aa r X i v : . [ m a t h . L O ] J a n CYCLIC HENKIN LOGIC
ALBERT VISSER
Abstract.
In this paper, we study Cyclic Henkin Logic
CHL , a logic thatcan be described as provability logic without the third L¨ob condition, to wit,that provable implies provably provable (aka principle 4). The logic
CHL doeshave full modalised fixed points. We implement these fixed points using cyclicsyntax, so that we can work just with the usual repertoire of connectives.The main part of the paper is devoted to developing the logic on cyclicsyntax. Many theorems, like the multiple fixed point theorem, become matterof course in this context. We submit that the use of cyclic syntax is of interesteven for the study of classical L¨ob’s Logic. We show that a version of thede Jongh-Sambin algorithm can be seen as one half of a synonymy between GL ◦ , i.e. CHL plus the third L¨ob Condition, and ordinary L¨ob’s Logic GL .Our development illustrates that an appropriate computation scheme for thealgorithm is guard recursion.We show how arithmetical interpretations work for the cyclic syntax. Inan appendix, we give some further information about the arithmetical side ofthe equation. Introduction
In the present paper, we study Cyclic Henkin Logic
CHL . Our original interestin this logic was triggered by the development, in our paper [Vis19], of a class ofprovability predicates for which we do have L¨ob’s Rule but for which the thirdL¨ob condition, to wit, provable implies provably provable (aka principle 4), fails.It turns out that, in this context, the de Jongh-Sambin-Bernardi Theorem aboutthe uniqueness of modalised fixed points does hold. Thus, a substantial amountof familiar reasoning from L¨ob’s Logic GL is still present. On the other hand, inthe absence of the third L¨ob condition, the de Jongh-Sambin Theorem about theexplicit definability of fixed points fails. In fact, the most salient fixed point ofthem all, the G¨odel fixed point, does not have an explicit definition.So, what is the appropriate form for an appropriately weakened version of L¨ob’sLogic? Our intuition is that the logic in question should have full modalised fixedpoints. Given that design choice, in the light of non-explicit definability, thereseem to be essentially three roads to follow: (i) we try to extend the repertoire ofordinary modal operators so that we do recover explicit definability for the enriched Date : January 28, 2021.2000
Mathematics Subject Classification.
Key words and phrases.
Provability Logic, Fixed Points, Cyclic Syntax, Second IncompletenessTheorem, Formal Theories.I am grateful to Johan van Benthem and to the participants of the Implicit Seminar Warsaw,the Proof Theory Virtual Seminar, and the Logic Online Seminar Moscow for their comments onand questions about various talks I gave on the subject of this paper or related matters. I thankTadeusz Litak with whom I am writing a parallel paper. Without our shared project the presentpaper would not be possible. repertoire; (ii) we add a variable-binding fixed point operator ̥ p.ϕ that is onlyallowed when ϕ is modalised in p ; (iii) we work with cyclic syntax. We have notlooked seriously at option (i), but it very well possible that it is not feasible in areasonable way. Option (ii) will be worked out in a forthcoming paper by TadeuszLitak and the author. In the present paper we study option (iii).We think that the cyclic syntax has a wider interest than just the study of aweaker version of L¨ob’s Logic. It also gives us a new way to look at L¨ob’s Logicitself. We will show, for example, that a version of the de Jongh-Sambin algorithmto compute explicit fixed points can be viewed as the specification of one half of asynonymy between GL ◦ , i.e. CHL plus the third L¨ob condition, and GL in its originalformulation. The specification of the algorithm shows that guard recursion is anappropriate way of thinking about this algorithm.The logic CHL is synonymous with a corresponding theory, the Henkin Calculusor HC , which is formulated in a language with a variable-binding fixed point opera-tor. This will be proved in a forth-coming paper by Tadeusz Litak and the author.In that same paper, we will prove that HC , and, thus, also CHL , is synonymous withthe well-founded part of the µ -calculus, i.e., the µ -calculus plus the minimal Henkinsentence µp. p . This last result is based on ideas from Johan van Benthem’s paper[VB06], which were extended in a paper by the author [Vis05]. Thus, CHL can beviewed as a treatment of the well-founded part of the µ -calculus on a quite differentsyntax.An obvious further step in the project of provability logic on cyclic syntax is toconnect circular syntax with circular proofs as studied in, e.g., [Sha14] and [Sha20].We have not explored this attractive possibility yet.1.1. Plan of the Paper.
