aa r X i v : . [ m a t h . L O ] A p r Modal Matters in Interpretability Logics
E. GorisandJ. J. Joosten2008
Samenvatting
This paper from 2008 is the first in a series of three related paperson modal methods in interpretability logics and applications. In this firstpaper the fundaments are laid for later results. These fundaments consistof a thorough treatment of a construction method to obtain modal models.This construction method is used to reprove some known results in thearea of interpretability like the modal completeness of the logic IL . Next,the method is applied to obtain new results: the modal completeness ofthe logic IL M , and modal completeness of IL W ∗ . Interpretability logics are primarily used to describe structural behavior of inter-pretability between formal mathematical theories. We shall see that the logicscome with a good modal semantics that naturally extends the regular modalsemantics giving it a dynamical flavor. In this introduction we shall informallydescribe the project of this paper. Formal definitions are postponed to latersections.The notion of interpretability that we are primarily interested in, is the no-tion of relativized interpretability as studied e.g. by Tarski et al in [26]. Roughly,a theory U interprets a theory V –we write U ✄ V – if U proves all theorems of V under some structure preserving translation. We allow for relativization ofquantifiers. It is defendable to say that U is as least as strong as V if U ✄ V . Wethink that it is clear that interpretations are worth to be studied, as they areomnipresent in both mathematics and meta-mathematics (Langlands Program,relative consistency proofs, undecidability results, Hilberts Programme and soforth).One approach to the study of interpretability is to study general structuralbehavior of interpretability. An example of such a structural rule is the transiti-vity of interpretability. That is, for any U , V and W we have that if U ✄ V and V ✄ W , then also U ✄ W . As we shall see, modal interpretability logics providean informative way to support this structural study. Interpretability logics, in a1ense, generate all structural rules. Many important questions on interpretabi-lity logics have been settled. One of the most prominent open questions at thistime is the question of the interpretability logic of all reasonable arithmeticaltheories. In this paper we make a significant contribution to a solution of thisproblem. However, a modal characterization still remains an open question.The main aim of this paper is to establish some modal techniques/toolkitfor interpretability logics. Most techniques are aimed at establishing modalcompleteness results. As we shall see, in the field of interpretability logics, modalcompleteness can be a sticky business compared to unary modal logics. In thispaper we make a first attempt at pulling some (more) thorns out. Significantprogress with this respect has also been made by de Jong and Veltman [8].We have a feeling that the general modal theory of interpretability logicsis getting more and more mature. For example, fixed point phenomena andinterpolation are quite well understood ([10], [1], [32]).Experience tells us that our modal semantics is quite informative and perspi-cuous. It is even the case that new arithmetical principles can be obtained frommodal semantical considerations. An example is our new principle R . We foundthis principle primarily by modal investigation. Thus, indeed, there is a closematch between the modal part and the arithmetical part. It is even possible toembed our modal semantics into some category of models of arithmetic.Although this paper is mainly a modal investigation, the main questionsare still inspired by the arithmetical meaning of our logics. Thus, our investi-gations will lead to applications concerning arithmetically informative notionslike, essentially Σ -sentences, self provers and the interpretability logic of allreasonable arithmetical theories. In this section we will define the basic notions that are needed throughout thepaper. We advise the reader to just skim through this section and use it to lookup definitions whenever they are used in the rest of the paper.
In this paper we shall be mainly interested in interpretability logics, the formulasof which, we write
Form IL , are defined as follows. Form IL := ⊥ | Prop | ( Form IL → Form IL ) | ( ✷ Form IL ) | ( Form IL ✄ Form IL )Here Prop is a countable set of propositional variables p, q, r, s, t, p , p , . . . .We employ the usual definitions of the logical operators ¬ , ∨ , ∧ and ↔ . Alsoshall we write ✸ ϕ for ¬ ✷ ¬ ϕ . Formulas that start with a ✷ are called box-formulas or ✷ -formulas. Likewise we talk of ✸ -formulas.From now on we will stay in the realm of interpretability logics. Unlessmentioned otherwise, formulas or sentences are formulas of Form IL . We will2rite p ∈ ϕ to indicate that the proposition variable p does occur in ϕ . A literalis either a propositional variable or the negation of a propositional variable.In writing formulas we shall omit brackets that are superfluous accordingto the following reading conventions. We say that the operators ✸ , ✷ and ¬ bind equally strong. They bind stronger than the equally strong binding ∧ and ∨ which in turn bind stronger than ✄ . The weakest (weaker than ✄ ) bindingconnectives are → and ↔ . We shall also omit outer brackets. Thus, we shallwrite A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C instead of (( A ✄ B ) → (( A ∧ ( ✷ C )) ✄ ( B ∧ ( ✷ C )))).A schema of interpretability logic is syntactically like a formula. They areused to generate formulae that have a specific form. We will not be specificabout the syntax of schemata as this is similar to that of formulas. Below, onecan think of A , B and C as place holders.The rule of Modus Ponens allows one to conclude B from premises A → B and A . The rule of Necessitation allows one to conclude ✷ A from the premise A . Definition 2.1.
The logic IL is the smallest set of formulas being closed underthe rules of Necessitation and of Modus Ponens, that contains all tautologicalformulas and all instantiations of the following axiom schemata. L1 ✷ ( A → B ) → ( ✷ A → ✷ B ) L2 ✷ A → ✷✷ A L3 ✷ ( ✷ A → A ) → ✷ A J1 ✷ ( A → B ) → A ✄ B J2 ( A ✄ B ) ∧ ( B ✄ C ) → A ✄ C J3 ( A ✄ C ) ∧ ( B ✄ C ) → A ∨ B ✄ C J4 A ✄ B → ( ✸ A → ✸ B ) J5 ✸ A ✄ A We will write IL ⊢ ϕ for ϕ ∈ IL . An IL -derivation or IL -proof of ϕ is a finitesequence of formulae ending on ϕ , each being a logical tautology, an instantiationof one of the axiom schemata of IL , or the result of applying either ModusPonens or Necessitation to formulas earlier in the sequence. Clearly, IL ⊢ ϕ iffthere is an IL -proof of ϕ .Sometimes we will write IL ⊢ ϕ → ψ → χ as short for IL ⊢ ϕ → ψ & IL ⊢ ψ → χ . Similarly for ✄ . We adhere to a similar convention when we employbinary relations. Thus, xRyS x z (cid:13) B is short for xRy & yS x z & z (cid:13) B , and soon. Sometimes we will consider the part of IL that does not contain the ✄ -modality. This is the well-known provability logic GL , whose axiom schemataare L1 - L3 . The axiom schema L3 is often referred to as L¨ob’s axiom. Lemma 2.2. IL ⊢ ✷ A ↔ ¬ A ✄ ⊥ . IL ⊢ A ✄ A ∧ ✷ ¬ A IL ⊢ A ∨ ✸ A ✄ A Bewijs.
All of these statements have very easy proofs. We give an informal proofof the second statement. Reason in IL . It is easy to see A ✄ ( A ∧ ✷ ¬ A ) ∨ ( A ∧ ✸ A ).By L3 we get ✸ A → ✸ ( A ∧ ✷ ¬ A ). Thus, A ∧ ✸ A ✄ ✸ ( A ∧ ✷ ¬ A ) and by J5 weget ✸ ( A ∧ ✷ ¬ A ) ✄ A ∧ ✷ ¬ A . As certainly A ∧ ✷ ¬ A ✄ A ∧ ✷ ¬ A we have that( A ∧ ✷ ¬ A ) ∨ ( A ∧ ✸ A ) ✄ A ∧ ✷ ¬ A and the result follows from transitivity of ✄ . Apart from the axiom schemata exposed in Definition 2.1 we will on occas-sion consider other axiom schemata too. M A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C P A ✄ B → ✷ ( A ✄ B ) M A ✄ B → ✸ A ∧ ✷ C ✄ B ∧ ✷ C W A ✄ B → A ✄ B ∧ ✷ ¬ A W ∗ A ✄ B → B ∧ ✷ C ✄ B ∧ ✷ C ∧ ✷ ¬ A P A ✄ ✸ B → ✷ ( A ✄ B ) R A ✄ B → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C If X is a set of axiom schemata we will denote by IL X the logic that arisesby adding the axiom schemata in X to IL . Thus, IL X is the smallest set offormulas being closed under the rules of Modus Ponens and Necessitation andcontaining all tautologies and all instantiations of the axiom schemata of IL ( L1 - J5 ) and of the axiom schemata of X . Instead of writing IL { M , W } we willwrite IL M W and so on.We write IL X ⊢ ϕ for ϕ ∈ IL X . An IL X -derivation or IL X -proof of ϕ is a finite sequence of formulae ending on ϕ , each being a logical tautology,an instantiation of one of the axiom schemata of IL X , or the result of applyingeither Modus Ponens or Necessitation to formulas earlier in the sequence. Again, IL X ⊢ ϕ iff there is an IL X -proof of ϕ . For a schema Y , we write IL X ⊢ Y if IL X proves every instantiation of Y . Definition 2.3.
Let Γ be a set of formulas. We say that ϕ is provable fromΓ in IL X and write Γ ⊢ IL X ϕ , iff there is a finite sequence of formulae endingon ϕ , each being a theorem of IL X , a formula from Γ, or the result of applyingModus Ponens to formulas earlier in the sequence.Clearly we have ∅ ⊢ IL X ϕ ⇔ IL X ⊢ ϕ . In the sequel we will often write justΓ ⊢ ϕ instead of Γ ⊢ IL X ϕ if the context allows us so. It is well known that wehave a deduction theorem for this notion of derivability. Lemma 2.4 (Deduction theorem) . Γ , A ⊢ IL X B ⇔ Γ ⊢ IL X A → B ewijs. “ ⇐ ” is obvious and “ ⇒ ” goes by induction on the length n of the IL X -proof σ of B from Γ , A .If n>
1, then σ = τ, B , where B is obtained from some C and C → B occurring earlier in τ . Thus we can find subsequences τ ′ and τ ′′ of τ such that τ ′ , C and τ ′′ , C → B are IL X -proofs from Γ , A . By the induction hypothesiswe find IL X -proofs from Γ of the form σ ′ , A → C and σ ′′ , A → ( C → B ). Wenow use the tautology ( A → ( C → B )) → (( A → C ) → ( A → B )) to get an IL X -proof of A → B from Γ. Definition 2.5.
A set Γ is IL X -consistent iff Γ IL X ⊥ . An IL X -consistent setis maximal IL X -consistent if for any ϕ , either ϕ ∈ Γ or ¬ ϕ ∈ Γ. Lemma 2.6.
Every IL X -consistent set can be extended to a maximal IL X -consistent one.Bewijs. This is Lindebaums lemma for IL X . We can just do the regular argu-ment as we have the deduction theorem. Note that there are countably manydifferent formulas.We will often abbreviate “maximal consistent set” by MCS and refrain fromexplicitly mentioning the logic IL X when the context allows us to do so. Wedefine three useful relations on MCS’s, the successor relation ≺ , the C -criticalsuccessor relation ≺ C and the Box-inclusion relation ⊆ ✷ . Definition 2.7.
Let Γ and ∆ denote maximal IL X -consistent sets. • Γ ≺ ∆ := ✷ A ∈ Γ ⇒ A, ✷ A ∈ ∆ • Γ ≺ C ∆ := A ✄ C ∈ Γ ⇒ ¬ A, ✷ ¬ A ∈ ∆ • Γ ⊆ ✷ ∆ := ✷ A ∈ Γ ⇒ ✷ A ∈ ∆It is clear that Γ ≺ C ∆ ⇒ Γ ≺ ∆. For, if ✷ A ∈ Γ then ¬ A ✄ ⊥ ∈ Γ. Also ⊥ ✄ C ∈ Γ, whence ¬ A ✄ C ∈ Γ. If now Γ ≺ C ∆ then A, ✷ A ∈ ∆, whenceΓ ≺ ∆. It is also clear that Γ ≺ C ∆ ≺ ∆ ′ ⇒ Γ ≺ C ∆ ′ . Lemma 2.8.
Let Γ and ∆ denote maximal IL X -consistent sets. We have Γ ≺ ∆ iff Γ ≺ ⊥ ∆ .Bewijs. Above we have seen that Γ ≺ A ∆ ⇒ Γ ≺ ∆. For the other directionsuppose now that Γ ≺ ∆. If A ✄ ⊥ ∈ Γ then, by Lemma 2.2.1, ✷ ¬ A ∈ Γ whence ¬ A, ✷ ¬ A ∈ ∆. Interpretability logics come with a Kripke-like semantics. As the signature ofour language is countable, we shall only consider countable models.
