Intermediate Logics 3.1 Overview
3.3. KRIPKE SEMANTICS
which we read as stating that formulaA is true in statex. (We abbreviate¬(M,xA) as
M,x1A.) When the specific model is clear from the context, we simply write xA(or
x1A).
We call a formulaAtruein a Kripke modelM, written asMA(or asAwhenK is clear from the context), ifffor allw∈W.wA.
We call a formulaAvalidin a classK of Kripke models, written asKA(or simply asA whenK is obvious from the context), to denote that the formula A is true in all Kripke models fromK.
Remark 3.6. An alternative definition of a model is to use a domain function D in
place of v such that P∈D(w) iff v(w,P)=1. This is why we distinguish between the interpretation functionvand the extended interpretation functionv0.
Definition3.7 (Completeness). LetSbe a propositional logic, and letK be a class of
Kripke models. SiscompleteforK when for allA∈Prop,A∈SiffKA.
Definition3.8 (Finite Kripke Frame). Let M=hW,Ribe a Kripke frame. Misfinite
iffW is finite.
Definition 3.9 (Finite Model Property). A logic S has the finite model property
(FMP) iff it is complete for a class of finite Kripke frames K. That is, a if a formula
A<S, then there exists a finite Kripke modelM∈ K such thatM2A.
3.3.2. Intuitionistic Kripke Frames and Models. We recall the definition of intu- itionistic Kripke frames and models below (cf. [CZ97, Min00, van02]).
Definition3.10 (Intuitionistic Kripke Models). The classKIntofintuitionistic Krip-
ke modelsis the class of all Kripke modelsM=hW,R,viwith the properties: (1) Ris apre-orderingonW—i.e. Risreflexiveandtransitive.
(2) Mispersistent2, that is, for allx,y∈WandP∈Var, if (x,y)∈Randv(x,P)=1, thenv(y,P)=1;
The extended interpretation functionv0for intuitionistic Kripke models is defined for all points x,y∈W and formulaeA,B∈Propas:
(1) v0(x,A)=v(x,A) iffAis a propositional variable; 2Persistence is also called “monotonicity” or “heredity”.
46 3. INTERMEDIATE LOGICS
(2) v0(x,⊥)=0;
(3) v0(x,A∧B)=1 iffv0(x,A)=1 andv0(x,B)=1; (4) v0(x,A∨B)=1 iffeitherv0(x,A)=1 orv0(x,B)=1;
(5) v0(x,A⊃B)=1 iffwhenever (x,y)∈R, ifv0(y,A)=1 thenv0(y,B)=1.
Proposition 3.11 (General Persistence). In the class of Intuitionistic Kripke frames
KInt, for all states x,y∈W, if Rxy and v0(x,A)= 1 then v0(y,A)= 1, for all compound formulae A.
Proof. By induction on the structure ofA. See [Min00].
Lemma3.12. Inthas the FMP.
Proof. From completeness with respect to finite intuitionistic Kripke frames,
e.g. [Min00].
Definition3.13 (Rooted frame and model). Arooted (Kripke) frame(also called a
pointed frame) M is a Kripke framehW,Ri, where there exists x∈W such that, for all
y∈W, (x,y)∈R. x is called the rootof a Kripke frame (also called the distinguished point).
Arooted (Kripke) modelis a Kripke model based on a rooted Kripke frame.
Lemma 3.14. A rooted intuitionistic Kripke frame is partially ordered—that is, it is
reflexive, transitive and antisymmetric.
Proof. [Min00, §7.2] or [CZ97, §2.3].
Theorem3.15. A formula A is valid in all intuitionistic Kripke models iffit is valid in
all rooted intuitionistic Kripke models.
Proof. [Min00, §7.2] or [CZ97, §2.3].
The existence of a distinguished point in an intuitionistic Kripke frame is generally used for completeness proofs for calculi, e.g. [Min00]. Later we will use Theorem 3.15 to justify the soundness of a root rule for relational sequent calculi. (For example, see Lemma 9.30 on page 205.)
