2. Modal Completeness: Introduction
Let K be the class of ILM–frames. As usual in completeness proofs, the modal completeness of ILM is proved by contraposition. Given a sentenceAs.t. 0ILM A,
we will find anILM-model MwithM1A.
The general idea of the proof is as follows. If 0ILM A, there is a maximal–ILM–
consistent set Γ with¬A∈Γ. We will build anILM-frame F, where every nodex
is labeled with a maximal–ILM–consistent set ν(x). We start with a node w and
ν(w) = Γ;wwill be the root ofF. Fwill be built step by step, using the information contained in the maximal consistent sets labeling the nodes. Finally, F will be transformed into a modelMby defining a valuation onFas:M, xp:⇔p∈ν(x).
We want to build F in such a way that the harmony between truth in M and membership in the maximal consistent sets labeling the nodes extends to a larger set than just the propositional formulas. In particular, we want to be able to conclude thatM, w¬Aon the basis that¬A∈ν(w).
Definition 79. A set D of formulas is adequate if it is finite1, and closed under
subformulas and single negations.
Let Dbe an adequate set containing A. If M is defined as above, our goal is to have
(85) M, xB⇔B ∈ν(x).
We call the equivalence in (85) atruth lemma w.r.t. D. When constructing F, we thus need to ensure:
1. a truth lemma holds inF w.r.t. D 2. F is anILM–frame
F will be constructed as the union of an infinite chain of ILM–frames {Fn}n∈ω.
With each Fn, we come closer to the truth lemma (w.r.t. D). In order to make
sure that each Fn is an ILM–frame, Fn itself is also constructed as the union of
an infinite chain{Gi}i∈ω, where eachGi is closer to being an ILM–frame than the
previous one.
3. Preparing the Construction
This section introduces the tools we will need for our construction. In modal completeness proofs, the nodes of the countermodel are often taken to be maximal consistent sets. However, in the context of interpretability logic it is sometimes necessary to use the same maximal consistent set in different places of the model. Therefore we will not identify a nodexwith a maximal consistent set, but rather label it with a maximal consistent setν(x). We will also label someR transitions with formulas: ifxRy andν(hx, yi) =B, theny leads into a B-critical cone above
x(the notion of aB-critical cone will be defined below).
1It is not necessary forDto be actually finite; it is also sufficient if it contains only finitely many formulas up to provable equivalence. See the discussion in Section 3.1 of Chapter 2.
3. PREPARING THE CONSTRUCTION 77
Definition 80. A labeled frame is a quadruple hW, R, S, νi. Here W is a non- empty set of worlds,R a binary relation onW, andS a set of binary relations on
W indexed by elements of W. The function ν assigns to each x∈ W a maximal
ILM–consistent set of sentences ν(x). To some pairs hx, yiwith xRy, ν assigns a formulaν(hx, yi).
Ifν(hx, yi) =B, we will writexRBy. Thus anRB transition is just anRtransition labeled by the formula B. Note that a labeled frame F does not have to be an
ILM–frame, or even anIL–frame.
When defining relationsRandS between nodesxandy of the frame, we want the maximal consistent sets ν(x) and ν(y) to be related in a coherent way. For this purpose, we will define the following relations between maximal consistent.
Definition 81. Let Γ and ∆ be maximalILM–consistent sets.
i. Γ≺∆ :=2A∈∆⇒A,2A∈∆, and there is some E∈ Ds.t.2E∈∆\Γ ii. Γ≺B∆ :=AB∈Γ⇒ ¬A2¬A∈∆ and there is someE∈ Ds.t.2E∈∆\Γ
iii. Γ⊆2∆ :=2A∈Γ⇒2A∈∆
Note that if Γ ≺B ∆, then also Γ ≺ ∆. Furthermore, Γ ≺B ∆ ≺ ∆0 implies
Γ≺B ∆0. We will refer to≺as thesuccessor relation, and to≺B as theB-critical successor relation.
The following definition helps us to enforce the truth lemma for formulas of the form ¬(AB). Ifx¬(AB), there has to be somey s.t.xRyAand for all
z, ifySxz, thenz 1B. The B-critical cone above xcontains all nodes which are Sx-accessible fromy. All of them have to beB-critical successors ofx.