Definition 2.9. An IL -frame is a triple h W, R, S i . Here W is a non-emptycountable universe, R is a binary relation on W and S is a set of binary re-lations on W , indexed by elements of W . The R and S satisfy the followingrequirements. 5. R is conversely well-founded xRy & yRz → xRz yS x z → xRy & xRz xRy → yS x y xRyRz → yS x z uS x vS x w → uS x w IL -frames are sometimes also called Veltman frames. We will on occasionspeak of R or S x transitions instead of relations. If we write ySz , we shall meanthat yS x z for some x . W is sometimes called the universe, or domain, of theframe and its elements are referred to as worlds or nodes. With x ↾ we shalldenote the set { y ∈ W | xRy } . We will often represent S by a ternary relationin the canonical way, writing h x, y, z i for yS x z . Definition 2.10. An IL -model is a quadruple h W, R, S, (cid:13) i . Here h W, R, S, i isan IL -frame and (cid:13) is a subset of W × Prop . We write w (cid:13) p for h w, p i ∈ (cid:13) . Asusual, (cid:13) is extended to a subset e (cid:13) of W × Form IL by demanding the following. • w e (cid:13) p iff w (cid:13) p for p ∈ Prop • w e (cid:13) ⊥• w e (cid:13) A → B iff w e (cid:13) A or w e (cid:13) B • w e (cid:13) ✷ A iff ∀ v ( wRv ⇒ v e (cid:13) A ) • w e (cid:13) A ✄ B iff ∀ u ( wRu ∧ u e (cid:13) A ⇒ ∃ v ( uS w v e (cid:13) B ))Note that e (cid:13) is completely determined by (cid:13) . Thus we will denote e (cid:13) also by (cid:13) . We call (cid:13) a forcing relation. The (cid:13) -relation depends on the model M . Ifnecessary, we will write M, w (cid:13) ϕ , if not, we will just write w (cid:13) ϕ . In this casewe say that ϕ holds at w , or that ϕ is forced at w . We say that p is in therange of (cid:13) if w (cid:13) p for some w .If F = h W, R, S i is an IL -frame, we will write x ∈ F to denote x ∈ W andsimilarly for IL -models. Attributes on F will be inherited by its constituentparts. For example F i = h W i , R i , S i i . Often however we will write F i | = xRy instead of F i | = xR i y and likewise for the S -relation. This notation is consis-tent with notation in first order logic where the symbol R is interpreted in thestructure F i as R i .If M = h W, R, S, (cid:13) i , we say that M is based on the frame h W, R, S i and wecall h W, R, S i its underlying frame.If Γ is a set of formulas, we will write M, x (cid:13)
Γ as short for ∀ γ ∈ Γ M, x (cid:13) γ .We have similar reading conventions for frames and for validity. A relation R on W is called conversely well-founded if every non-empty subset of W hasan R -maximal element. efinition 2.11 (Generated Submodel) . Let M = h W, R, S, (cid:13) i be an IL -modeland let m ∈ M . We define m ↾ ∗ to be the set { x ∈ W | x = m ∨ mRx } . By M ↾ m we denote the submodel generated by m defined as follows. M ↾ m := h m ↾ ∗ , R ∩ ( m ↾ ∗ ) , [ x ∈ m ↾ ∗ S x ∩ ( m ↾ ∗ ) , (cid:13) ∩ ( m ↾ ∗ × Prop ) i Lemma 2.12 (Generated Submodel Lemma) . Let M be an IL -model and let m ∈ M . For all formulas ϕ and all x ∈ m ↾ ∗ we have that M ↾ m, x (cid:13) ϕ iff M, x (cid:13) ϕ. Bewijs.
By an easy induction on the complexity of ϕ .We say that an IL -model makes a formula ϕ true, and write M | = ϕ , if ϕ isforced in all the nodes of M . In a formula we write M | = ϕ : ⇔ ∀ w ∈ M w (cid:13) ϕ. If F = h W, R, S i is an IL -frame and (cid:13) a subset of W × Prop , we denote by h W, (cid:13) i the IL -model that is based on F and has forcing relation (cid:13) . We say thata frame F makes a formula ϕ true, and write F | = ϕ , if any model based on F makes ϕ true. In a second-order formula: F | = ϕ : ⇔ ∀ (cid:13) h F, (cid:13) i | = ϕ We say that an IL -model or frame makes a scheme true if it makes all itsinstantiations true. If we want to express this by a formula we should havea means to quantify over all instantiations. For example, we could regard aninstantiation of a scheme X as a substitution σ carried out on X resulting in X σ . We do not wish to be very precise here, as it is clear what is meant. Ourdefinitions thus read F | = X iff ∀ σ F | = X σ for frames F , and M | = X iff ∀ σ M | = X σ for models M . Sometimes we will also write F | = IL X for F | = X .It turns out that checking the validity of a scheme on a frame is fairlyeasy. If X is some scheme , let τ be some base substitution that sends differentplaceholders to different propositional variables. Lemma 2.13.
Let X be a scheme, and τ be a corresponding base substitutionas described above. Let F be an IL -frame. We have F | = X τ ⇔ ∀ σ F | = X σ . Or a set of schemata. All of our reasoning generalizes without problems to sets of sche-mata. We will therefore no longer mention the distinction. ewijs. If ∀ σ F | = X σ , then certainly F | = X τ , thus we should concentrate onthe other direction. Thus, assuming F | = X τ we fix some σ and (cid:13) and set outto prove h F, (cid:13) i | = X σ . We define another forcing relation (cid:13) ′ on F by sayingthat for any place holder A in X we have w (cid:13) ′ τ ( A ) : ⇔ h F, (cid:13) i | = σ ( A )By induction on the complexity of a subscheme Y of X we can now prove h F, (cid:13) ′ i , w (cid:13) ′ Y τ ⇔ h F, (cid:13) i , w (cid:13) Y σ . By our assumption we get that h F, (cid:13) i , w (cid:13) X σ .If χ is some formula in first, or higher, order predicate logic, we will evaluate F | = χ in the standard way. In this case F is considered as a structure of firstor higher order predicate logic. We will not be too formal about these mattersas the context will always dict us which reading to choose. Definition 2.14.
Let X be a scheme of interpretability logic. We say that aformula C in first or higher order predicate logic is a frame condition of X if F | = C iff F | = X . The C in Definition 2.14 is also called the frame condition of the logic IL X .A frame satisfying the IL X frame condition is often called an IL X -frame. Incase no such frame condition exists, an IL X -frame resp. model is just a frameresp. model, validating X .The semantics for interpretability logics is good in the sense that we havethe necessary soundness results. Lemma 2.15 (Soundness) . IL ⊢ ϕ ⇒ ∀ F F | = ϕ Bewijs.
By induction on the length of an IL -proof of ϕ . The requirements on R and S in Definition 2.9 are precisely such that the axiom schemata hold.Note that all axiom schemata have their semantical counterpart except for theschema ( A ✄ C ) ∧ ( B ✄ C ) → A ∨ B ✄ C . Lemma 2.16 (Soundness) . Let C be the frame condition of the logic IL X . Wehave that IL X ⊢ ϕ ⇒ ∀ F ( F | = C ⇒ F | = ϕ ) . Bewijs.
As that of Lemma 2.15, plugging in the definition of the frame conditionat the right places. Note that we only need the direction F | = C ⇒ F | = X inthe proof. Corollary 2.17.
Let M be a model satisfying the IL X frame condition, and let m ∈ M . We have that Γ := { ϕ | M, m (cid:13) ϕ } is a maximal IL X -consistent set. It is clear what this notion should be. ewijs. Clearly ⊥ / ∈ Γ. Also A ∈ Γ or ¬ A ∈ Γ. By the soundness lemma,Lemma 2.16, we see that Γ is closed under IL X consequences. Lemma 2.18.
Let M be a model such that ∀ w ∈ M w (cid:13) IL X then IL X ⊢ ϕ ⇒ M | = ϕ .Bewijs. By induction on the derivation of ϕ .A modal logic IL X with frame condition C is called complete if we have theimplication the other way round too. That is, ∀ F ( F | = C ⇒ F | = ϕ ) ⇒ IL X ⊢ ϕ. A major concern of this paper is the question whether a given modal logic IL X is complete. Definition 2.19. Γ (cid:13) IL X ϕ iff ∀ M M | = IL X ⇒ ( ∀ m ∈ M [ M, m (cid:13) Γ ⇒ M, m (cid:13) ϕ ]) Lemma 2.20.
Let Γ be a finite set of formulas and let IL X be a complete logic.We have that Γ ⊢ IL X ϕ iff Γ (cid:13) IL X ϕ .Bewijs. Trivial. By the deduction theorem Γ ⊢ IL X ϕ ⇔⊢ IL X V Γ → ϕ . By ourassumption on completeness we get the result. Note that the requirement thatΓ be finite is necessary, as our modal logics are in general not compact (see alsoSection 3.1).Often we shall need to compare different frames or models. If F = h W, R, S i and F ′ = h W ′ , R ′ , S ′ i are frames, we say that F is a subframe of F ′ and write F ⊆ F ′ , if W ⊆ W ′ , R ⊆ R ′ and S ⊆ S ′ . Here S ⊆ S ′ is short for ∀ w ∈ W ( S w ⊆ S ′ w ). As with (almost) all interesting occurrences of modal logic, interpretability logicsare used to study a hard mathematical notion. Interpretability logics, as theirname slightly suggests, are used to study the notion of formal interpretability.In this subsection we shall very briefly say what this notion is and how modallogic is used to study it.We are interested in first order theories in the language of arithmetic. Alltheories we will consider will thus be arithmetical theories. Moreover, we wantour theories to have a certain minimal strength. That is, they should contain acertain core theory, say I ∆ + Ω from [13]. This will allow us to do reasonablecoding of syntax. We call these theories reasonable arithmetical theories.Once we can code syntax, we can write down a decidable predicate Proof T ( p, ϕ )that holds on the standard model precisely when p is a T -proof of ϕ . Weget a provability predicate by quantifying existentially, that is,
Prov T ( ϕ ) := ∃ p Proof T ( p, ϕ ). We take the liberty to not make a distinction between a syntactical object and its code.
9e can use these coding techniques to code the notion of formal interpreta-bility too. Roughly, a theory U interprets a theory V if there is some sort oftranslation so that every theorem of V is under that translation also a theoremof U . Definition 2.21.
Let U and V be reasonable arithmetical theories. An inter-pretation j from V in U is a pair h δ, F i . Here, δ is called a domain specifier. Itis a formula with one free variable. The F is a map that sends an n -ary relationsymbol of V to a formula of U with n free variables. (We treat functions andconstants as relations with additional properties.) The interpretation j inducesa translation from formulas ϕ of V to formulas ϕ j of U by replacing relationsymbols by their corresponding formulas and by relativizing quantifiers to δ .We have the following requirements. • ( R ( ~x )) j = F ( R )( ~x ) • The translation induced by j commutes with the boolean connectives.Thus, for example, ( ϕ ∨ ψ ) j = ϕ j ∨ ψ j . In particular ( ⊥ ) j = ( ∨ ∅ ) j = ∨ ∅ = ⊥• ( ∀ x ϕ ) j = ∀ x ( δ ( x ) → ϕ j ) • V ⊢ ϕ ⇒ U ⊢ ϕ j We say that V is interpretable in U if there exists an interpretation j of V in U . Using the Prov T ( ϕ ) predicate, it is possible to code the notion of formalinterpretability in arithmetical theories. This gives rise to a formula Int T ( ϕ, ψ ),to hold on the standard model precisely when T + ψ is interpretable in T + ϕ .This formula is related to the modal part by means of arithmetical realizations. Definition 2.22.
An arithmetical realization ∗ is a mapping that assigns toeach propositional variable an arithmetical sentence. This mapping is extendedto all modal formulas in the following way.- ( ϕ ∨ ψ ) ∗ = ϕ ∗ ∨ ψ ∗ and likewise for other boolean connectives. In particular ⊥ ∗ = ( ∨ ∅ ) ∗ = ∨ ∅ = ⊥ .- ( ✷ ϕ ) ∗ = Prov T ( ϕ ∗ )- ( ϕ ✄ ψ ) ∗ = Int T ( ϕ ∗ , ψ ∗ )From now on, the ∗ will always range over realizations. Often we will write ✷ T ϕ instead of Prov T ( ϕ ) or just even ✷ ϕ . The ✷ can thus denote both a modalsymbol and an arithmetical formula. For the ✄ -modality we adopt a similarconvention. We are confident that no confusion will arise from this. Definition 2.23.
An interpretability principle of a theory T is a modal formula ϕ that is provable in T under any realization. That is, ∀ ∗ T ⊢ ϕ ∗ . The inter-pretability logic of a theory T , we write IL (T), is the set of all interpretabilityprinciples. 10ikewise, we can talk of the set of all provability principles of a theory T ,denoted by PL (T). Since the famous result by Solovay, PL (T) is known for alarge class of theories T . Theorem 2.24 (Solovay [25]) . PL (T) = GL for any reasonable arithmeticaltheory T . For two classes of theories, IL (T) is known. Definition 2.25.
A theory T is reflexive if it proves the consistency of any ofits finite subtheories. It is essentially reflexive if any finite extension of it isreflexive. Theorem 2.26 (Berarducci [3], Shavrukov [24]) . If T is an essentially reflexivetheory, then IL (T) = IL M . Theorem 2.27 (Visser [29]) . If T is finitely axiomatizable, then IL (T) = IL P . Definition 2.28.
The interpretability logic of all reasonable arithmetical the-ories, we write IL (All), is the set of formulas ϕ such that ∀ T ∀ ∗ T ⊢ ϕ ∗ . Herethe T ranges over all the reasonable arithmetical theories.For sure IL (All) should be in the intersection of IL M and IL P . Up to now, IL (All) is unknown. In [19] it is conjectured to be IL P W ∗ . It is one of the majoropen problems in the field of interpretability logics, to characterize IL (All) in amodal way.We conclude this subsection with a definition of the arithmetical hierarchy.This definition is needed in Section ?? . Definition 2.29.
Inductively the following classes of arithmetical formulae aredefined. • Arithmetical formulas with only bounded quantifiers in it are called ∆ ,Σ or Π -formulas. • If ϕ is a Π n or Σ n +1 -formula, then ∃ x ϕ is a Σ n +1 -formula. • If ϕ is a Σ n or Π n +1 -formula, then ∀ x ϕ is a Π n +1 -formula. Definition 2.30.
