A new principle in the interpretability logic of all reasonable arithmetical theories
aa r X i v : . [ m a t h . L O ] A p r A New Principle in the Interpretability Logic ofall Reasonable Arithmetical Theories
Evan GorisandJoost J. Joosten2011
Samenvatting
The interpretability logic of a mathematical theory describes the struc-tural behavior of interpretations over that theory. Different theories havedifferent logics. This paper revolves around the question what logic de-scribes the behavior that is present in all theories with a minimum amountof arithmetic; the intersection over all such theories so to say. We denotethis target logic by IL (All).In this paper we present a new principle R in IL (All). We show that R does not follow from the logic IL P W ∗ that contains all previouslyknown principles. This is done by providing a modal incompleteness proofof IL P W ∗ : showing that R follows semantically but not syntacticallyfrom IL P W ∗ . Apart from giving the incompleteness proof by elementarymethods, we also sketch how to work with so-called Generalized VeltmanSemantics as to establish incompleteness. To this extent, a new versionof this Generalized Veltman Semantics is defined and studied. Moreover,for the important principles the frame correspondences are calculated.After the modal results it is shown that the new principle R is indeedvalid in any arithmetically theory. The proof employs some elementaryresults on definable cuts in arithmetical theories. Interpretations of one theory or structure into another are omnipresent in (meta-) mathematical practice. Interpretability logics study the structural behaviorof interpretations. Below we shall provide precise definitions. The structuralbehavior of interpretations is different for different kind of theories thus yieldingdifferent interpretability logics.For example, for finitely axiomatized theories, the corresponding logic turnedout to be IL P as defined below. For theories like Peano Arithmetic with fullinduction the interpretability logic is IL M . It is a long standing open problem Technically speaking the property of so-called essential reflexivity is sufficient. A theory is IL (All).This paper revolves around this question and presents some major results.In [19] a conjecture was posed that IL (All) = IL W ∗ P . In this paper we refutethis conjecture by exposing a new arithmetically valid principle. We shall provethe modal incompleteness of the logic IL P W ∗ by introducing a new principle R . Next, we show that the principle R follows semantically from IL P W ∗ but isnot provable in IL P W ∗ .We shall expose two proof methods here to prove that R does not follow syn-tactically from IL P W ∗ . The first, in Section 3, develops some general theory forproving incompleteness via so-called Generalized Veltman Semantics . However,it will turn out that the frame condition for the principle W is so ghastly andcumbersome that however possible to work with, proofs become too involved.The second proof method, in Section 4, uses the regular Veltman semanticsand some sort of bisimulation argument and a full proof is given that R doesnot follow syntactically from IL P W ∗ .To conclude, we shall prove in Section 5 that actually R is sound in anyreasonable arithmetical theory. In particular, this implies that IL (All) can notbe IL P W ∗ .We found the principle R by trying to formulate a sufficient condition forthe logic IL P W ∗ to be modally complete. We think that this illustrates nicelythat a modal formulation of an arithmetical phenomenon can be very useful toobtain new arithmetical results. This paper is the third and final in a series of three. All definitions and moti-vations behind the definitions were already included in [20]. For completenessand readability we include the main definitions and issues also in this paper.
All ) 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 want essentially reflexive if any of its finite extensions proves the consistency of any finite sub-theorythereof. 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 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 2.1.
Let U and V be reasonable arithmetical theories. An interpre-tation j from V in U is a pair h δ, F i . Here, δ is called a domain specifier. It isa 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 . Wehave 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.The modal language of interpretability logics is the same as that of prova-bility logics but now augmented by a binary modality ✄ to denote interpreta-bility.Thus, we define the interpretability formulas as 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 ↔ . Also We take the liberty to not make a distinction between a syntactical object and its code. ✸ ϕ for ¬ ✷ ¬ ϕ . We refer the reader to [20] for more details andstandard reading conventions.Now we can define the link between the modal language and the arithmeticalcounterpart. Definition 2.2.
An arithmetical realization ∗ is a mapping that assigns to eachpropositional variable an arithmetical sentence. This mapping is extended to allmodal 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 2.3.
