ⇒ A∈∆or B∈∆.
Proof.Everything is similar to Theorem 2.24, except that we consider an
enumera-tionB1, B2, ..., Bn of the elements of Φ.
Now we make the modelMΦ=hWΦ, g, NBΦ, VΦias follows,
1.WΦ:={∆⊆Φ|∆ satisfies conditions 2 to 5 in Lemma 4.2}, 2.g:=WFΦ={A|A∈Φand `A},
3. For each ∆∈WΦ,NBΦ(∆) ={(JAKΦ,JBKΦ)|A→B ∈∆}where
JAKΦ={∆∈WΦ|A∈∆},
4.VΦ(p) =JpKΦ={∆|∆∈WΦandp∈∆}.
It is easy to show thatWFΦ∈WΦ and thatWFΦis an omniscient world. So,MΦis in fact a neighborhood model.
Lemma 4.3
Theorem 4.4
(Finite Model Property)If0A, then there exists a finite countermodel forA.
Proof.We consider the finite adequate set Φ such that A ∈ Φ and consider the
model MΦ. By Lemma 2.29, there exists a set ∆⊆Φ such that A /∈∆ and ∆ is a node of MΦ. Hence MΦ,∆ 1 A and so MΦ is a finite neighborhood model which does not force A.
5
Relation of
WF
to
F
In [5] we obtained conservativity results forIPCoverF. In this section, we prove such results for WFas well. This clarifies the difference in strength betweenFandWF.
Definition 5.1
Let us call a formula A → B with A and B containing only ∧ and ∨ a simple implication.
The following is a uniform substitution theorem forWF. It states that we can always replace logically equivalent formulas by each other.
Theorem 5.2
(Uniform Substitution) If ` A ↔ B, then ` E[A/p] ↔ E[B/p], where p is an atom.
Proof.The proof is easy by induction on E. We only check the disjunction and implication cases. LetE=C∨D, then
1.`C[A/p]→C[B/p] By induction hypothesis 2.`C[B/p]→(C∨D) [B/p] axioms 1
3.`C[A/p]→(C∨D) [B/p] Follows from 1 and 2 using rule 12 4.`D[A/p]→D[B/p] By induction hypothesis
5.`D[B/p]→(C∨D) [B/p] axioms 1
6.`D[A/p]→(C∨D) [B/p] Follows from 4 and 5 using rule 12 7.`(C∨D) [A/p]→(C∨D) [B/p] Follows from 3 and 6 using rule 6
The other direction is similar to this. So `(C∨D) [A/p]↔(C∨D) [B/p]. Now LetE=C→D, then
1.`C[A/p]↔C[B/p] By induction hypothesis 2.`D[B/p]↔D[A/p] By induction hypothesis
3.`(C[A/p]→D[A/p])↔(C[B/p]→D[B/p]) Follows from rule 13
That is,`E[A/p]↔E[B/p].
Theorem 5.3
Let Abe constructed by applying only ∧and ∨to atoms. Then there are formulas
A0, A00 such that
Proof.The proof is straightforward.
Now by the theorem just proved, a simple implication A→B can be replaced by a WF- andIPC-equivalentA0→B0 such thatA0 is a disjunction of conjunctions and
B0 is a conjunction of disjunctions.
Lemma 5.4
For allpi,1≤i≤k andqj,1≤j≤mwe have,
`WFp1∨...∨pk →q1∧...∧qm
iff
`WFpi→qj for all i, j.
Proof.Easy.
Definition 5.5
A formula A→ B called avery simple implicationif A is conjunction of atoms and B is disjunction of atoms.
By the previous lemma we can conclude that to show that IPC is conservative over WFwith respect to simple implications it is sufficient to do so for very simple implications. We can do so now for very simple implications, and in fact even forCPC instead ofF.
Theorem 5.6
IfIPC(or CPC) proves a very simple implication, thenWFproves it as well.
Proof.Let A →B be a very simple implication, so A =∧i(Pi) and B =
∨
j(qj).
Assume 0WF A→B then by the completeness theorem there exists a neighborhood
model M and w∈ M, such that M, w 1 A →B. Hence (A)M *(B)M. So there
exists v ∈ M such that M, v A and M, v 1 B. Now we select this point v
from M and then we make the one pointIPC model MIPC =hv, NB(v),i such that
NB(v) = {(v, v),(∅, v)} and for all propositional variable p, MIPC, vpif and only
ifM, vp. Clearly
MIPC, vpi,for alli,
MIPC, v2qj,for allj.