Let ϕ be an arithmetical formula.- ϕ ∈ Π n ( T ) iff ∃ π ∈ Π n T ⊢ ϕ ↔ π - ϕ ∈ Σ n ( T ) iff ∃ σ ∈ Σ n T ⊢ ϕ ↔ σ - ϕ ∈ ∆ n ( T ) iff ∃ π ∈ Π n & ∃ σ ∈ Σ n T ⊢ ( ϕ ↔ π ) ∧ ( ϕ ↔ σ )Sometimes, if no confusion can arise, we will write Σ n !-formulas instead ofΣ n -formulas and Σ n -formulas instead of Σ n ( T )-formulas.11 General exposition of the construction me-thod
A central result in this paper is given by a construction method that shall beworked out in the next section. Most of the applications of this constructionmethod deal with modal completeness of a certain logic IL X . More precisely,showing that a logic IL X is modally complete amounts to constructing, or fin-ding, whenever IL X ϕ , a model M of IL X and an x ∈ M such that M, x (cid:13) ¬ ϕ .We will employ our construction method for this particular model construction.In this section, we shall lay out the basic ideas which are involved in theconstruction method. In particular, we will not always give precise definitionsof the notions we work with. All the definitions can be found in Section 4. As we mentioned above, a modal completeness proof of a logic IL X amounts to auniform model construction to obtain M, x (cid:13) ¬ ϕ for IL X ϕ . If IL X ϕ , then {¬ ϕ } is an IL X -consistent set and thus, by a version of Lindenbaum’s Lemma(Lemma 2.6), it is extendible to a maximal IL X -consistent set. On the otherhand, once we have an IL X -model M, x (cid:13) ¬ ϕ , we can find, by Corollary 2.17a maximal IL X -consistent set Γ with ¬ ϕ ∈ Γ. This Γ can simply be defined asthe set of all formulas that hold at x .To go from a maximal IL X -consistent set to a model is always the hardpart. This part is carried out in our construction method. In this method, themaximal consistent set is somehow partly unfolded to a model.Often in these sort of model constructions, the worlds in the model areMCS’s. For propositional variables one then defines x (cid:13) p iff. p ∈ x . In thesetting of interpretability logics it is sometimes inevitable to use the same MCSin different places in the model. Therefore we find it convenient not to identifya world x with a MCS, but rather label it with a MCS ν ( x ). However, we willstill write sometimes ϕ ∈ x instead of ϕ ∈ ν ( x ).One complication in unfolding a MCS to a model lies in the incompactnessof the modal logics we consider. This, in turn, is due to the fact that someframe conditions are not expressible in first order logic. As an example we canconsider the following set. Γ := { ✸ p } ∪ { ✷ ( p i → ✸ p i +1 ) | i ∈ ω } Clearly, Γ is a GL -consistent set, and any finite part of it is satisfiable in someworld in some model. However, it is not hard to see that in no IL -model all ofΓ can hold simultaneously in some world in it. As the truth definition of A ✄ B has a ∀∃ character, the corresponding notion of bisimu-lation is rather involved. As a consequence there is in general no obvious notion of a minimalbisimular model, contrary to the case of provability logics. This causes the necessity of severaloccurrences of MCS’s. This example comes from Fine and Rautenberg and is treated in Chapter 7 of [5]. M is an IL X -model and x ∈ M , then { ϕ | M, x (cid:13) ϕ } is a MCS. Bydefinition (and abuse of notation) we see that ∀ x [ x (cid:13) ϕ iff. ϕ ∈ x ] . We call this equivalence a truth lemma. (See for example Definition 4.5 fora more precise formulation.) In all completeness proofs a model is defined orconstructed in which some form of a truth lemma holds. Now, by the observedincompactness phenomenon, we can not expect that for every MCS, say Γ, wecan find a model “containing” Γ for which a truth lemma holds in full generality.There are various ways to circumvent this complication. Often one considerstruncated parts of maximal consistent sets which are finite. In choosing how totruncate, one is driven by two opposite forces.On the one hand this truncated part should be small. It should be at leastfinite so that the incompactness phenomenon is blocked. The finiteness is alsoa desideratum if one is interested in the decidability of a logic.On the other hand, the truncated part should be large. It should be largeenough to admit inductive reasoning to prove a truth lemma. For this, oftenclosure under subformulas and single negation suffices. Also, the truncatedpart should be large enough so that MCS’s contain enough information to dothe required calculation. For this, being closed under subformulas and singlenegations does not, in general, suffice. Examples of these sort of calculation areLemma ?? and Lemma 6.17.In our approach we take the best of both opposites. That is, we do nottruncate at all. Like this, calculation becomes uniform, smooth and relativelyeasy. However, we demand a truth lemma to hold only for finitely many formu-las.The question is now, how to unfold the MCS containing ¬ ϕ to a model where ¬ ϕ holds in some world. We would have such a model if a truth lemma holdsw.r.t. a finite set D containing ¬ ϕ .Proving that a truth lemma holds is usually done by induction on the com-plexity of formulas. As such, this is a typical “bottom up” or “inside out”activity. On the other hand, unfolding, or reading off, the truth value of aformula is a typical “top down” or “outside in” activity.Yet, we do want to gradually build up a model so that we get closer andcloser to a truth lemma. But, how could we possibly measure that we comecloser to a truth lemma? Either everything is in place and a truth lemma holds,or a truth lemma does not hold, in which case it seems unclear how to measureto what extend it does not hold.The gradually building up a model will take place by consecutively addingbits and pieces to the MCS we started out with. Thus somehow, we do want tomeasure that we come closer to a truth lemma by doing so. Therefore, we switchto an alternative forcing relation k∼ that follows the “outside in” direction thatis so characteristic to the evaluation of x (cid:13) ϕ , but at the same time incorporatesthe necessary elements of a truth lemma.13 k∼ p iff. p ∈ x for propositional variables px k∼ ϕ ∧ ψ iff. x k∼ ϕ & x k∼ ψ and likewise forother boolean connectives x k∼ ϕ ✄ ψ iff. ∀ y [ xRy ∧ ϕ ∈ x → ∃ z ( yS x z ∧ ψ ∈ z )]If D is a set of sentences that is closed under subformulas and single negations,then it is not hard to see that (see Lemma 4.9) ∀ x ∀ ϕ ∈D [ x k∼ ϕ iff. ϕ ∈ x ] ( ∗ )is equivalent to ∀ x ∀ ϕ ∈D [ x (cid:13) ϕ iff. ϕ ∈ x ] . ( ∗∗ )Thus, if we want to obtain a truth lemma for a finite set D that is closed undersingle negations and subformulas, we are done if we can obtain ( ∗ ). But now itis clear how we can at each step measure that we come closer to a truth lemma.This brings us to the definition of problems and deficiencies.A problem is some formula ¬ ( ϕ ✄ ψ ) ∈ x ∩ D such that x k6∼¬ ( ϕ ✄ ψ ). Wedefine a deficiency to be a configuration such that ϕ ✄ ψ ∈ x ∩D but x k6∼ ϕ ✄ ψ . Itnow becomes clear how we can successively eliminate problems and deficiencies.A deficiency ϕ ✄ ψ ∈ x ∩ D is a deficiency because there is some y (or maybemore of them) with xRy , and ϕ ∈ y , but for no z with yS x z , we have ψ ∈ z .This can simply be eliminated by adding a z with yS x z and ψ ∈ z .A problem ¬ ( ϕ ✄ ψ ) ∈ x ∩ D can be eliminated by adding a completelyisolated y to the model with xRy and ϕ, ¬ ψ ∈ y . As y is completely isolated, yS x z ⇒ z = y and thus indeed, it is not possible to reach a world where ψ holds. Now here is one complication.We want that a problem or a deficiency, once eliminated, can never re-occur.For deficiencies this complication is not so severe, as the quantifier complexityis ∀∃ . Thus, any time “a deficiency becomes active”, we can immediately dealwith it.With the elimination of a problem, things are more subtle. When we in-troduced y ∋ ϕ, ¬ ψ to eliminate a problem ¬ ( ϕ ✄ ψ ) ∈ x ∩ D , we did indeedeliminate it, as for no z with yS x z we have ψ ∈ z . However, this should hold forany future expansion of the model too. Thus, any time we eliminate a problem ¬ ( ϕ ✄ ψ ) ∈ x ∩ D , we introduce a world y with a promise that in no futuretime we will be able to go to a world z containing ψ via an S x -transition. So-mehow we should keep track of all these promises throughout the constructionand make sure that all the promises are indeed kept. This is taken care of byour so called ψ -critical cones (see for example also [6]). As ψ is certainly notallowed to hold in R -successors of y , it is reasonable to demand that ✷ ¬ ψ ∈ y .(Where y was introduced to eliminate the problem ¬ ( ϕ ✄ ψ ) ∈ x ∩ D .)Note that problems have quantifier complexity ∃∀ . We have chosen to callthem problems due to their prominent existential nature.14 .2 Some methods to obtain completeness For modal logics in general, quite an arsenal of methods to obtain completenessis available. For instance the standard operations on canonical models like path–coding (unraveling), filtrations and bulldozing (see [4]). Or one can mentionuniform methods like the use of Shalqvist formulas or the David Lewis theorem[5]. A very secure method is to construct counter models piece by piece. Anice example can be found in [5], Chapter 10. In [15] and in [14] a step-by-stepmethod is exposed in the setting of universal algebras. New approximations ofthe model are given by moves in an (infinite) game.For interpretability logics the available methods are rather limited in num-ber. In the case of the basic logic IL a relatively simple unraveling works.Although IL M does allow a same treatment, the proof is already much lessclear. (For both proofs, see [6]). However, for logics that contain IL M but not IL M it is completely unclear how to obtain completeness via an unraveling andwe are forced into more secure methods like the above mentioned building ofmodels piece by piece. And this is precisely what we do in this paper.Decidability and the finite model property are two related issues that more orless seem to divide the landscape of interpretability logics into the same classes.That is, the proof that IL has the finite model property is relatively easy. Thesame can be said about IL M . For logics like IL M the issue seems much moreinvolved and a proper proof of the finite model property, if one exists at all,has not been given yet. Alternatively, one could resort to other methods forshowing decidability like the Mosaic method [4]. In this section we describe our construction method in full detail. Sections 5-7are applications of the construction method. An IL X -labeled frame is just a Veltman frame in which every node is labeled by amaximal IL X -consistent set and some R -transitions are labeled by a formula. R -transitions labeled by a formula C indicate that some C -criticallity is essentiallypresent at this place. Definition 4.1. An IL X -labeled frame is a quadruple h W, R, S, ν i . Here h W, R, S i is an IL -frame and ν is a labeling function. The function ν assigns to each x ∈ W a maximal IL X -consistent set of sentences ν ( x ). To some pairs h x, y i with xRy , ν assigns a formula ν ( h x, y i ).If there is no chance of confusion we will just speak of labeled frames or evenjust of frames rather than IL X -labeled frames. Labeled frames inherit all theterminology and notation from normal frames. Note that an IL X -labeled frameneed not be, and shall in general not be, an IL X -frame. If we speak about a15abeled IL X -frame we always mean an IL X -labeled IL X -frame. To indicate that ν ( h x, y i ) = A we will sometimes write xR A y or ν ( x, y ) = A .Formally, given F = h W, R, S, ν i , one can see ν as a subset of ( W ∪ ( W × W )) × ( Form IL ∪{ Γ | Γ is a maximal IL X consistent set } ) such that the followingproperties hold.- ∀ x ∈ W ( h x, y i ∈ ν ⇒ y is a MCS)- ∀ h x, y i∈ W × W ( hh x, y i , z i ∈ ν ⇒ z is a formula)- ∀ x ∈ W ∃ y h x, y i ∈ ν - ∀ x, y, y ′ ( h x, y i ∈ ν ∧ h x, y ′ i ∈ ν → y = y ′ )We will often regard ν as a partial function on W ∪ ( W × W ) which is total on W and which has its values in Form IL ∪ { Γ | Γ is a maximal IL X consistent set } Remark 4.2.
Every IL X -labeled frame F = h W, R, S, ν i can be transformedto an IL -model F in a uniform way by defining for propositional variables p thevaluation as F , x (cid:13) p iff. p ∈ ν ( x ). By Corollary 2.17 we can also regard anymodel M satisfying the IL X frame condition as an IL X -labeled frame M bydefining ν ( m ) := { ϕ | M, m (cid:13) ϕ } .We sometimes refer to F as the model induced by the frame F . Alternativelywe will speak about the model corresponding to F . Note that for IL X -modelsM, we have M = M , but in general F = F for IL X -labeled frames F . Definition 4.3.
Let x be a world in some IL X -labeled frame h W, R, S, ν i . The C -critical cone above x , we write C Cx , is defined inductively as • ν ( h x, y i ) = C ⇒ y ∈ C Cx • x ′ ∈ C Cx & x ′ S x y ⇒ y ∈ C Cx • x ′ ∈ C Cx & x ′ Ry ⇒ y ∈ C Cx Definition 4.4.
Let x be a world in some IL X -labeled frame h W, R, S, ν i . The generalized C -cone above x , we write G Cx , is defined inductively as • y ∈ C Cx ⇒ y ∈ G Cx • x ′ ∈ G Cx & x ′ S w z ⇒ z ∈ G Cx for arbitrary w • x ′ ∈ G Cx & x ′ Ry ⇒ y ∈ G Cx It follows directly from the definition that the C -critical cone above x ispart of the generalized C -cone above x . So, if G Bx ∩ G Cx = ∅ , then certainly C Bx ∩ C Cx = ∅ .We also note that there is some redundancy in Definitions 4.3 and 4.4. Thelast clause in the inductive definitions demands closure of the cone under R -successors. But from Definition 2.9.5 closure of the cone under R follows from We could even say, any IL X -model. S x . We have chosen to explicitly adopt the closureunder the R . In doing so, we obtain a notion that serves us also in the envi-ronment of so-called quasi frames (see Definition 5.1) in which not necessarily( x ↾ ) ∩ R ⊆ S x . Definition 4.5.