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. For two classes of theories, IL (T) is known. Definition 2.4.
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 2.5 (Berarducci [3], Shavrukov [24]) . If T is an essentially reflexivetheory, then IL (T) = IL M . Theorem 2.6 (Visser [29]) . If T is finitely axiomatizable, then IL (T) = IL P . Now we have all in place to define our central subject of interest.
Definition 2.7.
The interpretability logic of all reasonable arithmetical theories,we write IL ( All ), is the set of formulas ϕ such that ∀ T ∀ ∗ T ⊢ ϕ ∗ . Here the 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. As IL P ∩ IL M is known to be a strict upper bound, we also knowsome strict lower bounds. We close off this section on preliminaries by definingthese lower bounds and providing modal semantics for interpretability logics.4 .2 A lower bound to IL( All ) We first define a core logic which will be part of all interpretability logics studied.
Definition 2.8.
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 isa finite sequence of formulae ending on ϕ , each being a logical tautology, aninstantiation of one of the axiom schemata of IL , or the result of applyingeither Modus Ponens or Necessitation to formulas earlier in the sequence.Apart from the axiom schemata exposed in Definition 2.8 we will need con-sider 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 .Now, with the results of this paper, we know that IL P W ∗ is a strict lowerbound for IL (All). That is, from ϕ you are allowed to conclude ✷ ϕ . .3 Semantics 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.1. 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) . A relation R on W is called conversely well-founded if every non-empty subset of W hasan R -maximal element. Generalized semantics
In [27], ˇSvejdar showed the independence of some extensions of IL . Some ofthese logics, however, had the same class of characteristic Veltman frames. Na-turally, frames alone are not sufficient to distinguish between such logics soˇSvejdar used models combined with some bisimulation arguments instead. Ageneralized Veltman semantics, intended to uniformize this method, was pro-posed by de Jongh. This generalized semantics was previously investigated byVukovic´c [34], Joosten [18] and Verbrugge and was successfully used to showindependence of certain extensions of IL .We will set both the generalized Veltman semantics and the model/bisimulationmethod to work in order to distinguish some extensions of IL , which are indistin-guishable using Veltman frames alone. We use a slight variation of the semanticsused in [34]. Any result in this section can be obtained with the old semantics,we think that nevertheless this might be a useful variation. Definition 3.1 ( IL set -frame) . A structure h W, R, S i is an IL set -frame iff.1. W is an non-empty set.2. R is a transitive and conversely well-founded binary relation on W .3. S ⊆ W × W × ( P ( W ) −{∅} ) , such that (where we write yS x Y for ( x, y, Y ) ∈ S )(a) if xS w Y then wRx and for all y ∈ Y , wRy ,(b) S is quasi-reflexive: wRx implies xS w { x } ,(c) S is quasi-transitive: If xS w Y then for all y ∈ Y we have that if y Z and yS w Z then xS w Z ,(d) wRxRy implies xS w { y } . Definition 3.2 ( IL set -model) . An IL set -model is a structure h W, R, S, (cid:13) i suchthat h W, R, S i is an IL set -frame and (cid:13) is a binary relation between elements of W and modal formulas such that the following cases apply.1. (cid:13) commutes with boolean connectives. For instance, w (cid:13) A ∧ B iff. w (cid:13) A and w (cid:13) B .2. w (cid:13) ✷ A iff. for all x such that wRx we have that x (cid:13) A .3. w (cid:13) A ✄ B iff. for all x such that wRx and x (cid:13) A there exists some Y ,such that xS w Y and for all y ∈ Y , y (cid:13) B .For IL set -models F = h W, R, S, (cid:13) i and Y ⊆ W we will write Y (cid:13) A for ∀ y ∈ Y, y (cid:13) A . As usual, we say that a formula A is valid on an IL set -frame F = h W, R, S i iffor any model F = h W, R, S, (cid:13) i , based on F , and any w ∈ W , we have F , w (cid:13) A . Lemma 3.3 (Soundness of IL ) . If IL ⊢ A then for any IL set -frame F , F | = A . ewijs. Validity is preserved under modus ponens and generalization and tri-vially any propositional tautology is valid on each IL set -frame. So it is enoughto show that all axioms of IL are valid on each IL set -frame. We only treat J A ✄ B ) ∧ ( B ✄ C ) → A ✄ C .Suppose w (cid:13) A ✄ B and w (cid:13) B ✄ C . Pick some x with wRx and suppose x (cid:13) A . There exists some Y with xS w Y and Y (cid:13) B . W.l.o.g. we can assumethat for some y ∈ Y , y (cid:13) C . Fix such a y . Since y (cid:13) B and wRy there existssome Z such that yS w Z and Z (cid:13) C . In particular, y Z . And thus we have xS w Z . ⊣ Theorem 3.4 (Completeness of IL ) . If A is valid on each IL set -frame, then IL ⊢ A .Bewijs. Suppose IL A . Then there exists an IL -model M = h W, R, S i , andsome m ∈ M such that M, m (cid:13) ¬ A . Let M ′ = h W, R, S ′ , (cid:13) ′ i , where (cid:13) ′ = (cid:13) onpropositional variables and is extended as usual, and S ′ = { ( w, x, Y ) | ∀ y ∈ Y xS w y } . It is easy to see that M ′ is an IL set -model. As an example let us see that S isquasi-transitive. Suppose xS ′ w X , y ∈ X and yS ′ w Y . (We can assume y Y ,but we won’t use this.) Pick y ′ ∈ Y . Then xS w y and yS w y ′ . Thus xS w y ′ . Since y ′ ∈ Y was arbitrary we conclude xS ′ w Y .A straightforward induction on B shows that for all B we have w (cid:13) ′ B ⇔ w (cid:13) B . Thus we have m (cid:13) ′ ¬ A and in particular A is not valid on the underlingframe of M ′ . ⊣ Definition 3.5 ( IL set M -frame) . An IL set -frame is an IL set M -frame iff. forall w, x, y, Y such that wRxRyS w Y there exists some Y ′ ⊆ Y such that1. xS w Y ′ and2. for all y ′ ∈ Y ′ we have that for all z , y ′ Rz → xRz . Lemma 3.6.
For any IL set -frame F = h W, R, S i we have F | = M iff. F is an IL set M -frame.Bewijs. ( ⇐ ) Suppose F is an IL set M -frame. Let F = h W, R, S, (cid:13) i be a modelbased on this frame. Pick w ∈ W and suppose w (cid:13) A ✄ B . Pick x ∈ W with wRx and x (cid:13) ✸ A ∧ ✷ C . Now there exists some y with xRy and y (cid:13) A . Thus,for some Y , yS w Y and Y (cid:13) B . Since F is an IL set M -frame, there exists some Y ′ ⊆ Y such that xS w Y ′ and for all y ′ ∈ Y ′ we have that for all z , y ′ Rz → xRz .So, in particular, Y ′ (cid:13) ✷ C .( ⇒ ) Suppose F | = M . Choose w, x, y, Y such that wRxRyS w Y . Let p, q, s be distinct proposition variables. Define an IL set -model F = h W, R, S, (cid:13) i asfollows. v (cid:13) p ⇔ v = yv (cid:13) q ⇔ v ∈ Yv (cid:13) s ⇔ xRv w (cid:13) p ✄ q and thus w (cid:13) ✸ p ∧ ✷ s ✄ q ∧ ✷ s . Also, x (cid:13) ✸ p ∧ ✷ s . So, thereexists some Y ′ such that xS w Y ′ and Y ′ (cid:13) q ∧ ✷ s . But the only candidates forsuch an Y ′ are the subsets of Y . Also, since Y ′ (cid:13) ✷ s , by definition of (cid:13) we have y ′ ∈ Y ′ and y ′ Rz implies xRz . ⊣ Definition 3.7 ( IL set P -frame) . An IL set -frame is an IL set P -frame iff. forall w, x, y, Y, Z such that1. wRxRyS w Y and2. for all y ∈ Y there exists some z ∈ Z with yRz ,we have that there exists some Z ′ ⊆ Z with yS x Z ′ . Lemma 3.8.