A substantial part of the paper is devoted to carefullydeveloping the system. We develop the syntax and introduce the appropriate princi-ples of definition and proof concerning the syntax in Section 2. Then, we introduce
CHL and work out the basic facts about the logic in Section 3. In Section 4, we provethe synonymy between GL ◦ and GL . We study some further inter-theory relationsin Section 5. Section 6 contains our development of arithmetical interpretations.Finally, Appendix A provides a somewhat closer look at the arithmetical side ofthe equation.1.2. Prerequisites.
Some knowledge of basic provability logic is helpful. Theclassical textbooks [Boo93] or [Smo85] are quite sufficient. However, there are manyother good expositions available nowadays, like [Lin96], [JdJ98], [ˇSve00], [AB04].It would also be good if the reader has at least seen the notion of bisimulation.Any modern introduction to modal logic or to computer science will explain thisnotion. The books [BE87] and [Acz88] also introduce the notion and also containsome material closely connected to the present paper.The appendix contains some more advanced material on arithmetic, but the restof the paper is independent of this. 2.
Syntax
In this Section we provide the basics of our cyclic syntax. A major inspirationof our approach is the treatment of the paradoxes in the book [BE87].In Subsection 2.1, we specify what kind of graphs we employ in our development.Then, in Subsection 2.2, we introduce the relevant notion of formula and develop
YCLIC HENKIN LOGIC 3 some basic proof methods and definition methods. Also, we prove a number ofbasic facts.2.1.
Graphs.
Let a non-empty set of labels L be fixed. We have an arity function ar : L → ω .We need the following notion of graph: a directed pointed labeled graph withordered successors. A graph G is given as a quadrupel h V, r, S, λ i . Here: • V is the set of vertices or nodes. In our paper this set will always be finite. • r ∈ V is the point or root. • λ : V → L is the labeling function. A vertex a with label a will be called an occurrence of a . • S : V → V ∗ , where V ∗ is the set of (finite) sequences of elements in V (including the empty sequence). We demand that length ( Sa ) = ar ( λa ) (in G ).We write S i ( a ) for ( S ( a )) i , where i < length ( Sa ). • Let ˆ S be the relation given by a ˆ Sb iff there is an i < length ( Sa ) such that b = S i a . A path in the a graph is a sequence of vertices a ˆ Sa ˆ S . . . ˆ Sa k .We demand that every vertex can be reached via a finite path from r .For many purposes the notion of path as defined here is sufficient. However itis also good to have the notion of directive path . To motivate this consider a graphwith nodes a and b and suppose Sa = h b, b i and Sb = h a i . We can have a path aba that takes the left turn and one that takes the right turn. To distinguish suchpossibilities, we define a directive path as a sequence a i . . . a k − i k − a k , where i j < ar ( λa j ) and a j +1 = S i j a j . Remark 2.1.
We opted for the present format for graphs since this is in accordancewith the representation as co-algebra. Of course other formats are possible. ❍ A bisimulation between graphs G and G ′ is a relation R between V and V ′ , suchthat:i. If aRa ′ , then λa = λ ′ a ′ .ii. If aRa ′ , then Sa and S ′ a ′ are sequences with the same length ℓ and we have S i a R S ′ i a ′ , for all i < ℓ .Two graphs are bisimilar if there is a bisimulation between them that relates theirroots. We note that is follows that the bisimulation is total and surjective. We write ≃ for bisimilarity. We remind the reader of the well-known fact that bisimulationsare closed under unions. Thus, there is a maximal bisimulation between any twographs. Of course, this does not need to relate the roots.An isomorphism between graphs is a bijective bisimulation that relates the roots.We write ∼ = for isomorphism. We will think of our graphs modulo isomorphism.Philosophically, we want to think about the graphs modulo bisimulation, however,it is technically convenient to have the more ‘concrete’ representations moduloisomorphism available. Remark 2.2.
If we would allow infinite graphs, we could define the canonicalunraveling of G as the graph G ′ with as domain the directive paths in G . Thenew successor and labeling functions are as expected. One can then show that twographs are bisimilar iff their canonical unravelings are isomorphic. ❍ Rooted graphs are our default. On occasion we will also consider unrooted graphs .Of course these are just graphs minus the root. We will allow such graphs to contain
ALBERT VISSER disconnected parts. The definition of bisimulation remains the same without thecondition for the roots. In stead we demand the relation to be total and surjective.A cycle in a graph is a set of vertices C such that we can arrange the elementsof C in a path a ˆ Sa ˆ S . . . ˆ Sa k − ˆ Sa . We demand that the a j are pairwise distinct.Note that in our definition a cycle has no designated starting point. A vertex is a cycle vertex if it is on a cycle.We will write c ( G ) for the number of cycles in G . Remark 2.3.