Let F = h W, R, S, ν i be a labeled frame and let F be theinduced IL -model. Furthermore, let D be some set of sentences. We say that atruth lemma holds in F with respect to D if ∀ A ∈D ∀ x ∈ FF , x (cid:13) A ⇔ A ∈ ν ( x ) . If there is no chance of confusion we will omit some parameters and justsay “a truth lemma holds at F ” or even “a truth lemma holds”. The followingdefinitions give us a means to measure how far we are away from a truth lemma. Definition 4.6 (Temporary definition) . Let D be some set of sentences and let F = h W, R, S, ν i be an IL X -labeled frame. A D -problem is a pair h x, ¬ ( A ✄ B ) i such that ¬ ( A ✄ B ) ∈ ν ( x ) ∩ D and for every y with xRy we have [ A ∈ ν ( y ) ⇒∃ z ( yS x z ∧ B ∈ ν ( z ))]. Definition 4.7 (Deficiencies) . Let D be some set of sentences and let F = h W, R, S, ν i be an IL X -labeled frame. A D -deficiency is a triple h x, y, C ✄ D i with xRy , C ✄ D ∈ ν ( x ) ∩ D , and C ∈ ν ( y ), but for no z with yS x z we have D ∈ ν ( z ).If the set D is clear or fixed, we will just speak about problems and deficien-cies. Definition 4.8.
Let A be a formula. We define the single negation of A , wewrite ∼ A , as follows. If A is of the form ¬ B we define ∼ A to be B . If A is nota negated formula we set ∼ A := ¬ A .The next lemma shows that a truth lemma w.r.t. D can be reformulated inthe combinatoric terms of deficiencies and problems. (See also the equivalenceof ( ∗ ) and ( ∗∗ ) in Section 3.) Lemma 4.9.
Let F = h W, R, S, ν i be a labeled frame, and let D be a set ofsentences closed under single negation and subformulas. A truth lemma holdsin F w.r.t. D iff. there are no D -problems nor D -deficiencies.Bewijs. The proof is really very simple and precisely shows they interplay bet-ween all the ingredients.The labeled frames we will construct are always supposed to satisfy someminimal reasonable requirements. We summarize these in the notion of ade-quacy.
Definition 4.10 (Adequate frames) . A frame is called adequate if the followingconditions are satisfied. We will eventually work with Definition 4.11. xRy ⇒ ν ( x ) ≺ ν ( y )2. A = B ⇒ G Ax ∩ G Bx = ∅ y ∈ C Ax ⇒ ν ( x ) ≺ A ν ( y )If no confusion is possible we will just speak of frames instead of adequatelabeled frames. As a matter of fact, all the labeled frames we will see from nowon will be adequate. In the light of adequacy it seems reasonable to work witha slightly more elegant definition of a D -problem. Definition 4.11 (Problems) . Let D be some set of sentences. A D -problem isa pair h x, ¬ ( A ✄ B ) i such that ¬ ( A ✄ B ) ∈ ν ( x ) ∩ D and for no y ∈ C Bx we have A ∈ ν ( y ).From now on, this will be our working definition. Clearly, on adequatelabeled frames, if h x, ¬ ( A ✄ B ) i is not a problem in the new sense, it is not aproblem in the old sense. Remark 4.12.
It is also easy to see that the we still have the interesting halfof Lemma 4.9. Thus, we still have, that a truth lemma holds if there are nodeficiencies nor problems.To get a truth lemma we have to somehow get rid of problems and deficien-cies. This will be done by adding bits and pieces to the original labeled frame.Thus the notion of an extension comes into play.
Definition 4.13 (Extension) . Let F = h W, R, S, ν i be a labeled frame. We saythat F ′ = h W ′ , R ′ , S ′ , ν ′ i is an extension of F , we write F ⊆ F ′ , if W ⊆ W ′ and the relations in F ′ restricted to F yield the corresponding relations in F .More formally, the requirements on the restrictions in the above definitionamount to saying that for x, y, z ∈ F we have the following.- xR ′ y iff. xRy - yS ′ x z iff. yS x z - ν ′ ( x ) = ν ( x )- ν ′ ( h x, y i ) is defined iff. ν ( h x, y i ) is defined, and in this case ν ′ ( h x, y i ) = ν ( h x, y i ).A problem in F is said to be eliminated by the extension F ′ if it is no longera problem in F ′ . Likewise we can speak about elimination of deficiencies. Definition 4.14 (Depth) . The depth of a finite frame F , we will write depth ( F )is the maximal length of sequences of the form x R . . . Rx n . (For conveniencewe define max( ∅ ) = 0.) 18 efinition 4.15 (Union of Bounded Chains) . An indexed set { F i } i ∈ ω of labeledframes is called a chain if for all i , F i ⊆ F i +1 . It is called a bounded chain iffor some number n , depth ( F i ) ≤ n for all i ∈ ω . The union of a bounded chain { F i } i ∈ ω of labeled frames F i is defined as follows. ∪ i ∈ ω F i := h∪ i ∈ ω W i , ∪ i ∈ ω R i , ∪ i ∈ ω S i , ∪ i ∈ ω ν i i It is clear why we really need the boundedness condition. We want the unionto be an IL -frame. So, certainly R should be conversely well-founded. This canonly be the case if our chain is bounded. We now come to the main motor behind many results. It is formulated in rathergeneral terms so that it has a wide range of applicability. As a draw-back, weget that any application still requires quite some work.
Lemma 4.16 (Main Lemma) . Let IL X be an interpretability logic and let C bea (first or higher order) frame condition such that for any IL -frame F we have F | = C ⇒ F | = X . Let D be a finite set of sentences. Let I be a set of so-called invariants of labeledframes so that we have the following properties. • F | = I U ⇒ F | = C , where I U is that part of I that is closed under boundedunions of labeled frames. • I contains the following invariant: xRy → ∃ A ∈ ( ν ( y ) \ ν ( x )) ∩ { ✷ ¬ D | D a subformula of some B ∈ D} . • For any adequate labeled frame F , satisfying all the invariants, we havethe following. – Any D -problem of F can be eliminated by extending F in a way thatconserves all invariants. – Any D -deficiency of F can be eliminated by extending F in a waythat conserves all invariants.In case such a set of invariants I exists, we have that any IL X -labeled ade-quate frame F satisfying all the invariants can be extended to some labeled ade-quate IL X -frame ˆ F on which a truth-lemma with respect to D holds.Moreover, if for any finite D that is closed under subformulas and singlenegations, a corresponding set of invariants I can be found as above and suchthat moreover I holds on any one-point labeled frame, we have that IL X is acomplete logic. ewijs. By subsequently eliminating problems and deficiencies by means of ex-tensions. These elimination processes have to be robust in the sense that everyproblem or deficiency that has been dealt with, should not possibly re-emerge.But, the requirements of the lemma almost immediately imply this.For the second part of the Main Lemma, we suppose that for any finite set D closed under subformulas and single negations, we can find a correspondingset of invariants I . If now, for any such D , all the corresponding invariants I hold on any one-point labeled frame, we are to see that IL X is a complete logic,that is, IL X A ⇒ ∃ M ( M | = X & M | = ¬ A ).But this just follows from the above. If IL X A , we can find a maximal IL X -consistent set Γ with ¬ A ∈ Γ. Let D be the smallest set that contains ¬ A andis closed under subformulas and single negations and consider the invariantscorresponding to D . The labeled frame F := h{ x } , ∅ , ∅ , h x, Γ ii can thus beextended to a labeled adequate IL X -frame ˆ F on which a truth lemma withrespect to D holds. Thus certainly ˆ F , x (cid:13) ¬ A , that is, A is not valid on themodel induced by ˆ F .The construction method can also be used to obtain decidability via the finitemodel property. In such a case, one should re-use worlds that were introducedearlier in the construction.The following two lemmata indicate how good labels can be found for theelimination of problems and deficiencies. Lemma 4.17.
Let Γ be a maximal IL X -consistent set such that ¬ ( A ✄ B ) ∈ Γ .Then there exists a maximal IL X -consistent set ∆ such that Γ ≺ B ∆ ∋ A, ✷ ¬ A .Bewijs. So, consider ¬ ( A ✄ B ) ∈ Γ, and suppose that no required ∆ exists. Wecan then find a formula C for which C ✄ B ∈ Γ such that ¬ C, ✷ ¬ C, A, ✷ ¬ A ⊢ IL X ⊥ . Consequently ⊢ IL X A ∧ ✷ ¬ A → C ∨ ✸ C and thus, by Lemma 2.2, also ⊢ IL X A ✄ C . But as C ✄ B ∈ Γ, also A ✄ B ∈ Γ.This clearly contradicts the consistency of Γ.For deficiencies there is a similar lemma.
Lemma 4.18.
Consider C ✄ D ∈ Γ ≺ B ∆ ∋ C . There exists ∆ ′ with Γ ≺ B ∆ ′ ∋ D, ✷ ¬ D .Bewijs. Suppose for a contradiction that C ✄ D ∈ Γ ≺ B ∆ ∋ C and there doesnot exist a ∆ ′ with Γ ≺ B ∆ ′ ∋ D, ✷ ¬ D . Taking the contraposition of Lemma4.17 we get that ¬ ( D ✄ B ) / ∈ Γ, whence D ✄ B ∈ Γ and also C ✄ B ∈ Γ. Thisclearly contradicts the consistency of ∆ as Γ ≺ B ∆ ∋ C . Writing out the definition and by compactness, we get a finite number of formulas C , . . . , C n with C i ✄ B ∈ Γ, such that ¬ C , . . . , ¬ C n , ✷ ¬ C , . . . , ✷ ¬ C n , A, ✷ ¬ A ⊢ IL X ⊥ .We can now take C := C ∨ . . . ∨ C n . Clearly, as all the C i ✄ B ∈ Γ, also C ✄ B ∈ Γ. .3 Completeness and the main lemma The main lemma provides a powerful method for proving modal completeness.In several cases it is actually the only known method available.
Remark 4.19.
A modal completeness proof for an interpretability logic IL X is by the main lemma reduced to the following four ingredients. • Frame Condition
Providing a frame condition C and a proof that F | = C ⇒ F | = IL X . • Invariants
Given a finite set of sentences D (closed under subformulasand single negations), providing invariants I that hold for any one-pointlabeled frame. Certainly I should contain xRy → ∃ A ∈ ( ν ( y ) \ ν ( x )) ∩{ ✷ D | D ∈ D} . • elimination– Problems Providing a procedure of elimination by extension for pro-blems in labeled frames that satisfy all the invariants. This procedureshould come with a proof that it preserves all the invariants. – Deficiencies
Providing a procedure of elimination by extension fordeficiencies in labeled frames that satisfy all the invariants. Alsothis procedure should come with a proof that it preserves all theinvariants. • Rounding up
A proof that for any bounded chain of labeled framesthat satisfy the invariants, automatically, the union satisfies the framecondition C of the logic.The completeness proofs that we will present will all have the same struc-ture, also in their preparations. As we will see, eliminating problems is moreelementary than eliminating deficiencies.As we already pointed out, we eliminate a problem by adding some newworld plus an adequate label to the model we had. Like this, we get a structurethat need not even be an IL -model. For example, in general, the R relation isnot transitive. To come back to at least an IL -model, we should close off thenew structure under transitivity of R and S et cetera. This closing off is inits self an easy and elementary process but we do want that the invariants arepreserved under this process. Therefore we should have started already with astructure that admitted a closure. Actually in this paper we will always wantto obtain a model that satisfies the frame condition of the logic.The preparations to a completeness proof in this paper thus have the follo-wing structure. • Determining a frame condition for IL X and a corresponding notion of an IL X -frame. 21 Defining a notion of a quasi IL X -frame. • Defining some notions that remain constant throughout the closing ofquasi IL X -frames, but somehow capture the dynamic features of this pro-cess. • Proving that a quasi IL X -frame can be closed off to an adequate labeled IL X -frame. • Preparing the elimination of deficiencies.The most difficult job in a the completeness proofs we present in this paper,was in finding correct invariants and in preparing the elimination of deficien-cies. Once this is fixed, the rest follows in a rather mechanical way. Especiallythe closure of quasi IL X -frames to adequate IL X -frames is a very laboriousenterprise. The modal logic IL has been proved to be modally complete in [8]. We shallreprove the completeness here using the main lemma.The completeness proof of IL can be seen as the mother of all our com-pleteness proofs in interpretability logics. Not only does it reflect the generalstructure of applications of the Main Lemma clearly, also it so that we can uselarge parts of the preparations to the completeness proof of IL in other proofstoo. Especially closability proofs are cumulative. Thus, we can use the lemmathat any quasi-frame is closable to an adequate frame, in any other completenessproof. Definition 5.1. A quasi-frame G is a quadruple h W, R, S, ν i . Here W is anon-empty set of worlds, and R a binary relation on W . S is a set of binaryrelations on W indexed by elements of W . The ν is a labeling as defined onlabeled frames. Critical cones and generalized cones are defined just in the sameway as in the case of labeled frames. G should posess the following properties.1. R is conversely well-founded2. yS x z → xRy & xRz xRy → ν ( x ) ≺ ν ( y )4. A = B → G Ax ∩ G Bx = ∅ y ∈C Ax → ν ( x ) ≺ A ν ( y )Clearly, adequate labeled frames are special cases of quasi frames. Quasi-frames inherit all the notations from labeled frames. In particular we can thusspeak of chains and the like. 22 emma 5.2 ( IL -closure) . Let G = h W, R, S, ν i be a quasi-frame. There is anadequate IL -frame F extending G . That is, F = h W, R ′ , S ′ , ν i with R ⊆ R ′ and S ⊆ S ′ .Bewijs. We define an imperfection on a quasi-frame F n to be a tuple γ havingone of the following forms.( i ) γ = h , a, b, c i with F n | = aRbRc but F n = aRc ( ii ) γ = h , a, b i with F n | = aRb but F n = bS a b ( iii ) γ = h , a, b, c, d i with F n | = bS a cS a d but not F n | = bS a d ( iv ) γ = h , a, b, c i with F n | = aRbRc but F n = bS a c Now let us start with a quasi-frame G = h W, R, S, ν i . We will define a chain ofquasi-frames. Every new element in the chain will have at least one imperfectionless than its predecessor. The union will have no imperfections at all. It will beour required adequate IL -frame.Let < be the well-ordering on C := ( { } × W ) ∪ ( { } × W ) ∪ ( { } × W ) ∪ ( { } × W )induced by the occurrence order in some fixed enumeration of C . We define ourchain to start with F := G . To go from F n to F n +1 we proceed as follows. Let γ be the < -minimal imperfection on F n . In case no such γ exists we set F n +1 := F n . Ifsuch a γ does exist, F n +1 is as dicted by the case distinctions.( i ) F n +1 := h W n , R n ∪ {h a, c i} , S n , ν n i ( ii ) F n +1 := h W n , R n , S n ∪ {h a, b, b i} , ν n i ( iii ) F n +1 := h W n , R n , S n ∪ {h a, b, d i} , ν n i ( iv ) F n +1 := h W n , R n ∪ {h a, c i} , S n ∪ {h a, b, c i} , ν n i By an easy but elaborate induction, we can see that each F n is a quasi-frame.The induction boils down to checking for each case ( i )-( iv ) that all the properties(1)-(5) from Definition 5.1 remain valid.Instead of proving (4) and (5), it is better to prove something stronger, thatis, that the critical and generalized cones remain unchanged.4’. ∀ n [ F n +1 | = y ∈G Ax ⇔ F n | = y ∈G Ax ]5’. ∀ n [ F n +1 | = y ∈C Ax ⇔ F n | = y ∈C Ax ]23ext, it is not hard to prove that F := ∪ i ∈ ω F i is the required adequate IL -frame.To this extent, the following properties have to be checked. All properties haveeasy proofs.( a. ) W is the domain of F ( g. ) F | = xRy → yS x y ( b. ) R ⊆ ∪ i ∈ ω R i ( h. ) F | = xRyRz → yS x z ( c. ) S ⊆ ∪ i ∈ ω S i ( i. ) F | = uS x vS x w → uS x w ( d. ) R is conv. wellfounded on F ( j. ) F | = xRy ⇒ ν ( x ) ≺ ν ( y )( e. ) F | = xRyRz → xRz ( k. ) A = B ⇒ F | = G Ax ∩ G Bx = ∅ ( f. ) F | = yS x z → xRy & xRz ( l. ) F | = y ∈C Ax ⇒ ν ( x ) ≺ A ν ( y )We note that the IL -frame F ⊇ G from above is actually the minimal oneextending G . If in the sequel, if we refer to the closure given by the lemma, weshall mean this minimal one. Also do we note that the proof is independent onthe enumeration of C and hence the order < on C . The lemma can also beapplied to non-labeled structures. If we drop all the requirements on the labelsin Definition 5.1 and in Lemma 5.2 we end up with a true statement about justthe old IL -frames.Lemma 5.2 also allows a very short proof running as follows. Any intersectionof adequate IL -frames with the same domain is again an adequate IL -frame.There is an adequate IL -frame extending G . Thus by taking intersections wefind a minimal one. We have chosen to present our explicit proof as they allowus, now and in the sequel, to see which properties remain invariant. Corollary 5.3.