3.3. KRIPKE SEMANTICS 47
3.3.3. Semantics of Intermediate Logics. We present the extensions for various in- termediate logics belows, but first define useful notation and terminology with respect to the properties of some of these logics.
Because extensions to intuitionistic Kripke frames can be considered rooted, and thus partially ordered, we can apply the terminology of trees and directed graphs (e.g. from [Die05]) to them:
Definition 3.16 (Subframe). Let M =hW,Ri be an intuitionistic Kripke frame. We
call M0=hW0,R0iasubframeofMifW0⊆W,R0⊆RandR0is a partial order.
Definition 3.17 (Chain). Let M=hW,Ribe an intuitionistic Kripke frame. A set of
pointsW0⊆Wforms achainin Mifffor all x,y∈W0, either (x,y)∈Ror (y,x)∈R. Definition3.18 (Depth of a Kripke Frame). Thedepthof a Kripke FrameM=hW,Ri
is the size of the largest chain in M.
Definition3.19 (Antichain). Let M=hW,Ribe an intuitionistic Kripke frame. A set
of pointsW0⊆Wforms anantichaininMifffor allx,y∈W0, both (x,y)<Rand (y,x)<R. Definition3.20 (Width of a Kripke Frame). Thewidthof a Kripke FrameM=hW,Ri
is the size of the largest antichain in M.
Definition3.21 (Cardinality of a Kripke Frame). Thecardinalityof a Kripke Frame
hW,Riis|W|(the size ofW).
We now give the semantic properties of the intermediate logics listed in Section 3.2.2 above, defined as extensions to an intuitionistic Kripke frameM=hW,Ri. We assume that quantified variables below belong to the set of pointsW, and that relations belong to the set of relationsR, but generally omit specific reference to them below for brevity.
The frame properties given below come from [CZ97] unless otherwise noted. All of these logics have the finite model property (FMP) [CZ97, van02].
Classical Logic: Clcorresponds to the class ofsymmetricKripke frames:
∀xy.Rxy⊃Ryx (42)
Because we can assume that the Kripke frames are rooted, from antisymmetry, we can conclude that∀xy.x=y, and thus|W|=1.
48 3. INTERMEDIATE LOGICS
Jankov-De Morgan Logic: Jancorresponds to the class ofdirected(or converg- ing) Kripke frames:
∀wxy.(Rwx∧Rwy)⊃ ∃z.(Rxz∧Ryz) (43) Because frames are rooted, we can simplify this to∃z.(Rxz∧Ryz).
Gödel-Dummett Logic: GDcorresponds to the class of linearKripke frames:
∀xy.Rxy∨Ryx (44)
Logics of Bounded Depth: BDk corresponds to the class of Kripke frames with bounded depthk: ∀x0,...,xk. k−1 ^ i=0 Rxixi+1⊃ _ i,j xi=zj (45)
Using antisymmetry, this can be simplified to
∀x0,...,xk. k−1 ^ i=0 Rxixi+1⊃ k−1 _ i=0 Rxi+1xi (46)
For example, the characteristic frame condition ofBD2is:
∀xyz.((Rxy∧Ryz)⊃(Ryx∨Rzy)) (47)
Gödel Logics: Gk corresponds to the class of linear Kripke frames (44) of bound- ed depthk−1 (46). For example,G3 (also called Sm) is the union ofGDand
BD2.
Logics of Bounded Width: BWkcorresponds to the class of Kripke frames where every rooted subframe is of a width bounded byk:
∀x,y0,...,yk. k ^ i=0 Rxyi⊃ _ j,i Ryiyj (48)
Because we can assume that the frames are rooted, this can be simplified to
∀x,y0,...,yk.