For any IL set -frame F = h W, R, S i we have F | = P iff. F is an IL set P -frame.Bewijs. ( ⇐ ) Suppose F is an IL set P -frame. And let F = h W, R, S, (cid:13) i be an IL set -model based on this frame. Let w ∈ W and suppose w (cid:13) A ✄✸ B . Pick x, y in W with wRxRy and y (cid:13) A . There exists some Y with yS w Y and Y (cid:13) ✸ B .Put Z = { z | z (cid:13) B } . Now for all y ∈ Y there exists some z ∈ Z such that yRz .So, there exists some Z ′ ⊆ Z with yS x Z ′ .( ⇒ ) Suppose F | = P . Choose w, x, y ∈ W and Y, Z ⊆ W such that wRxRyS w Y and for all y ∈ Y there exists some z ∈ Z with yRz . Let p, q be distinct propositional variables. Define the IL set -model F = h W, R, S, (cid:13) i asfollows. v (cid:13) p ⇔ v = yv (cid:13) q ⇔ v ∈ Z Now, Y (cid:13) ✸ q . So, w (cid:13) p ✄ ✸ q and thus, since w (cid:13) P , w (cid:13) ✷ ( p ✄ q ). So forsome Z ′ we have yS x Z and Z ′ (cid:13) q . But the only candidates for such Z ′ are thesubsets of Z . ⊣ Lemma 3.9.
There exists an IL set P -frame which is not an IL set M -frame.Bewijs. Consider Figure 1. It represents an IL set -frame. For clarity we haveomitted the following arrows. Those needed for the transitivity of R . Thoseneeded for the quasi-reflexivity of S . Those needed for the inclusion of S in R .Additionally, quasi-transitivity dictates that we need xS w { z } , yS w { z } and yS w { z } . All the other ones are drawn.Let us first see that we actually have an IL set P -frame. So suppose vRaRbS v B .And let Z be such that for all b ′ ∈ B there exists some z ∈ Z such that b ′ Rz .It is not hard to see that only for v = w , a = x , b = y and B = { y , y } sucha Z exists. And that moreover this Z must equal { z , z } . According to the P -condition we must find a Z ′ ⊆ Z such that yS x Z . And { z } is such a Z ′ .Now let us see that we do not have an IL set M -frame. Put Y = { y , y } . Wehave wRxRyS w Y . So, if we do have an IL set M -frame then for some Y ′ ⊆ Y z y y y x S w S x w Figuur 1: An IL set P -frame which is not an IL set M -frame.we have xS w Y ′ and for all y ′ ∈ Y ′ we have that for all z , y ′ Rz implies xRz .But the only Y ′ ⊆ Y for which xS w Y ′ is Y itself. We have y ∈ Y , y Rz butnot xRz . ⊣ Theorem 3.10. IL P M .Bewijs. If IL P ⊢ M then M is valid on any IL set P -frame. But then any IL set P -frame is an IL set M -frame. Which, by Lemma 3.9 is not so. ⊣ Corollary 3.11. IL P R .Bewijs. By Theorem 3.10 and the fact that M follows from R . ⊣ Definition 3.12.
Let F = h W, R, S i be an IL set -frame. For any wRx we saythat Γ ⊆ W is a choice set for ( w, x ) iff. for all X such that xS w X , X ∩ Γ = ∅ . Definition 3.13.