Let D be a finite set of sentences, closed under subformulasand single negations. Let G = h W, R, S, ν i be a quasi-frame on which xRy → ∃ A ∈ (( ν ( y ) \ νx ) ∩ { ✷ D | D ∈ D} ) ( ∗ ) holds. Property ( ∗ ) does also hold on the IL -closure F of G .Bewijs. We can just take the property along in the proof of Lemma 5.2. In Case( i ) and ( iv ) we note that aRbRc → ν ( a ) ⊆ ✷ ν ( c ). Thus, if A ∈ (( ν ( c ) \ ν ( b )) ∩{ ✷ D | D ∈ D} ), then certainly A ν ( a ).We have now done all the preparations for the completeness proof. Normally,also a lemma is needed to deal with deficiencies. But in the case of IL , Lemma4.18 suffices. Theorem 5.4. IL is a complete logicBewijs.
We specify the four ingredients mentioned in Remark 4.19.
Frame Condition
For IL , the frame condition is empty, that is, everyframe is an IL frame. 24 nvariants Given a finite set of sentences D closed under subformulas andsingle negation, the only invariant is xRy → ∃ A ∈ ( ν ( y ) \ ν ( x )) ∩ { ✷ D | D ∈ D} .Clearly this invariant holds on any one-point labeled frame. Elimination
So, let F := h W, R, S, ν i be a labeled frame satisfying theinvariant. We will see how to eliminate both problems and deficiencies whileconserving the invariant. Problems
Any problem h a, ¬ ( A ✄ B ) i of F will be eliminated in two steps.1. With Lemma 4.17 we find ∆ with ν ( a ) ≺ B ∆ ∋ A, ✷ ¬ A . We fix some b / ∈ W . We now define G ′ := h W ∪ { b } , R ∪ {h a, b i} , S, ν ∪ {h b, ∆ i , hh a, b i , B i}i . It is easy to see that G ′ is actually a quasi-frame. Note that if G ′ | = xRb ,then x must be a and whence ν ( x ) ≺ ν ( b ) by definition of ν ( b ). Also itis not hard to see that if b ∈ C Cx for x = a , that then ν ( x ) ≺ C ν ( b ). For, b ∈ C Cx implies a ∈ C Cx whence ν ( x ) ≺ C ν ( a ). By ν ( a ) ≺ ν ( b ) we get that ν ( x ) ≺ C ν ( b ). In case x = a we see that by definition b ∈ C Ba . But, we havechosen ∆ so that ν ( a ) ≺ B ν ( b ). We also see that G ′ satisfies the invariantas ✷ ¬ A ∈ ν ( b ) \ ν ( a ) and ∼ A ∈ D .2. With Lemma 5.2 we extend G ′ to an adequate labeled IL -frame G . Corol-lary 5.3 tells us that the invariant indeed holds at G . Clearly h a, ¬ ( A ✄ B ) i is no longer a problem in G . Deficiencies . Again, any deficiency h a, b, C ✄ D i in F will be eliminated intwo steps.1. We first define B to be the formula such that b ∈ C Ba . If such a B doesnot exist, we take B to be ⊥ . Note that if such a B does exist, it must beunique by Property 4 of Definition 5.1. By Lemma 4.18 we can now finda ∆ ′ such that ν ( a ) ≺ B ∆ ′ ∋ D, ✷ ¬ D . We fix some c W and define G ′ := h W, R ∪ { a, c } , S ∪ { a, b, c } , ν ∪ { c, ∆ ′ }i . Again it is not hard to see that G ′ is a quasi-frame that satisfies theinvariant. Clearly R is conversely well-founded. The only new S in G ′ is bS a c , but we also defined aRc . We have chosen ∆ ′ such that ν ( a ) ≺ B ν ( c ).Clearly ✷ ¬ D ν ( a ).2. Again, G ′ is closed off under the frame conditions with Lemma 5.2. Againwe note that the invariant is preserved in this process. Clearly h a, b, C ✄ D i is not a deficiency in G . Rounding up
Clearly the union of a bounded chain of IL -frames is againan IL -frame. 25t is well known that IL has the finite model property and whence is deci-dable. With some more effort however we could have obtained the finite modelproperty using the Main Lemma. We have chosen not to do so, as for ourpurposes the completeness via the construction method is sufficient.Also, to obtain the finite model property, one has to re-use worlds duringthe construction method. The constraints on which worlds can be re-used isper logic differently. One aim of this section was to prove some results on aconstruction that is present in all other completeness proofs too. Therefore weneeded some uniformity and did not want to consider re-using of worlds. M This section is devoted to showing the following theorem.
Theorem 6.1. IL M is a complete logic. It turns out that the modal frame condition of IL M gives rise to a bewil-dering structure of possible models that seems very hard to tame. As M is in IL (All), it is important that the class of IL M -frames is well understood. Fora long time IL W ∗ has been conjectured ([30]) to be IL (All). A first step inproving this conjecture would have been a modal completeness proof of IL W ∗ .It is well known that IL W ∗ is the union of IL W and IL M , see Lemma 7.3.The modal completeness of IL W was proved in [9]. So, the missing link was amodal completeness proof for IL M . In [18] a proof sketch of this completenessresult was given. In this paper we give for the first time a fully detailed proof.In the light of Remark 4.19 a proof of Theorem 6.1 boils down to givingthe four ingredients mentioned there. Sections 6.3, 6.4, 6.5, 6.6 and 6.7 belowcontain those ingredients. Before these main sections, we have in Section 6.2some preliminaries. We start in Section 6.1 with an overview of the difficultieswe encounter during the application of the construction method to IL M . In the construction method we repeatedly eliminate problems and deficienciesby extensions that satisfy all the invariants. During these operations we needto keep track of two things.1. If x has been added to solve a problem in w , say ¬ ( A ✄ B ) ∈ ν ( w ). Thenfor all y such that xS w y we have ν ( w ) ≺ B ν ( y ).2. If wRx then ν ( w ) ≺ ν ( x )Item 1. does not impose any direct difficulties. But some do emerge whenwe try to deal with the difficulties concerning Item 2. So let us see why it isdifficult to ensure 2. Suppose we have wRxRyS w y ′ Rz . The M –frame condition(Theorem 6.20) requires that we also have xRz . So, from 2. and the M –frame26 wy y x x C ✄ DCC ✄ DC xx
Figuur 1: A deficiency in w w.r.t. y condition we obtain wRxRyS w y ′ Rz → ν ( x ) ≺ ν ( z ). A sufficient (and in certainsense necessary) condition is, wRxRyS w y ′ → ν ( x ) ⊆ ✷ ν ( y ′ ) . Let us illustrate some difficulties concerning this condition by some examples.Consider the left model in Figure 1. That is, we have a deficiency in w w.r.t. y . Namely, C ✄ D ∈ ν ( w ) and C ∈ ν ( y ). If we solve this deficiency by adding aworld y ′ , we thus require that for all x such that wRxRy we have ν ( x ) ⊆ ✷ ν ( y ′ ).This difficulty is partially handled by Lemma 6.2 below. We omit a proof,but it can easily be given by replacing in the corresponding lemma for IL M ,applications of the M -axiom by applications of the M -axiom. Lemma 6.2.
Let Γ , ∆ be MCS ’s such that C ✄ D ∈ Γ , Γ ≺ A ∆ and ✸ C ∈ ∆ .Then there exists some ∆ ′ with Γ ≺ A ∆ ′ , ✷ ¬ D, D ∈ ∆ ′ and ∆ ⊆ ✷ ∆ ′ . Let us now consider the right most model in Figure 1. We have at least fortwo different worlds x , say x and x , that wRxRy . Lemma 6.2 is applicableto ν ( x ) and ν ( x ) separately but not simultaneously. In other words we find y ′ and y ′ such that ν ( x ) ⊆ ✷ ν ( y ′ ) and ν ( x ) ⊆ ✷ ν ( y ′ ). But we actually wantone single y ′ such that ν ( x ) ⊆ ✷ ν ( y ′ ) and ν ( x ) ⊆ ✷ ν ( y ′ ). We shall handlethis difficulty by ensuring that it is enough to consider only one of the worlds inbetween w and y . To be precise, we shall ensure ν ( x ′ ) ⊆ ✷ ν ( x ) or ν ( x ) ⊆ ✷ ν ( x ′ ).But now some difficulties concerning Item 1. occur. In the situations inFigure 1 we were asked to solve a deficiency in w w.r.t. y . As usual, if w ≺ A y then we should be ably to choose a solution y ′ such that w ≺ A y ′ . But Lemma6.2 takes only criticallity of x w.r.t. w into account. This issue is solved byensuring that wRxRy ∈ C Aw implies ν ( w ) ≺ A ν ( x ).We are not there yet. Consider the leftmost model in Figure 2. That is, wehave a deficiency in w w.r.t. y ′ . Namely, C ✄ D ∈ ν ( w ) and C ∈ ν ( y ′ ). If we adda world y ′′ to solve this deficiency, as in the middle model, then by transitivity27 ′′ y ′′ S w S w S w S w S w S w wy xC ✄ Dy ′ C ¬ ✸ C wy xC ✄ Dy ′ C ¬ ✸ C wy xC ✄ Dy ′ C ¬ ✸ CD D
Figuur 2: A deficiency in w w.r.t. y ′ of S w we have yS w y ′′ , as shown in the rightmost model. So, we require that ν ( x ) ⊆ ✷ ν ( y ′′ ). But we might very well have ✸ C ν ( x ). So the Lemma 6.2 isnot applicable.In Lemma 6.17 we formulate and prove a more complicated version of theLemma 6.2 which basically says that if we have chosen ν ( y ′ ) appropriately, thenwe can choose ν ( y ′′ ) such that ν ( x ) ⊆ ✷ ν ( y ′′ ). And moreover, Lemma 6.17ensures us that we can, indeed, choose ν ( y ′ ) appropriate. Definition 6.3 ( T t r , T ∗ , T ; T ′ , T , T ≥ , T ∪ T ′ ) . Let T and T ′ be binaryrelations on a set W . We fix the following fairly standard notations. T t r isthe transitive closure of T ; T ∗ is the transitive reflexive closure of T ; xT ; T ′ y ⇔∃ t xT tT ′ y ; xT y ⇔ xT y ∧¬∃ t xT tT y ; xT ≥ y ⇔ xT y ∧¬ ( xT y ) and xT ∪ T ′ y ⇔ xT y ∨ xT ′ y . Definition 6.4 ( S w ) . Let F = h W, R, S, ν i be a quasi–frame. For each w ∈ W we define the relation S w , of pure S w transitions, as follows. x S w y ⇔ xS w y ∧ ¬ ( x = y ) ∧ ¬ ( x ( S w ∪ R ) ∗ ; R ; ( S w ∪ R ) ∗ y ) Definition 6.5 (Adequate IL M –frame) . Let F = h W, R, S, ν i be an adequateframe. We say that F is an adequate IL M –frame iff. the following additionalproperties hold. wRxRyS w y ′ Rz → xRz wRxRyS w y ′ → ν ( x ) ⊆ ✷ ν ( y ′ ) One might think that 6. is superfluous. In finite frame this is indeed the case, but in thegeneral case we need it as an requirement. xS w y → x ( S w ∪ R ) ∗ y xRy → x ( R ) t r y As usual, when we speak of IL M –frames we shall actually mean an adequate IL M –frame. Below we will construct IL M –frames out of frames belonging to acertain subclass of the class of quasi–frames. (Namely the quasi– IL M –frames,see Definition 6.10 below.) We would like to predict on forehand which extra R relations will be added during this construction. The following definition doesjust that. Definition 6.6 ( K ( F ), K ) . Let F = h W, R, S, ν i be a quasi–frame. We define K = K ( F ) to be the smallest binary relation on W such that1. R ⊆ K ,2. K = K tr ,3. wKxK y ( S w ) t r y ′ K z → xKz .Note that for IL M –frames we have K = R . The following lemma showsthat K satisfies some stability conditions. The lemma will mainly be used toshow that whenever we extend R within K , then K does not change. Lemma 6.7.