_
j,i
3.3. KRIPKE SEMANTICS 49
Logics of Bounded Top Width: BTWkcorresponds to the class of Kripke frames where there are at mostkpoints at the top of the frame:
∀x,y0,...,yk. k ^ i=0 Rxyi⊃ ∃z. _ j,i (Ryiz∧Ryjz) (50)
Because we can assume that the frames are rooted, we can simplify this to
∀y0,...,yk∃z. k
_
i=0,j,i
(Ryiz∧Ryjz) (51)
Logics of Bounded Cardinality: BCk corresponds to the class of Kripke frames where|W|=k, which can be expressed as the frame condition
∀x0,x1,...,xk. k ^ i=1 Rx0xi⊃ _ j,i xj=xi (52)
Using antisymmetry, this can be simplified to
∀x0,x1,...,xk. k ^ i=1 Rx0xi⊃ k _ i=1 Rxix0 (53)
Greatest Semiconstructive Logic: GSccorresponds to the class of Kripke frames that have a depth of at most 2 (47) and a bounded top width (51) of 2. That is for any three distinct points, there is a point accessible from two of them . This corresponds to (47) and (51) fork=2:
∀wxy.∃z.((Rwz∧Rxz)∨(Rxz∧Ryz)∨(Rwz∧Ryz)) (54)
Kreisel-Putnam Logic: KP corresponds to the class of Kripke frames with the following condition:
∀xyz.(Rxy∧Rxz∧ ¬Ryz∧ ¬Rzy⊃ ∃u.(Rxu∧Ruy∧Ruz∧fin2(u,y,z))) (55) where we adapt (51) to be the function
fink(x,y0,...,yk)=de f ∀w.Rxw⊃ ∃z.Rwz∧ k
_
i=0,j,i
50 3. INTERMEDIATE LOGICS
Remark3.22. We have not discussed other well-known intermediate logics, such as
the logic of bounded branching (BBk) [GD74], Scott logic (SL) or Anti-Scott logic (ASL) [KP57] because they do not have Kripke frames that are axiomatised by first-order for- mulae. (See [FM93] for further discussion.)
3.3.4. Beth Semantics. We give a brief outline of Beth semantics for Int here, as it is an alternative relational semantics (that predates Kripke semantics), and inspired the hypersequent calculus by Beth from [Bet59] (shown in Figure 4.1 on page 83), and possibly other calculi such as Maslov’sO[Mas67, Mas69] (Figure 5.1 on page 112).
The description of Beth semantics given here is based on [van02]. The reader is also referred to [Tv88, Ch. 13], and to [Tv99] for the historical context.
Definition3.23 (Beth Frames and Models). We define Beth Frames and Models sim-
ilarly to Kripke Frames and Models (Definition 3.5 on page 44). A Beth frame M is a tree (orspread)hW,Ri, whereW is a non-empty set ofpoints,R⊆W2is a binary relation between points We may abbreviate (x,y)∈R asRxy. M is rooted, that is, there exists a distinguished point x∈W such that for ally∈W, (x,y)∈R.
Let M=hW,Ribe a Beth frame. Apath P⊆W through a point x∈W is a maximal, linearly ordered set (onR). A barfor x∈W is a subset B⊆W such that if P is a path through x (that is, x∈P), then there exists y∈B such that y∈P. Informally, all paths throughxpass through the barB.
The definitions relating to Kripke models are extended naturally to Beth models. Definition 3.24 (Intuitionistic Beth Models). The class BInt of intuitionistic Beth
modelsis the class of all Beth modelsM=hW,R,viwith the properties: (1) Ris apre-orderingonW—i.e. Risreflexiveandtransitive.
(2) Mispersistent, that is, for allx∈WandP∈Var, ifv(x,P)=1, then there exists a barB⊆W forxsuch that for ally∈B,v(y,P)=1;;
(3) For all x∈W,v(x,⊥)=0;
(4) For all x∈W,v(x,A∧B)=1 iffv(x,A)=1 andv(x,B)=1;
(5) v(x,A∨B)=1 iff there exists a bar B⊆ W for x such that for all y∈B, either
v(y,A)=1 orv(y,B)=1;