Definition82. Letxbe a node in a labeled frame. TheB-critical cone abovex, we
writeCB
x, containsywithxRBy, and is closed underR,SxandR◦Strtransitions2.
Thegeneralized B-cone above x, we writeGB
x, contains CxB, and is closed underR
andSw (for arbitraryw) transitions.
The above definition is redundant forIL–frames, where closure underRtransitions follows from closure under Sx transitions. However, we want to use the notion of B-critical cones also in the context of frames which arenot IL–frames. Demanding closure under R, Sx and R◦Str transitions is motivated by the fact that in an ILM–frame, allSxtransitions will then remain insideCBx. Consequently, if we want
to ensure that x ¬(AB), then CB
x will be a “good” place for having an R
successor y of xwith y A — given that we can guarantee that all labels inCB x
contain¬B, and notB. Note that sinceCB
x ⊆ GxB by definition,GxB∩ GxC=∅ impliesCxB∩ CxC=∅for allB
andC.
The notion of adequacy will help us to guarantee that the labels of nodes related via an RorS transition are coherently related themselves. Clauses iii and iv help us to enforce the truth lemma for formulas of the form¬(AB). All frames we construct will be adequate in this sense.
2IfSis a relation, we writeStrfor the transitive closure ofS. The◦is the composition operator
4. OVERVIEW 78
Definition83. Anadequateframe is a labeled frame with the following properties: i. xRy⇒ν(x)≺ν(y) ii. ySxz⇒ν(y)⊆2ν(z) iii. y∈ CB x ⇒ν(x)≺B ν(y) iv. A6=B⇒ GA x ∩ GxB =∅
The notions of problems and deficiencies allow us to approximate the truth lemma step by step. Whenever we eliminate aD-problem or aD-deficiency, we get closer to the truth lemma w.r.t. D. If the setD is clear or fixed, we will just speak of problems and deficiencies.
Definition 84. A D-problem is a pair hx,¬(AB)is.t. ¬(AB) ∈ν(x)∩ D,
but there is noy∈ CB
x s.t.A∈ν(y).
Definition85. AD-deficiency is a triplehx, y, CDis.txRy,CD∈ν(x)∩ D,
C∈ν(y)∩ D, but for no zs.t. ySxz we haveD∈ν(z).
4. Overview
This section gives an overview of the construction. LetAbe a sentence s.t.0ILMA,
let Γ be a maximal consistent set containing¬A, and letDbe the smallest adequate set containing¬A. Define a labeled frameF0 :=h{w} ∅,{∅},hw,ΓiiNote thatF0 is adequate. We will now extend F0 to an adequateILM–frame F containing no problems or deficiencies. It is easy to see that then the truth lemma holds onF.
Lemma 4.1. Let hW, R, S, νi be an adequate labeled frame. Let M be the model induced by lettingM, xp:⇔p∈ν(x). Then a truth lemma holds inMw.r.t.D
iff there are noD-problems or D-deficiencies inF.
Proof. Immediate. The only non-trivial part of the truth lemma could be
reformulated as “there are no problems or deficiencies”. 2 As said above, we will constructFas the limit of a possibly infinite chain{Fn}n∈ω,
ofILM–frames. Furthermore, we require eachFn to be adequate. Fix an ordering
on the setP of possibleD-problems and -deficiencies in current and future worlds3. When going from Fn to Fn+1, we will eliminate the problem or deficiency in Fn
which is minimal w.r.t. this ordering, guaranteeing that it will not recur in the future. By constructionF :=S
n∈ωFn, does not contain problems or deficiencies.
Apart from eliminating problems and deficiencies, we have to guarantee that each
Fn is anILM–frame. As we will see, a problem or deficiency inFn is eliminated by
adding toFn a new node together with an appropriate label, as well as a newRor
a newSrelation. E.g. to eliminate the problemha,¬(AB)i, we add toW a new node b withaRb, A∈ν(b), andb ∈ CB
a. The resulting frameGis not necessarily
anILM–frame. For example, wRa does not implywRbin G, i.e. theR relation is not transitive. In order to come back to an adequateILM–frameFn+1, we have to
close off under the frame conditions ofILM.