Let F = h W, R , S, ν i and F = h W, R , S, ν i be quasi–frames. If R ⊆ K ( F ) and R ⊆ K ( F ) . Then K ( F ) = K ( F ) . In a great deal of situations we have a particular interest in K . To determinesome of its properties the following lemma comes in handy. It basically showsthat we can compute K by first closing of under the M –condition and thentake the transitive closure. Lemma 6.8 (Calculation of K ) . Let F = h W, R, S, ν i be a quasi–frame. Let K = K ( F ) and suppose K conversely well–founded. Let T be a binary relationon W such that1. R ⊆ T t r ⊆ K ,2. wT t r xT y ( S w ) t r y ′ T z → xT t r z .Then we have the following.(a) K = T t r (b) xK y → xT y Bewijs.
To see (a), it is enough to see that T t r satisfies the three properties ofthe definition of K (Definition 6.6). Item (b) follows from (a).Another entity that changes during the construction of an IL M –frame outof a quasi–frame is the critical cone In accordance with the above definition of K ( F ), we also like to predict what eventually becomes the critical cone.29 efinition 6.9 ( N Cw ) . For any quasi–frame F we define N Cw to be the smallestset such that1. ν ( w, x ) = C ⇒ x ∈ N Cw ,2. x ∈ N Cw ∧ x ( K ∪ S w ) y ⇒ y ∈ N Cw .In accordance with the notion of a quasi–frame we introduce the notionof a quasi– IL M –frame. This gives sufficient conditions for a quasi–frame tobe closeable, not only under the IL –frameconditions, but under all the IL M –frameconditions. Definition 6.10 (Quasi– IL M –frame) . A quasi– IL M –frame is a quasi–framethat satisfies the following additional properties.6. K is conversely well–founded.7. xKy → ν ( x ) ≺ ν ( y )8. x ∈ N Aw → ν ( w ) ≺ A ν ( x )9. wKxKy ( S w ∪ K ) ∗ y ′ → ν ( x ) ⊆ ✷ ν ( y ′ )10. xS w y → x ( S w ∪ R ) ∗ y wKxK y ( S w ) t r y ′ K z → x ( K ) t r z xRy → x ( R ) t r y Lemma 6.11. If F is a quasi– IL M –frame, then K = ( K ) t r .Bewijs. Using Lemma 6.8.
Lemma 6.12.
Suppose that F is a quasi– IL M –frame. Let K = K ( F ) . Let K ′ , K ′′ and K ′′′ the smallest binary relations on W satifying 1. and 2. of 6.6and additionaly we have the following.3 ′ . wK ′ xK ′ y ( S w ∪ K ′ ) ∗ y ′ K ′ z → xK ′ z ′′ . wK ′′ xK ′′ y ( S w ) t r y ′ K ′′ z → xK ′′ z ′′′ . wK ′′′ xK ′′′ y ( S w ∪ K ′′′ ) ∗ y ′ K ′′′ z → xK ′′′ z Then K = K ′ = K ′′ = K ′′′ .Bewijs. Using Lemma 6.11.Before we move on, let us first sum up a few comments.
Corollary 6.13. If F = h W, R, S, ν i is an adequate IL M –frame. Then wehave the following.1. K ( F ) = R . F | = x ∈ N Aw ⇔ F | = x ∈ C Aw F is a quasi– IL M –frame Lemma 6.14 ( IL M –closure) . Any quasi– IL M –frame can be extended to anadequate IL M –frame.Bewijs. Given a quasi– IL M –frame F we construct a sequence F = F ⊆ F ⊆ · · · very similar to the sequence constructed for the IL closure of a quasi–frame(Lemma 5.2). The only difference is that we add a fifth entry to the list ofimperfections.(v) γ = h , w, a, b, b ′ , c i with F n | = wRaRbS w b ′ Rc but F n = aRc In this case we set, of course, F n +1 := h W n , R n ∪ h a, c i , S n , ν n i . First we willshow by induction that each F n is a quasi– IL M –frame. Then we show that theunion ˆ F = S n ≥ F n , is quasi and satisfies all the IL M frame conditions.We assume that F n is a quasi- IL M -frame and define K n := K ( F n ), R n := R F n and S n := S F n . Quasi-ness of F n +1 will follow from Claim 1, and fromClaim 2 we may conlude that F n +1 is indeed a quasi- IL M -frame. Claim 1.
For all w, x, y and A we have the following.(a) R n +1 ⊆ K n (b) x ( S n +1 w ∪ R n +1 ) ∗ y ⇒ x ( S nw ∪ K n ) ∗ y (c) F n +1 | = x ∈ C Aw ⇒ F n | = x ∈ N Aw . Bewijs.
We distinguish cases according to which imperfection is dealt with inthe step from F n to F n +1 . The only interesting case is the new imperfectionwhich is dealt with by Lemma 6.12, Item 3 ′′ . Claim 2.
For all w, x and A we have the following.1. K n +1 ⊆ K n .2. x ( S n +1 w ∪ K n +1 ) ∗ y ⇒ x ( S nw ∪ K n ) ∗ y F n +1 | = x ∈ N Aw ⇒ F n | = x ∈ N Aw . Bewijs.
Item 1. follows by Claim 1 and Lemma 6.7. Item 2. follows from Item1. and Claim 1-(b). Item 3. is an immediate corollary of item 2.Again, it is not hard to see that ˆ F = S n ≥ F n is an adequate IL M -frame. Lemma 6.15.
Let F = h W, R, S, ν i be a quasi– IL M –frame and K = K ( F ) .Then xKy → ∃ z ( ν ( x ) ⊆ ✷ ν ( z ) ∧ x ( R ∪ S ) ∗ zRy ) . ewijs. We define T := { ( x, y ) | ∃ z ( ν ( x ) ⊆ ✷ ν ( z ) ∧ x ( R ∪ S ) ∗ zRy ) } . It is nothard to see that T is transitive and that { ( x, y ) | ∃ t ( ν ( x ) ⊆ ✷ ν ( t ) ∧ xT ; ( S ∪ K ) ∗ tT y ) } ⊆ T . We now define K ′ = K ∩ T . We have to show that K ′ = K .As K ′ ⊆ K is trivial, we will show K ⊆ K ′ .It is easy to see that K ′ satisfies properties 1., 2. and 3. of Definition 6.6;It follows on the two observations on T we just made. Since K is the smallestbinary relation that satisfies these properties we conclude K ⊆ K ′ .The next lemma shows that K is a rather stable relation. We show that ifwe extend a frame G to a frame F such that from worlds in F − G we cannotreach worlds in G , then K on G does not change. Lemma 6.16.
Let F = h W, R, S, ν i be a quasi– IL M –frame. And let G = h W − , R − , S − , ν − i be a subframe of F (which means W − ⊆ W , R − ⊆ R , S − ⊆ S and ν − ⊆ ν ). If(a) for each f ∈ W − W − and g ∈ W − not f ( R ∪ S ) g and(b) R ↾ W − ⊆ K ( G ) .Then K ( G ) = K ( F ) ↾ W − .Bewijs. Clearly K ( F ) ↾ W − satisfies the properties 1., 2. and 3. of the definitionof K ( G ) (Definition 6.6). Thus, since K G is the smallest such relation, we getthat K ( G ) ⊆ K ( F ) ↾ W − .Let K ′ = K ( F ) − ( K ( F ) ↾ W − − K ( G )). Using Lemma 6.15 one can showthat K ( F ) ⊆ K ′ . From this it immediately follows that K ( F ) ↾ W − ⊆ K ( G ).We finish the basic preliminaries with a somewhat complicated variation ofLemma 4.18. Lemma 6.17.
Let Γ and ∆ be MCS ’s. Γ ≺ C ∆ . P ✄ Q, S ✄ T , . . . , S n ✄ T n ∈ Γ and ✸ P ∈ ∆ . There exist k ≤ n . MCS ’s ∆ , ∆ , . . . , ∆ k such that • Each ∆ i lies C -critical above Γ , • Each ∆ i lies ⊆ ✷ above ∆ (i.e. ∆ ⊆ ✷ ∆ i ), • Q ∈ ∆ , • For all ≤ j ≤ n , S j ∈ ∆ h ⇒ for some i ≤ k , T j ∈ ∆ i .Bewijs. First a definition. For each I ⊆ { , . . . , n } put S I : ⇔ ^ {¬ S i | i ∈ I } . The lemma can now be formulated as follows. There exists I ⊆ { , . . . , n } suchthat { Q, S I } ∪ {¬ B, ✷ ¬ B | B ✄ C ∈ Γ } ∪ { ✷ A | ✷ A ∈ ∆ } 6⊢ ⊥ i I , { T i , S I } ∪ {¬ B, ✷ ¬ B | B ✄ C ∈ Γ } ∪ { ✷ A | ✷ A ∈ ∆ } 6⊢ ⊥ . So let us assume, for a contradiction, that this is false. Then there existfinite sets
A ⊆ { A | ✷ A ∈ ∆ } and B ⊆ { B | B ✄ C ∈ Γ } such that, if we put A : ⇔ ^ A , and B : ⇔ _ B , then, for all I ⊆ { , . . . , n } , Q, S I , ✷ A, ¬ B ∧ ✷ ¬ B ⊢ ⊥ (1)or, for some i I , T i , S I , ✷ A, ¬ B ∧ ✷ ¬ B ⊢ ⊥ . (2)We are going to define a permutation i , . . . , i n of { , . . . , n } such that if weput I k = { i j | j < k } then T i k , S I k , ✷ A, ¬ B ∧ ✷ ¬ B ⊢ ⊥ . (3)Additionally, we will verify that for each k (1) does not hold with I k for I. We will define i k with induction on k . We define I = ∅ . And by Lemma 4.18,(1) does not hold with I = ∅ . Moreover, because of this, (2) must be true with I = ∅ . So, there exists some i ∈ { , . . . , n } such that T i , ✷ A, ¬ B ∧ ✷ ¬ B ⊢ ⊥ . It is thus sufficient to take for i , for example, the least such i .Now suppose i k has been defined. We will first show that Q, S I k +1 , ✷ A, ¬ B ∧ ✷ ¬ B
6⊢ ⊥ . (4)Let us suppose that this is not so. Then ⊢ ✷ ( Q → ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ∨ · · · ∨ S i k ) . (5)So,Γ ⊢ P ✄ Q ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ∨ · · · ∨ S i k − ∨ S i k by (5) ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ∨ · · · ∨ S i k − ∨ T i k ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ∨ · · · ∨ S i k − ∨ ( T i k ∧ ✷ A ∧ ¬ B ∧ ✷ ¬ B ∧ S I k ) ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ∨ · · · ∨ S i k − by (3)... ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ T i ✄ ✸ ¬ A ∨ B ∨ ✸ B ∨ ( T i ∧ ✷ A ∧ ¬ B ∧ ✷ ¬ B ) ✄ ✸ ¬ A ∨ B ∨ ✸ B. by (3), with k = 1 .
33o by M , ✸ P ∧ ✷ A ✄ ( ✸ ¬ A ∨ B ∨ ✸ B ) ∧ ✷ A ∈ Γ . But ✸ P ∧ ✷ A ∈ ∆. So, by Lemma 4.18 there exists some MCS ∆ with Γ ≺ C ∆that contains B ∨ ✸ B . This is a contradiction, so we have shown (4).But now, since (4) is indeed true, and thus (1) with I k +1 for I is false, (2)must hold. Thus there must exist some i I k +1 such that T i , S I k +1 , ✷ A, ¬ B ∧ ✷ ¬ B ⊢ ⊥ . So we can take for i k +1 , for example, the smallest such i .It is clear that for I = { , , . . . , n } , (2) cannot be true. Thus, for I = { , , . . . , n } , (1) must be true. This implies ⊢ ✷ ( Q → ✸ ¬ A ∨ B ∨ ✸ B ∨ S i ∨ · · · ∨ S i n ) . Now exactly as above we can show Γ ⊢ P ✄ ✸ ¬ A ∨ B ∨ ✸ B . And again as above,this leads to a contradiction.In order to formulate the invariants needed in the main lemma applied to IL M , we need one more definition and a corollary. Definition 6.18 ( ⊂ , ⊂ ) . Let F = h W, R, S, ν i be a quasi–frame. Let K = K ( F ). We define ⊂ and ⊂ as follows.1. x ⊂ y ⇔ ∃ wy ′ wKxK y ′ ( S w ) t r y x ⊂ y ⇔ x ( ⊂ ∪ K ) ∗ y Corollary 6.19.