Let F = h W, R, S i be an IL set -frame. We say that F is an IL set R -frame iff. wRxRyS w Y implies that for all choice sets Γ for ( x, y ) thereexists some Y ′ = Y ′ (Γ) ⊆ Y such that xS w Y ′ and for all y ′ ∈ Y ′ we have thatfor all z , y ′ Rz implies z ∈ Γ . Lemma 3.14. An IL set -frame F = h W, R, S i is an IL set R -frame iff. F | = R .Bewijs. ( ⇒ ) Suppose F is an IL set R -frame. Let F = h W, R, S, (cid:13) i be a modelbased on F . Choose w, x ∈ W and suppose wRx , w (cid:13) A ✄ B and x (cid:13) ¬ ( A ✄ C ).We have to find some Y ′ with xS w Y ′ and Y ′ (cid:13) B ∧ ✷ ¬ C . There exists some y ∈ W such that xRy , y (cid:13) A and for all U such that yS x U there exists some u ∈ U with u (cid:13) ¬ C . Let Γ be a choice set for ( x, y ) such that Γ (cid:13) ¬ C andΓ ⊆ S yS x U U . Since w (cid:13) A ✄ B we can find some Y such that yS w Y and10 (cid:13) B . By the R frame condition we can find some Y ′ ⊆ Y such that xS w Y ′ and for all y ′ ∈ Y ′ we have that for all z , y ′ Rz implies z ∈ Γ. So since Γ (cid:13) ¬ C we conclude Y ′ (cid:13) B ∧ ✷ ¬ C .( ⇐ ). Suppose F | = R . Let w, x, y, Y ∈ W and suppose wRxRyS w Y . Let Γbe a choice set for ( x, y ). Let p, q, s, t be distinct propositional variables. Definethe IL set -model F = h W, R, S, (cid:13) i as follows. v (cid:13) p ⇔ v = yv (cid:13) q ⇔ v ∈ Yv (cid:13) s ⇔ v ΓNow, w (cid:13) p ✄ q . So, w (cid:13) ¬ ( p ✄ s ) ✄ q ∧ ✷ ¬ s . Also, Γ (cid:13) ¬ s . So, x (cid:13) ¬ ( p ✄ s ) andtherefore there exists some Y ′ such that xS w Y ′ and Y ′ (cid:13) q ∧ ✷ s . Since Y ′ (cid:13) q we must have Y ′ ⊆ Y . Now let y ′ ∈ Y ′ and pick some z for which y ′ Rz . Then z (cid:13) ¬ s and thus by definition of (cid:13) , z ∈ Γ. ⊣ Theorem 3.15. IL P M R .Bewijs. Now that we have all the frame conditions at hand, we will provide aframe that is both a P and a M frame but not an R frame. We define therequired frame F as follows. F = h W, R, S i with W = { w, x, y, a , a , b , b } R and S are defined by the minimal requirements below: wRxRyS w AyS x B where A = { a , a } and B = { b , b } a i Rb i for i = 0 , xRb i for i = 0 , . We conclude the proof with a series of easy observations.1. F is an IL set M frame is clear: let Y ′ = Y = A in Definition 3.5.2. F is an IL set P frame is clear: let Z ′ = Z = B in Definition 3.7.3. F is not an IL R frame: Let Γ be a choice set for ( x, y ) that omits b . Asfor any Y ′ we have yS w Y ′ ⊆ Y implies Y ′ = Y , we see that a ∈ Y ′ . But a Rb and b / ∈ Γ. ⊣ We can also formulate a frame condition for W . However it shall turn outthat this frame condition becomes so intricate that it is not efficient to workwith even over finite frames. However, one can check that the exposed counterframes above are indeed also IL set W frames. We choose not to do so and rathergive direct proofs that include W in Section 4. We start by defining a higherorder property Not − W on frames. In this definition, capital letters shall rangeover subsets and lower case to elements of the domain. The index i is supposedto run over the natural numbers. 11 efinition 3.16. Not − W := ∃ w, z , { Y i } i ∈ ω , { y i } i ∈ ω, y i ∈ Y i , Z, { z i +1 } i ∈ ω, z i +1 ∈ Z [ ∀ i ∈ ω ( z i S w Y i ∋ y i Rz i + i ) & ∀ z ∈ Z ∃ i ∈ ωzS w Y i & ∀ z ∈ Z ∀ Y ( zS w Y ∧ Y ⊆ ( ∪ i ∈ ω Y i ) → ∃ z ′ ∈ Z ∃ y ∈ Y yRz ′ )] Lemma 3.17.