So, ((A)MIPC,(B)MIPC) = (v,∅) and (v,∅) ∈/ NB(v). That is M
IPC, v A → B, so
0IPC A→B.
In [5] IPCwas proved to be conservative overFwith respect to basic implications, formulas of the formA→B withAandB conjunction/disjunctions of simple impli-cations. For example (p→q)∧(p→r)→(p→q∧r) is a basic implication provable in Ffor which0WF(p→q)∧(p→r)→(p→q∧r). This shows a clear difference in
strength betweenFandWF.
6
WF
and Modal Logic
1. p=p;
2. (A∧B)=A∧B;
3. (A∨B)=A∨B;
4. (A→B)=(A→B).
Definition 6.1
A system of modal logic is classical iff it is closed under RE (AA↔↔BB) [1].
E is the smallest classical modal logic. The logic EN extends E by adding the axiom scheme >. Completeness holds for EN with respect neighborhood frames that contain the unit, i.e. for all w ∈W, W ∈ N(w) [1]. The clause for ϕin the
N-neighborhood models is: w ϕiffV(ϕ)∈N(w).
We will show that if `WF A, then `ENA. We conjecture that the reverse
direc-tion holds as well but we were unable to prove it because of the difference between neighborhoods andN-neighborhoods.
Proposition 6.2
If M =hW, N, Vi is an intuitionisticN-neighborhood model without an omniscient world thenM can be extended by adding an omniscient world to obtain a modelM0
such that for all formulas A and for all w∈W,
M, wA iff M0, wA.
Proof.We add a worldg to W and make a new model M0 =hW0, g, N0, V0i, with
W0 =W∪ {g}, for all propositional letters p, (p)M0 = (p)M and for allw∈W and
X ∈P(W),
N0(w) ={X∪ {g}, X |X ∈N(w)}, N0(g) ={W0}.
Then we will show that for all formulas D, (D)M0 ∩W = (D)M, in other words
(D)M0 = (D)M or (D)M0 = (D)M∪ {g}. The proof is by induction onD. The case
whereDis a proposition letter follows by definition. Conjunction and disjunction are
easy, and if (E)M0 ∩W = (E)M, then ((E)M0)c∩W = ((E)M)c. So letD=E→F
andw∈W. Then,
M, wE→F ⇐⇒ ((E)M)c∪(F)M ∈N(w) (by induction hypothesis) ⇐⇒∗ ((E)M0)c∪(F)M0 ∈N0(w)
⇐⇒ M0, w0 E→F.
The explanation of⇐⇒∗ is that by induction hypothesis,
(EM
0
)c= (EM)c or (EM
0
)c= (EM)c∪ {g},
FM
0
=FM orFM
0
=FM ∪ {g}.
Lemma 6.3
LetM =hW, N, Vibe anN-neighborhood model. Then for allw∈W,
M, wNA iff M, wMA.
Proof.The proof is by induction onA. The atomic case holds by induction and the conjunction and disjunction cases are easy. We only check the implication case. So letA=C→D, then
M, wNC→D ⇐⇒ {v |v1NC} ∪ {v |vND} ∈N(w)
(by induction hypothesis) ⇐⇒ {v |v1MC
}
∪{v|vMD
}
∈N(w)
⇐⇒ {v |vM¬C
}
∪{v|vMD
}
∈N(w)
⇐⇒ {v |vM¬C∨D
}
∈N(w)
⇐⇒ M, wM(¬C∨D)
⇐⇒ M, wM(C→D).
Theorem 6.4
For all formulas A, if `WFA, then`ENA.
Proof.Assume0ENA, then by the completeness theorem there existsM =hW, N, Vi
andw∈W, such that M, w1ENA. Then, by Lemma 6.3,M, w1NA. To transfer
M into a properN-neighborhood model forWFit needs to have an omniscient world. In case M =hW, N, Vidoes not have one, by Proposition 6.2, there exists M0 with an omniscient world such that M0, w1NA. By Lemma 2.7 there is then a pointwise
equivalent neighborhood model MNB0 = hW, g, NB, Vi, such that MNB0 , w 1A. So, by soundness,0WFA.