Let F = h W, R, S, ν i be a quasi–frame. And let K = K ( F ) .1. x ⊂ y ∧ yKz → xKz
2. If F is a quasi– IL M –frame, then x ⊂ y ⇒ ν ( x ) ⊆ ✷ ν ( y ) . The following theorem is well known.
Theorem 6.20.
For an IL -frame F = h W, R, S, ν i we have ∀ wxyy ′ z ( wRxRyS w y ′ Rz → xRz ) ⇔ F | = M . Let D be some finite set of formulas, closed under subformulas and single nega-tion.During the construction we will keep track of the following main–invariants. I ✷ for all y , { ν ( x ) | xK y } is linearly ordered by ⊆ ✷ d wK x ∧ wK ≥ x ′ ( S w ∪ K ) ∗ x → ‘there does not exists a deficiency in w w.r.t. x ’ I S wKxKy ( S w ∪ K ) ∗ y ′ → ‘the ⊆ ✷ -max of { ν ( t ) | wKtK y ′ } , if it exists, is ⊆ ✷ -larger than ν ( x )’ I N wKxKy ∧ y ∈ N Aw → x ∈ N Aw I D xRy → ∃ A ∈ ( ν ( y ) \ ν ( x )) ∩ { ✷ D | D ∈ D}I M All conditions for an adequate IL M –frame holdIn order to ensure that the main–invariants are preserved during the con-struction we need to consider the following sub–invariants. J u wK ≥ x ( S w ) t r y ∧ wK ≥ x ′ ( S w ) t r y → x = x ′ J K wKxK y ( S w ) t r y ′ K z → xK z J ⊂ y ⊂ x ∧ x ⊂ y → y = x J N x ( S v ) t r y ∧ wKy ∧ x ∈ N Aw → y ∈ N Aw J N x ( S w ) t r y ∧ y ∈ N Aw → x ∈ N Aw J ν ‘ ν ( w, y ) is defined’ ∧ vKy → v ⊂ w J ν ‘ ν ( w, y ) is defined’ → wK y J ν If x ( S w ) t r y , then ν ( w, y ) is defined J ν If ν ( v, y ) and ν ( w, y ) are defined then w = v What can we say about these invariants? I ✷ , I S , I N and I d were discussed inSection 6.1. I M is there to ensure that our final frame is an IL M –frame. Aboutthe sub–invariants there is not much to say. They are merely technicalities thatensure that the main–invariants are invariant.Let us first show that if we have a quasi– IL M –frame that satisfies all theinvariants, possibly I M excluded, then we can assume, nevertheless, that I M holds as well. Corollary 6.21.
Any quasi– IL M –frame that satisfies all of the above invari-ants, except possibly I M , can be extended to an IL M –frame that satisfies allthe invariants.Bewijs. Only I D and I d need some attention. All the other invariants are givenin terms of relations that do not change during the construction of the IL M -closure (Lemma 6.14). Lemma 6.22.
Let F = h W, R, S, ν i be a quasi– IL M –frame. Then F | = x ∈N Aw iff. one of the following cases applies. We call them sub–invariants since they merely serve the purpose of showing that themain-invariants are, indeed, invariant. . ν ( w, x ) = A
2. There exists t ∈ N Aw such that tKx
3. There exists t ∈ N Aw such that t S w x Corollary 6.23.
Let F be a quasi– IL M –frame that satisfies J ν . Let w, x ∈ F and let A be a formula. Then x ∈ N Aw implies ν ( w, x ) = A or there exists some t ∈ N Aw such that tKx . Lemma 6.24.
Let F be a quasi–frame which satisfies J N , J ν , J ν and J ν .Then x S v y, y ∈ N Aw ⇒ x ∈ N Aw .Bewijs. Suppose x S v y and y ∈ N Aw . Then, by Corollary 6.23, ν ( w, y ) = A or,for some t ∈ N Aw , tKy . In the first case we obtain w = v by J ν and J ν . Andthus by J N , x ∈ N Aw . In the second case we have, by J ν and J ν that t ⊂ v .Which implies, by Lemma 6.19–1., tKx . Let F = h W, R, S, ν i be a quasi– IL M –frame that satisfies all the invariants.Let ( a , ¬ ( A ✄ B )) be a D -problem in F . We fix some b W . Using Lemma4.17 we find a MCS ∆ b , such that ν ( a ) ≺ B ∆ b and A, ✷ ¬ A ∈ ∆ b . We putˆ F = h ˆ W , ˆ R, ˆ S, ˆ ν i = h W ∪ { b } , R ∪ {h a , b i} , S, ν ∪ {h b , ∆ b i , hh a , b i , B i}i , and define ˆ K = K ( ˆ F ). The frames F and ˆ F satisfy the conditions of Lemma6.16. Thus we have ∀ xy ∈ F xKy ⇔ x ˆ Ky. (6)Since ˆ S = S , this implies that all simple enough properties expressed in ˆ K and ˆ S using only parameters from F are true if they are true with ˆ K replaced by K . Claim 3. ˆ F is a quasi– IL M –frame. Bewijs.
A simple check of Properties (1.–5.) of Definition 5.1 (quasi–frames) andProperties (6.–10.) of Definition 6.10 (quasi– IL M –frames) and the remainingones in Definition 5.1 (quasi–frames). Let us comment on two of them. x ˆ Ky → ˆ ν ( x ) ≺ ˆ ν ( y ) follows from Lemma 6.15 and (6).Let us show ˆ F | = x ∈ N Cw ⇒ ˆ ν ( w ) ≺ C ˆ ν ( x ). We have ∀ xw ∈ F F | = x ∈N Cw ⇔ ˆ F | = x ∈ N Cw . So we only have to consider the case ˆ F | = b ∈ N Cw . If w = a then we are done by choice of ˆ ν ( b ). Otherwise, by Lemma 6.24, we havefor some x ∈ F , F | = x ∈ N Cw and x ˆ K b . By the first property we proved, weget ˆ ν ( x ) ≺ ˆ ν ( b ). So, since ˆ ν ( w ) ≺ C ˆ ν ( x ) we have ˆ ν ( w ) ≺ C ˆ ν ( b ).Before we show that ˆ F satisfies all the invariants we prove some lemmata. Lemma 6.25.
If for some x = a , x ˆ K b . Then there exist unique u and w (independent of x ) such that wK ≥ u ( S w ) t r a . ewijs. If such w and u do not exists then T = K ∪{ a , b } satisfies the conditionsof Lemma 6.8. In which case xK b gives xT b which implies x = a . Theuniqueness of w follows from J ν and J ν . The uniqueness of u follows from J u and the uniqueness of w .In what follows we will denote these w and u , if they exist, by w and u . Lemma 6.26.
For all x . If x ˆ K b then x ⊂ a .Bewijs. Let K ′ = K ∪ { ( x, b ) | x ˆ K b ∧ x ⊂ a } . It is not hard to show that K ′ satisfies the conditions of T in Lemma 6.8. Lemma 6.27.
Suppose the conditions of Lemma 6.25 are satisfied and let u bethe u asserted to exist. Then for all x = a , if x ˆ K b , then xK u .Bewijs. By Lemma 6.26 we have x ⊂ a . Let x = x ( ⊂ ∪ K ) x ( ⊂ ∪ K ) · · · ( ⊂ ∪ K ) x n = a . First we show x = x ⊂ x ⊂ · · · ⊂ x n = a . Suppose, for a contradiction,that for some i < n , x i Kx i +1 . Then, by Lemma 6.19, xKx i +1 K b . So, xK ≥ b .A contradiction. The lemma now follows by showing, with induction on i andusing F | = J K , that for all i ≥ x n − ( i +1) K u . Lemma 6.28. ˆ F satisfies all the sub-invariants.Bewijs. We only comment on J K and J ν . Let K = K ( ˆ F ). J ν follows from Lemma 6.26, so let us treat J K . Suppose w ˆ Kx ˆ K y ( ˆ S w ) t r y ′ ˆ K z .We can assume that at least one of w, x, y, y ′ , z is not in F and the only candi-date for this is z . So we have z = b . We can assume that x = y ′ (otherwise weare done at once), so the conditions of Lemma 6.25 are fulfilled and thus w and u as stated there exist.Suppose now, for a contradiction, that for some t , x ˆ Kt ˆ K b . Then by Lemma6.27, t = a or t ˆ K u . Suppose we are in the case t = a . Since ν ( w , a ) is definedand x ˆ K a we obtain by J ν , that x ⊂ w . Since w ˆ K ≥ u we obtain by Lemma6.19 that x ˆ K ≥ u . In the case t ˆ K u we have x ˆ K ≥ u trivially. So in any casewe have x ˆ K ≥ u . However, by Lemma 6.27 and since y ′ ˆ K z we have y ′ ˆ K u or y ′ = a . In thefirst case, since F | = J K , we have x ˆ K u . In the second case we obtain, by theuniqueness of u , that y = u and thus x ˆ K u . So in any case we have x ˆ K u . A contradiction. 37 emma 6.29.
Possibly with the exception of I M , ˆ F satisfies all the main-invariants.Bewijs. Let K = K ( ˆ F ). We only comment on I ✷ and I N .First we treat I ✷ . So we have to show that for all y , { ˆ ν ( x ) | x ˆ K y } is linearlyordered by ⊆ ✷ . We only need to consider the case y = b . If { a } = { x | x ˆ K b } then the claim is obvious. So we can assume that the condition of Lemma 6.25is fulfilled and we fix u as stated. The claim now follows by F | = I ✷ (with y = u ) and noting that, by Lemma 6.15, x ˆ K b ⇒ x ⊆ ✷ a .Now we look at I N : w ˆ Kx ˆ Ky ∧ ˆ F | = y ∈ N Aw → ˆ F | = x ∈ N Aw . Suppose w ˆ Kx ˆ Ky and ˆ F | = y ∈ N Aw . We only have to consider the case y = b . Then, byLemma 6.22, ˆ ν ( w, b ) = A or for some t ∈ N Aw we have t ˆ S w b or t ˆ K b . The firstcase is impossible by J ν . The second is also clearly not so. Thus we have t ˆ K b . (7)We suppose that the conditions of Lemma 6.25 are fulfilled (the other caseis easy). If t ˆ K u and x ˆ K ∗ u then we are done simmilarly as the case above.So assume t ˆ K a or x ˆ K ∗ a . Since wRt and wRx in any case we have w ˆ K a .Now by Lemma 6.24 and J N we have u ∈ N Aw ⇔ a ∈ N Aw . Also, by (7), u ∈ N Aw ∨ a ∈ N Aw . So since x ˆ K u or x = a or x ˆ K a we obtain x ∈ N Aw by F | = I N .To finish this subsection we note that by Lemma 6.14 and Corollary 6.21 wecan extend ˆ F to an adequate IL M –frame that satisfies all invariants. Let F = h W, R, S, ν i be an IL M –frame satisfing all the invariants. Let ( a , b , C ✄ D )be a D -deficiency in F .Suppose a R ≥ b (the case a R b is easy). Let x be the ⊆ ✷ -maximum of { x | a KxK b } . This maximum exists by I ✷ . Pick some A such that b ∈ N A a .(If such an A exists, then by adequacy of F , it is unique. If no such A exists,take A = ⊥ .) By I N and adequacy we have ν ( a ) ≺ A ν ( x ). So we have C ✄ D ∈ ν ( a ) ≺ A ν ( x ) ∋ ✸ C . We apply Lemma 6.17 to obtain, for some set Y , disjointfrom W , a set { ∆ y | y ∈ Y } of MCS’s with all the properties as stated in thatlemma. We defineˆ F = h W ∪ Y, R ∪ {h a , y i | y ∈ Y } ,S ∪ {h a , b , y i | y ∈ Y } ∪ {h a , y, y ′ i | y, y ′ ∈ Y, y = y ′ } ,ν ∪ {h y, ∆ y i , hh a , y i , A i | y ∈ Y }i . Claim 4. ˆ F is a quasi– IL M –frame. Bewijs.
An easy check of Properties (1.–5.) of Definition 5.1 (quasi–frames) andProperties (6.–10.) of Definition 6.10 (quasi– IL M –frames). Let us commenton two cases. 38irst we see that x ˆ Ky → ˆ ν ( x ) ≺ ˆ ν ( y ). We can assume y ∈ Y . By Lemma6.15 we obtain some z with ˆ ν ( x ) ⊆ ✷ ˆ ν ( z ) and x ( ˆ R ∪ ˆ S ) ∗ z ˆ Ry . This z can onlybe a . By choice of ˆ ν ( y ) we have ˆ ν ( a ) ≺ ˆ ν ( y ). And thus ˆ ν ( x ) ≺ ˆ ν ( y ).We now see that w ˆ Kx ˆ Ky ( ˆ S w ∪ ˆ K ) ∗ y ′ → ˆ ν ( x ) ⊆ ✷ ˆ ν ( y ′ ). We can assume atleast one of w, x, y, y ′ is in Y . The only candidates for this are y and y ′ . If bothare in Y then w = a and an x as stated does not exists. So only y ′ ∈ Y andthus in particular y = y ′ . Now there are two cases to consider.The first case is that for some t , w ˆ Kx ˆ Ky ( ˆ S w ∪ ˆ K ) ∗ t ˆ Ky ′ . But, ˆ ν ( y ′ ) is ⊆ ✷ -larger than ˆ ν ( t ) by x ˆ Ky → ˆ ν ( x ) ≺ ˆ ν ( y ). Also we have wKxKy ( S w ∪ K ) ∗ t . So,ˆ ν ( x ) = ν ( x ) ⊆ ✷ ν ( t ) = ˆ ν ( t ).The second case is w ˆ Kx ˆ Ky ( ˆ S w ∪ ˆ K ) ∗ b ˆ S w y ′ . In this case we have w = a . y ′ is chosen to be ⊆ ✷ –larger than the ⊆ ✷ -maximum of { ν ( r ) | a KrK b } . Wehave wKxKy ( S w ∪ K ) ∗ b So, by F | = I S , this ⊆ ✷ –maximum is ⊆ ✷ –larger than ν ( x ). Lemma 6.30.