For any IL set frame F we have that F | = Not − W ⇔ F = W. Bewijs. ” ⇒ ”: Suppose Not − W holds. We use the same notation as in thedefinition and set out to define a valuation so that the instance p ✄ q → p ✄ q ∧ ✷ ¬ p of W fails.We define a (cid:13) p ⇔ a ∈ Za (cid:13) q ⇔ ∃ i ∈ ω a ∈ Y i Now clearly w (cid:13) p ✄ q as ∀ z ∈ Z ∃ i ∈ ωzS w Y i and p is only true at points in Z and the Y i make q true. However, w (cid:13) p ✄ q ∧ ✷ ¬ p can never hold. For, supposethat some Y and some z ∈ Z we have that zS w Y and Y (cid:13) q . By the definitionof (cid:13) clearly, Y ⊆ ( ∪ i ∈ ω Y i ), whence ∃ z ′ ∈ Z ∃ y ∈ Y yRz ′ ) and Y (cid:13) ✷ ¬ p as z ′ (cid:13) p .” ⇐ ”: Suppose that A ✄ B → A ✄ B ∧ ✷ ¬ A fails to hold in w in some model ba-sed on F . We will set out to find the required z , { Y i } i ∈ ω , { y i } i ∈ ω, y i ∈ Y i , Z, { z i +1 } i ∈ ω, z i +1 ∈ Z .As w (cid:13) ¬ ( A ✄ B ∧ ✷ ¬ A ), we can find z with wRz (cid:13) A such that for no Y with z S w Y we have Y (cid:13) B ∧ ✷ ¬ A . To find our other entities, we will need atechnical definition of R w,z of those those worlds that are reachable from w bymeans of R and S w successors. z ∈ R w,z ; y ∈ R w,z & yS w Y ⇒ Y ⊆ R w,z . Next, we define Z := { x ∈ R w,z | x (cid:13) A ∧ ( ∃ Y ∃ y ∈ Y ( z S w Y ∧ yRx ) ∨ x = z ) } and Y := { Y ⊆ R w,z | zS w Y (cid:13) B for some z ∈ Z } . Now it is easy to pick { Y i } i ∈ ω with Y i ∈ Y and to pick { z i } i ∈ ω ⊆ Z such that z i S w Y i ∋ y i Rz i +1 : as the z i (cid:13) A we can go via S w to some Y (cid:13) B whence bydefinition Y ∈ Y ; as Y (cid:13) ✷ ¬ A , at some R successor ˜ z of some y ∈ Y we have˜ z (cid:13) A whence ˜ z ∈ Z .By the definitions of Z and the Y i we have that ∀ z ∈ Z ∃ i ∈ ωzS w Y i . Thuswe only need to check that ∀ z ∈ Z ∀ Y ( zS w Y ∧ Y ⊆ ( ∪ i ∈ ω Y i ) → ∃ z ′ ∈ Z ∃ y ∈ Y yRz ′ ). But this is also not hard. If we consider any z ∈ Z and Y for which zS w Y ∧ Y ⊆ ( ∪ i ∈ ω Y i ), we see that Y (cid:13) B . But, as also z S w Y , we need to have Y (cid:13) B ∧ ✷ ¬ A whence we can find some y ∈ Y and z ′ ∈ Z with yRz ′ (cid:13) B . ⊣ Incompleteness of IL P W ∗ Let us first calculate the frame condition of R where R := A ✄ B → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C. It turns out to be the same frame condition as for P (see [18]). Lemma 4.1. F | = R ⇔ [ xRyRzS x uRv → zS y v ] Bewijs. “ ⇐ ” Suppose that at some world x (cid:13) A ✄ B . We are to show x (cid:13) ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C . Thus, if xRy (cid:13) ¬ ( A ✄ ¬ C ) we need to go via an S x to a u with u ⊢ B ∧ ✷ C .As y (cid:13) ¬ ( A ✄ ¬ C ), we can find z with yRz (cid:13) A . Now, by x (cid:13) A ✄ B , wecan find u with yS x u (cid:13) B . We shall now see that u (cid:13) B ∧ ✷ C . For, if uRv ,then by our assumption, zS y v , and by y (cid:13) ¬ ( A ✄ ¬ C ), we must have v ⊢ C .Thus, u (cid:13) B ∧ ✷ C and clearly yS x u .“ ⇒ ” We suppose that R holds. Now we consider arbitrary a, b, c, d and e with aRbRcS a dRe . For propositional variables p, q and r we define a valuation (cid:13) as follows. x (cid:13) p : ⇔ x = cx (cid:13) q : ⇔ x = dx (cid:13) r : ⇔ cS b x Clearly, a (cid:13) p ✄ q and b (cid:13) ¬ ( p ✄ ¬ r ). By R we conclude a (cid:13) ¬ ( p ✄ ¬ r ) ✄ q ∧ ✷ r .Thus, d (cid:13) q ∧ ✷ r which implies cS b e . ⊣ As P and R have the same frame condition we can never find an R -frame onwhich P fails to hold. However, the following elementary lemma tells us thatit is not necessary to work with frames. Lemma 4.2.