There is also the notion of directive cycle . A directive cycle is afunction γ from a set of vertices C to numbers, such that γ ( a ) < ar ( λa ). We demandthat we can arrange the elements of C in a directive path a i a . . . a k i k − a , where i j = γ ( a j ). Here the a j are pairwise distinct. ❍ We define a number of operations on graphs. • g ( a, G , . . . , G k − ) is the result of taking the disjoint sum of the G , . . . , G k − and adding a fresh root r with label a to this sum. We allow that k = 0here. • G ↓ a =: G ′ is subgraph of G generated by a . It is defined as follows. ◦ V ′ is the set of vertices that can be reached via a (possibly empty)path from a . ◦ r ′ := a . ◦ λ ′ is λ restricted to V ′ . ◦ S ′ is S restricted to V ′ . • Suppose the length of Sr is n and i < n . Then su i ( G ) := G ↓ S i r . • min ( G ) is the result of dividing out the maximal auto-bisimulation of G .We have the following obvious lemmas. Lemma 2.4.
Suppose G ≃ G via R and a Ra . Then G ↓ a ≃ G ↓ a via therestriction of R to the nodes of G ↓ a and G ↓ a . Lemma 2.5. min ( G ) is bisimulation minimal: all bisimulations on it are subsetsof the identity relation on V . Moreover, G ≃ G ′ iff min ( G ) ∼ = min ( G ′ ) . Consider a graph G . Let W ⊆ V . We say that W is a guard (for G ) if everycycle contains an element of W .Thus, we have guard-induction and guard-recursion in a guarded graph: Lemma 2.6.
Suppose W is a guard of G . We have:i. Suppose we have a property P of vertices such that all g ∈ W have P . Supposefurther that if a W and all a ’s successors have P , then a has P . Then allvertices have P .ii. Suppose for every label a , we have a function G a : D ar ( a ) → D and suppose F : W → D . Then there is a unique function H : V → D such that H ( a ) = F ( a ) if a ∈ W and H ( a ) = G a ( H ( S a ) , . . . , H ( S k − a )) if a W and λa = a and k = ar ( a ) .Proof. Ad (i): Consider any a in G . Let B be the union of W with the set of leavesof G . Consider any vertex a . Clearly in G ↓ a , the intersection C of B with thevertices in G ↓ a forms a bar, i.e. every indefinitely prolonged path must eventuallypass through an element of C . So by bar-induction, the root a has property P .The proof of (ii) is similar using bar recursion. ❑ YCLIC HENKIN LOGIC 5
Suppose W is a guard for G and W ′ is a guard for G ′ . A bisimulation R between G and G ′ is a W, W ′ -bisimulation if whenever aRa ′ , then a ∈ W iff a ′ ∈ W ′ . Lemma 2.7.
Suppose for every label a , we have a function G a : D ar ( a ) → D . Let G be a gard for G and let G ′ be a guard for G ′ . Suppose R is a G, G ′ -bisimulationbetween G and G ′ . ( R does not need to be root-preserving.) Consider F : G → D and F ′ : G ′ → D such that whenever a ∈ W and aRa ′ , we have F ( a ) = F ′ ( a ′ ) . Let H and H ′ be the functions guaranteed to exist by Lemma 2.6. Then, for all a ∈ V and a ′ ∈ V ′ , if aRa ′ , then H ( a ) = H ( a ′ ) .Proof. Let P ( a ) be the property: for all a ′ if aRa ′ , then H ( a ) = H ( a ′ ). Clearly wehave P on W . Moreover, it is easy to see that P is preserved from the successorsof a to a . ❑ Formulas.
We define our formulas. The totality of formulas will constituteour full language L ◦ .The set of labels L for formulas is given by ⊥ , ⊤ , ¬ , , ∧ , ∨ , → , p , p , . . . ,where ar ( ⊤ ) = ar ( ⊥ ) = ar ( p i ) = 0, ar ( ¬ ) = ar ( ) = 1, ar ( ∧ ) = ar ( ∨ ) = ar ( → ) = 2.As usual we use also p, q, r, r ′ , . . . for propositional variables. A formula ϕ is agraph for which bo ( ϕ ), the set of -occurrences, is a guard. We note that the setof -occurrences on a cycle bo ◦ ( ϕ ) also forms a guard.If the label of the root is not a variable, we call it the main connective . We saythat ϕ is modalised in p if every path from the root to an occurrence of p containsa -occurrence. Remark 2.8.