For any x ∈ ˆ F and y ∈ Y we have x ˆ K y → x ⊂ a .Bewijs. We put K ′ = K ∪ { ( x, y ) | y ∈ Y, x ˆ Ky, x ⊂ a } . By showing that K ′ satisfies the conditions of T in Lemma 6.8. we obtain x ˆ K y → xK ′ y . Soif x ˆ K y then xK ′ y . But if y ∈ Y then xKy does not hold. Thus we have x ⊂ a . Lemma 6.31.
Suppose y ∈ Y and a ˆ K z . Then for all x , x ˆ K y → x ˆ K z .Bewijs. Suppose xK y . By Lemma 6.30 we have x ⊂ a . There exist x , x , x , . . . , x n such that x = x ( ⊂ ∪ K ) x ( ⊂ ∪ K ) · · · ( ⊂ ∪ K ) x n = a . First we show that x = x ⊂ x ⊂ · · · ⊂ a . Suppose, for a contradiction that for some i < n ,we have x i Kx i +1 . Then xKx i +1 Ky and thus xK ≥ y . A contradiction. Thelemma now follows by showing, with induction on i , using J K , that for all i ≤ n , x n − i K z . Lemma 6.32. ˆ F satisfies all the sub-invariants.Bewijs. The proofs are rather straightforward. We give two examples.First we show J u : w ˆ K ≥ x ( ˆ S w ) t r y ∧ w ˆ K ≥ x ′ ( ˆ S w ) t r y → x = x ′ . Suppose that w ˆ K ≥ x ( ˆ S w ) t r y and w ˆ K ≥ x ′ ( ˆ S w ) t r y . We can assume that y ∈ Y . (Otherwiseall of w, x, x ′ , y are in F and we are done by F | = J u .) We clearly have w ∈ F .If x ∈ Y then w = a and thus w ˆ K x . So, x Y . Next we show that both x, x ′ = b .Assume, for a contradiction, that at least one of them equals b . W.l.o.g. weassume it is x . But then wK ≥ b and wK ≥ x ′ ( S w ) t r b . By F | = J ν we nowobtain that ν ( w, b ) is defined. And thus by F | = J ν , wK b . A contradiction.So, both x, x ′ = b . But now wK ≥ x ( S w ) t r b and wK ≥ x ′ ( S w ) t r b . So, by F | = J u , we obtain x = x ′ .Now let us see that J K holds, that is w ˆ Kx ˆ K y ( ˆ S w ) t r y ′ ˆ K z → x ˆ K z .Suppose w ˆ Kx ˆ K y ( ˆ S w ) t r y ′ ˆ K z . We can assume that z ∈ Y . (Otherwise all of w, x, y, y ′ , z are in F and we are done by F | = J K .) Fix some a ∈ F for which39 K a . By Lemma 6.31 we have y ′ K a and thus, since F | = J K , xK a .By definition of ˆ K we have x ˆ Kz . Now, if for some t , we have x ˆ Kt ˆ K z , thensimilarly as above, tK a . So, this implies xK ≥ a . A contradiction, conclusion: xK z . Lemma 6.33.
Except for I M , ˆ F satisfies all main-invariants.Bewijs. We only comment on I ✷ and I N .First we show I ✷ : For all y , { ˆ ν ( x ) | x ˆ K y } is linearly ordered by ⊆ ✷ . Let y ∈ ˆ F and consider the set { x | xK y } . Since ˆ K ↾ F = K and for all y ∈ Y theredoes not exists z with y ˆ K z we only have to consider the case y ∈ Y . Fix some a such that a K a K ∗ b . By Lemma 6.30 for any such y we have { x | xK y } ⊆ { x | xK a } . And by F | = I ✷ with a for y , we know that { ν ( x ) | xK a } is linearly orderedby ⊆ ✷ .Now let us see I N : w ˆ Kx ˆ Ky ∧ ˆ F | = y ∈ N Aw → ˆ F | = x ∈ N Aw . Suppose w ˆ Kx ˆ Ky ˆ F | = y ∈ N Aw . We can assume y ∈ Y . By Lemma 6.30, x ⊂ a . So, wKxK b . By Lemma 6.24, F | = b ∈ N Aw and thus ˆ F | = x ∈ N Aw .To finish this section we noting that by Lemma 6.14 and Corollary 6.21 wecan extend ˆ F to an adequate IL M –frame that satisfies all invariants. It is clear that the union of a bounded chain of IL M –frames is itself an IL M –frame. W ∗ In this section we are going to prove the following theorem.
Theorem 7.1. IL W ∗ is a complete logic. For a long time IL W ∗ has been conjectured ([30]) to be IL (All). A firststep in proving this conjecture would have been a modal completeness result.However, the modal completeness of IL W ∗ resisted many attempts as the modalcompleteness of IL M , which is an essential part of IL W ∗ , was so hard andinvolved. (In [9] a completeness proof for IL W was given.)Finally, now that all the machinery has been developed, a modal complete-ness proof for IL W ∗ can be given. The completeness proof of IL W ∗ lifts almostcompletely along with the completeness proof for IL M . We only need someminor adaptations. 40 .1 Preliminaries The frame condition of W is well known. Theorem 7.2.
For any IL -frame F we have that F | = W ⇔ ∀ w ( S w ; R ) isconversely well-founded. We can define a new principle M ∗ that is equivalent to W ∗ , as follows. M ∗ : A ✄ B → ✸ A ∧ ✷ C ✄ B ∧ ✷ C ∧ ✷ ¬ A Lemma 7.3. IL M W = IL W ∗ = IL M ∗ Bewijs.
The proof we give consists of four natural parts.First we see IL W ∗ ⊢ M . We reason in IL W ∗ and assume A ✄ B . Thus, also A ✄ ( B ∨ ✸ A ). Applying the W ∗ axiom to the latter yields ( B ∨ ✸ A ) ∧ ✷ C ✄ ( B ∨ ✸ A ) ∧ ✷ C ∧ ✷ ¬ A . From this we may conclude ✸ A ∧ ✷ C ✄ ( B ∨ ✸ A ) ∧ ✷ C ✄ ( B ∨ ✸ A ) ∧ ✷ C ∧ ✷ ¬ A ✄ B ∧ ✷ C Secondly, we see that IL W ∗ ⊢ W . Again, we reason in IL W ∗ . We assume A ✄ B and take the C in the W ∗ axiom to be ⊤ . Then we immediately see that A ✄ B ✄ B ∧ ✷ ⊤ ✄ B ∧ ✷ ⊤ ∧ ✷ ¬ A ✄ B ∧ ✷ ¬ A .We now easily see that IL M W ⊢ M ∗ . For, reason in IL M W as follows.By W ∗ , A ✄ B ✄ B ∧ ✷ ¬ A . Now an application of M on A ✄ B ∧ ✷ ¬ A yields ✸ A ∧ ✷ C ✄ B ∧ ✷ C ∧ ✷ ¬ A .Finally we see that IL M ∗ ⊢ W ∗ . So, we reason in IL M ∗ and assume A ✄ B .Thus, we have also ✸ A ∧ ✷ C ✄ B ∧ ✷ C ∧ ✷ ¬ A . We now conclude B ∧ ✷ C ✄ B ∧ ✷ C ∧ ✷ ¬ A easily as follows. B ∧ ✷ C ✄ ( B ∧ ✷ C ∧ ✷ ¬ A ) ∨ ( ✷ C ∧ ✸ A ) ✄ B ∧ ✷ C ∧ ✷ ¬ A . Corollary 7.4.
For any IL -frame we have that F | = W ∗ iff. both (for each w , ( S w ; R ) is conversely well-founded) and ( ∀ w, x, y, y ′ , z ( wRxRyS w y ′ Rz → xRz ) ). The frame condition of W ∗ tells us how to correctly define the notions ofadequate IL W ∗ -frames and quasi- IL W ∗ -frames. Definition 7.5 ( ( D ✷ ) . Let D be a finite set of formulas. Let ( D ✷ be a binaryrelation on MCS’s defined as follows. ∆ ( D ✷ ∆ ′ iff.1. ∆ ⊆ ✷ ∆ ′ ,2. For some ✷ A ∈ D we have ✷ A ∈ ∆ ′ − ∆. Lemma 7.6.
Let F be a quasi-frame and D be a finite set of formulas. If wRxRyS w y ′ → ν ( x ) ( D ✷ ν ( y ′ ) then ( R ; S w ) is conversely well-founded.Bewijs. By the finiteness of D . 41 emma 7.7. Let F be a quasi- IL M -frame. If wRxRyS w y ′ → ν ( x ) ( D ✷ ν ( y ′ ) then wRxRy ( S w ∪ R ) ∗ y ′ → ν ( x ) ( D ✷ ν ( y ′ ) Bewijs.
Suppose wRxRy ( S w ∪ R ) ∗ y ′ . ν ( x ) ( D ✷ ν ( y ′ ) follows with induction onthe minimal number of R -steps in the path from y to y ′ . Definition 7.8 (Adequate IL W ∗ -frame) . Let D be a set of formulas. We saythat an adequate IL M -frame is an adequate IL W ∗ -frame (w.r.t. D ) iff. thefollowing additional property holds.8. wRxRy ( S w ) t r y ′ → x ( D ✷ y ′ Definition 7.9 (Quasi- IL W ∗ -frame) . Let D be a set of formulas. We say that aquasi- IL M -frame is a quasi- IL W ∗ -frame (w.r.t. D ) iff. the following additionalproperty holds.13. wKxKy ( S w ) t r y ′ → x ( D ✷ y ′ In what follows we might simply talk of adequate IL W ∗ -frames and quasi- IL W ∗ In these cases D is clear from context. Lemma 7.10.
Any quasi- IL W ∗ -frame can be extended to an adequate IL W ∗ -frame. (Both w.r.t. the same set of formulas D .)Bewijs. Let F be a quasi- IL W ∗ -frame. Then in particular F is a quasi- IL M -frame. So consider the proof of Lemma 6.14. There we constructed a sequenceof quasi- IL M -frames F = F ⊆ F ⊆ S i<ω F i = ˆ F . What we have to do,is to show that if F (= F ) is a quasi- IL W ∗ -frame, then each F i is as well.Additionally we have to show that ˆ F is an adequate IL W ∗ -frame.But this is rather trivial. As noted in the proof of Lemma 6.14, The relation K and the relations ( S w ) t r are constant throughout the whole process. Soclearly each F i is a quasi- IL W ∗ -frame.Also the extra property of quasi- IL W ∗ -frames is preserved under unions ofbounded chains. So, ˆ F is an adequate IL W ∗ -frame. Lemma 7.11.
Let Γ and ∆ be MCS ’s with Γ ≺ C ∆ , P ✄ Q, S ✄ T , . . . , S n ✄ T n ∈ Γ and ✸ P ∈ ∆ . There exist k ≤ n . MCS ’s ∆ , ∆ , . . . , ∆ k such that • Each ∆ i lies C -critical above Γ , • Each ∆ i lies ⊆ ✷ above ∆ , • Q ∈ ∆ , • For each i ≥ , ✷ ¬ P ∈ ∆ i , • For all ≤ j ≤ n , S j ∈ ∆ h ⇒ for some i ≤ k , T j ∈ ∆ i .Bewijs. The proof is a straightforward adaptation of the proof of Lemma 6.17.In that proof, a trick was to postpone an application of M as long as possible.We do the same here but let an application of M on P ✄ ✸ P ∨ ψ be precededby an application of W to obtain P ✄ ψ .42 .2 Completeness Again, we specify the four ingredients from Remark 4.19. The
Frame condi-tion is contained in Corollary 7.4.The
Invariants are all those of IL M and additionally I w ∗ wKxKy ( S w ) t r y ′ → x ( D ✷ y ′ Here, D is some finite set of formulas closed under subformulas and single ne-gation. Problems . We have to show that we can solve problems in an adequate IL W ∗ -frame in such a way that we end up with a quasi- IL W ∗ -frame. If we havesuch a frame then in particular it is an IL M -frame. So, as we have seen wecan extend this frame to a quasi- IL M -frame. It is easy to see that wheneverwe started with an adequate IL W ∗ -frame we end up with a quasi IL W ∗ -frame.(This is basically Lemma 7.10.) Deficiencies . We have to show that we can solve any deficiency in anadequate IL W ∗ -frame such that we end up with an quasi- IL W ∗ -frame. It iseasily seen that the process as described in the case of IL M works if we useLemma 7.11 instead of Lemma 6.17. Rounding up . We have to show that the union of a bounded chain ofquasi- IL W ∗ -frames that satisfy all the invariants is an IL W ∗ -frame. The onlynovelty is that we have to show that in this union for each w we have that( R ; S w ) is conversely well-founded. But this is ensured by I w ∗ and Lemma 7.6. Acknowledgement
We dedicate this series of three papers to Dick de Jongh. Dick supervised theMasters theses of both authors and suggested to study a step-by-step construc-tion method to obtain modal completeness results.Furthermore, we thank Lev Beklemishev, Marta Bilkova, Rosalie Iemhoff,Pavel Pudl´ak, Volodya Shavrukov and Albert Visser for questions, discussionsand suggestions.
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