Definition 6.5
AnN-neighborhood frameF =hW, g, Ni, is closed undersupersetif and only if
∀w∈W, ∀X, Y ∈P(W) : X ⊆Y, X∈N(w) ⇒ Y ∈N(w).
Definition 6.6
A system of modal logic is monotone iff it is closed under RM ( A→B
A→B) [4].
M is the smallest monotonic modal logic. Completeness holds for M with respect monotonic neighborhood frames, i.e. in F =hW, Ni,N is closed under superset [4].
Lemma 6.7
For every N-neighborhood model MN =hW, g, N, Viclosed under superset there is
a pointwise equivalent neighborhood modelMNB=hW, g, NB, Viclosed under upset
and downset.
Proof.The proof is easy if, by considering, as in Lemma 2.7, for each w∈W,
NB(w) ={(X, Y)|(X)c∪Y ∈N(w)}.
Theorem 6.8
For all formulas A, if`WFIRIL A, then`MA.
Proof.Assume0MA. Then by the completeness theorem there existM =hW, N, Vi,
closed under superset, and w ∈ W such that M, w 1M A. So, by Lemma 6.3,
Proposition 6.2, there exists M0 with an omniscient world such that M0, w 1N A.
Moreover, from the proof of Lemma 2.7, it is clear that, ifM is closed under superset,
M0is as well. By Lemma 6.7 then, there is a pointwise equivalent neighborhood model
MNB0 =hW, g, NB, Vi, closed under upset and downset, such thatMNB0 , w 1A. So by soundness,0WFIRIL A.
Also in this case we were not able to prove the converse.
7
Conclusion
In this article we constructed a neighborhood semantics for weak subintuitionistic logics extending a basic logicWF. It uses pairs of subsets of the set of worlds instead of just subsets. This definitely seems the right choice, especially in view of the results of Section 3 where the various obvious extensions of WFcan be neatly separated by frame properties. And in that way it does become clear what the right logics to be studies in this area are. It does make the connection to non-normal modal logic less clear than one could hope for because of the different semantics, and this lead to some open problems, for example the relationship between WF andEN. The relationship between the models used in modal logic and subintuitionistic logic does need clearing up. The conservativity result in Section 5 makes clear what kind of implications can be expected to be provable inWFand separatesWFfromF, for which we obtained a conservativity result before. It is not clear how this relates to the complexity of the decision problem of the logics. This may be an object for further study. Finally, it seems worthwhile to study interpolation forWFand other logics.
References
[1] B. Chellas,Modal logic: An Introduction, Cambridge University Press, 1980.
[2] G. Corsi, Weak Logics with strict implication,Zeitschrift fur Mathematische Logik und Grund-lagen der Mathematic, 33:389-406, 1987.
[3] K. Doˇsen. Duality between modal algebras and neighborhood frames,Studia Logica, 48;219-234, 1989.
[4] H. H. Hansen,Monotonic Modal Logics, Master thesis, University of Amsterdam, 2003. [5] D. de Jongh, F. Shirmohammadzadeh Maleki, Subintuitionistic Logics with Kripke Semantics,
submitted.
[6] S. C. Kleene, Disjunction and existence under implication in elementary intuitionistic formalisms. Journal of Symbolic Logic, 27:11-18, 1962. addendum ibidem 28,154-156.
[7] M. Kracht, F. Wolter, Normal monomodal logics can simulate all others,Journal of Symbolic Logic, 64(1): 99-138, 1999.
[8] M. Moniri, F. Shirmohammadzadeh Maleki, Neighborhood Semantics for Basic and Intuitionistic Logic,Logic and Logical Philosophy, Volume 24 (2015), 339-355.
[9] G. Restall, Subintuitionistic Logics,Notre Dame Journal of Formal Logic, Volume 35, Number 1, Winter 1994.
[10] K. Sano, M. Ma, Alternative semantics for Visser’s propositional logics. M. Aher et al (Eds.): 10th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2013, pp. 257-275, 2015, LNCS 8984, Springer.
[11] A. Visser, A propositional logic with explicit fixed points,Studia Logica, 40, 2 (1981): 155-175.