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 . ⊣ We can now prove the main theorem of this section.
Theorem 4.3. IL P W ∗ R Bewijs.
We consider the model M from Figure 2 and shall see that M | = IL P W ∗ but M, a (cid:13) R . By Lemma 4.2 we conclude that IL P W ∗ R .As M satisfies the frame condition for W ∗ , it is clear that M | = W ∗ . Weshall now see that M | = A ✄ ✸ B → ✷ ( A ✄ B ) for any formulas A and B .A formula ✷ ( A ✄ B ) can only be false at some world with at least twosuccessors. Thus, in M , we only need to consider the point a . So, supppose A ✄ ✸ B . For which x with aRx can we have x (cid:13) A ?As we have to be able to go via an S x -transition to a world where ✸ B holds,the only candidates for x are b, c and d . But clearly, c and f make true the samemodal formulas. From f it is impossible to go to a world where ✸ B holds.13 w S w ap, r qf g bc edqp, r M : Figuur 2: IL P W ∗ is incompleteThus, if a (cid:13) A ✄✸ B , the A can only hold at b or at d . But this automaticallyimplies that a (cid:13) ✷ ( A ✄ B ) and M | = P .It is not hard to see that a (cid:13) R . Clearly, a (cid:13) p ✄ q and b (cid:13) ¬ ( p ✄ ¬ r ).However, d (cid:13) q ∧ ✷ r and thus a (cid:13) ¬ ( p ✄ ¬ r ) ✄ q ∧ ✷ r . ⊣ The following lemma tells us that IL R is a proper extension of IL M P . Lemma 4.4. IL R ⊢ M , P Bewijs. As IL ⊢ ✸ A ∧ ✷ C → ¬ ( A ✄ ¬ C ) we get that A ✄ B → ✸ A ∧ ✷ C ✄ ¬ ( A ✄ ¬ C ) and M follows from R .The principle P follows directly from R by taking C = ¬ B . ⊣ We can consider the principle R ∗ that can be seen, in a sense, as the unionof W and R . R ∗ : A ✄ B → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C ∧ ✷ ¬ A Lemma 4.5. IL RW = IL R ∗ Bewijs. ⊇ : A ✄ B → A ✄ B ∧ ✷ ¬ A → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C ∧ ✷ ¬ A . ⊆ : A ✄ B → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C ∧ ✷ ¬ A ✄ B ∧ ✷ C ; and if A ✄ B , then A ✄ B ✄ (( B ∧ ✷ ¬ A ) ∨ ✸ A ) ✄ B ∧ ✷ ¬ A , as A ✄ B → ¬ ( A ✄ ⊥ ) ✄ B ∧ ✷ ⊤∧ ✷ ¬ A . ⊣ Arithmetical soundness of R Let us first recall Definition 2.7, that is, the definition of the interpretabilitylogic of all reasonable arithmetical theories. We shall write IL (All). We defined IL (All) to be the set of modal formulas that are interpretability principles inany reasonable arithmetical theory. That is, the set of ϕ for which ∀ T ∀ ∗ T ⊢ ϕ ∗ . In [28] IL (All) was conjectured to be IL W . In [30] this conjecture wasfalsified and strengthened to a new conjecture. There it was conjectured that IL W ∗ , which is a proper extension of IL W , is IL (All).In [18] it was proved that the logic IL W ∗ P is a proper extension of IL W ∗ ,and that IL W ∗ P is a subsystem of IL (All). This falsified the conjecture from[30]. In [18] it is also conjectured that IL W ∗ P is not the same as IL (All).In [19] it is conjectured that IL W ∗ P = IL (All). As we will see below we havethat the logic IL RW is a subsystem of IL (All) and a proper extension of IL W ∗ P .This rejects the conjecture pronounced in [19]. With all this conjecturing andrefuting of conjectures we are rather hesitant in proposing as a new conjecturethat IL RW = IL (All). We shall now give the proof that the new principle R is arithmetically va-lid in all reasonable theories. In the proof we shall employ some well-knownarithmetical facts. We will now first briefly summarize these facts. Definition 5.1.
A definable T -cut is a formula I ( x ) with one free variable,such that T ⊢ I (0) ∧ ∀ x ( I ( x ) → I ( x + 1)) . Cut ( · ) will denote the function thatassigns to the code of a formula ϕ , the code of the formula expressing that ϕ isa cut, that is, ϕ (0) ∧ ∀ x ( ϕ ( x ) → ϕ ( x + 1)) (whenever ϕ is of the right format). The function
Cut ( · ) is a very easy function. It is certainly provably total in I ∆ + Ω . In this section we shall denote the translation of a formula ϕ underan interpretation j by j ( ϕ ). If I is a cut and ϕ a formula, we shall by ϕ I denotethe formula ϕ , where all the quantifiers in ϕ are relativized to the cut I . Thefollowing lemma is mentioned (as an exercise) in [23]. It is central to manyarguments in the field of formalized interpretability. Lemma 5.2 (Pudl´ak) . There exists a function f , provably total in I ∆ + Ω ,such that for any reasonable arithmetical theory T , the following holds. T ⊢ j : α ✄ T β → [ ✷ T Cut ( f ( j )) ∧ ∀ σ ∈ Σ ! j : α ∧ σ f ( j ) ✄ T β ∧ σ ]Another fact from arithmetic that we shall need, is that we can performthe Henkin construction using numbers from a cut. This is expressed by thefollowing lemma. Lemma 5.3.
For any reasonable arithmetical theory T we have that T ⊢ ✷ T ( Cut ( I )) → ✸ IT α ✄ T α. In fact, we have strong evidence that actually IL RW = IL (All). R . Note that the j , I , α and β in Lemma 5.2 en 5.3 are parametersand hence could be universally quantified within the theory. Theorem 5.4 (Soundness of R ) . For any reasonable arithmetical theory T wehave the following. T ⊢ α ✄ β → ¬ ( α ✄ ¬ γ ) ✄ β ∧ ✷ γ Bewijs.
Let f denote the function from Lemma 5.2. To prove our theorem, wereason in T and assume α ✄ β . Thus, for some interpretation j we have j : α ✄ β .We now claim that ¬ ( α ✄ ¬ γ ) → ✸ ( α ∧ ✷ f ( j ) γ ) . (+)Let us first see that this claim, indeed entails the result. ¬ ( α ✄ ¬ γ ) ✄ By (+) ✸ ( α ∧ ✷ f ( j ) γ ) ✄ By J5 α ∧ ✷ f ( j ) γ ✄ By Lemma 5.2 and j : α ✄ ββ ∧ ✷ γ Thus, now we only need to prove the claim. We will prove (+) by showing thelogical equivalent ✷ ( α → ✸ f ( j ) ¬ γ ) → α ✄ ¬ γ. (++)We reason as follows. ✷ ( α → ✸ f ( j ) ¬ γ ) → By J1 α ✄ ✸ f ( j ) ¬ γ → By Lemma 5.3 and J2 α ✄ ¬ γ ⊣ Referenties [1] C. Areces, D. de Jongh, and E. Hoogland. The interpolation theorem for IL and ILP . In
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