We can very well model our formulas in the hereditarily finite setsof non-well-founded set theory
AFA with the labels as ur-elements. Only the guard-edness condition is perhaps somewhat unnatural in this context. ❍ We define some operations on formulas: • We identify ⊤ with g ( ⊤ ) and, similarly, for ⊥ and the p i . • ¬ ϕ := g ( ¬ , ϕ ) and, similarly, for . • ( ϕ ∧ ψ ) := g ( ∧ , ϕ, ψ ) and, similarly, for ∨ and → . • ̥ p.ϕ is the result of identifying the root with all vertices labeled p , whereone keeps the label of the root. This is only allowed when ϕ is modalised in p , since otherwise the resulting graph will not be a formula. It is easily seenthat in the resulting graph all cycles contain a -occurrence, as desired. • ϕ [ p : ψ , . . . , p k − : ψ k − ] is the result of the following operation. First, weform the disjoint union of ϕ and the ψ i . Then, simultaneously, we identifythe vertices labeled p i in (the disjoint copy of) ϕ with the root of (thedisjoint copy of) ψ i , where we keep the label of the root of ψ i and discardthe label p i . Remark 2.9.
Suppose ϕ is modalised in p and q . Then ̥ p. ̥ q.ϕ ∼ = ̥ p.ϕ [ q : p ].Thus, we see that one of the costs of the graph approach is that this principle isbuilt in. Of course, one may also consider it as a bonus. ❍ The operations are safe for bisimulation: The convenient confusion between labels and operations has its limits should be treated withsome care. We will warn the reader when to tread carefully.
ALBERT VISSER
Lemma 2.10.
Bisimilarity between formulas is a congruence relation for the op-erations in the above list.Proof.
We just do the case of ̥ p . Suppose ϕ is bisimilar to ϕ ′ . Let R be the wit-nessing bisimulation. We define R ′ between ̥ p.ϕ and ̥ p.ϕ ′ simply as R restrictedto the vertices not labeled by p in ϕ . We claim that R ′ is a root-preserving bisim-ulation. Consider any a, a ′ with aR ′ a ′ . Consider Sa in ̥ p.ϕ . The only differencewith Sa in ϕ is that all vertices labeled p are now replaced by the root with the labelof the root. Similarly, for S ′ a ′ in ̥ p.ϕ ′ . However, since R was a root-preservingbisimulation we see that now R ′ is. ❑ We will sometimes write σ , τ , . . . , for substitutions. If the substitution σ is[ q : ψ , . . . , q k − : ψ k − ] and τ is [ r : χ , . . . , r m − : χ m − ], then σ ⋆ τ := [ q : ψ , . . . , q k − : ψ k − , r : χ , . . . , r m − : χ m − ] . This only makes sense if the q i and the r j are pairwise disjoint. We note animportant insight. Lemma 2.11. i. ⋆ is associative, assuming that the three domains of the substi-tutions are pairwise disjoint.ii. Suppose the q i are disjoint from the domain of τ . Then, ϕ [ q : ψ , . . . , q k − : ψ k − ] τ ∼ = ϕ ([ q : ψ τ, . . . q k − : ψ k − τ ] ⋆ τ ) . Suppose p does not occur in ϕ . We define the following operation: • ( ϕ p ) =: ϕ ′ is obtained as follows. If the root of ϕ is not on a cycle ϕ ′ := ϕ . Otherwise, we take an r ⋆ that is not in V , the set of vertices of ϕ . Let V ′ := V ∪ { r ⋆ } and λ ′ := λ ∪ {h r ⋆ , p i} . We define f : V → V ′ by f ( r ) = r ⋆ and f ( a ) = a , if a = r . As usual, we write f h a , . . . , a k − i for h f a , . . . , f a n − i . We take S ′ a := f Sa if a = r ⋆ and Sr ⋆ := ε .So, ( ϕ p ) is the result of redirecting all incoming arrows of the root, if there areany, to a new vertex labeled p . The new vertex, of course, does not have outgoingarrows. We note that, whether p occurs in ϕ or not, ( ϕ p ) is modalised in p .A good heuristic, in case the root is on a cycle, is to view ( ϕ p ) as a non-deterministic sub-formula of ϕ . The number of nodes increases, so in the sense this‘subformula’ is larger than the original formula. On the other hand, trivially, thenumber of cycles decreases, so in that sense the ‘subformula’ is smaller. This lastfeature will be quite useful in the paper.It will be convenient to write ( ϕ ψ ) for ( ϕ p )[ p : ψ ].We enumerate some useful facts about the operation ( · · ). Lemma 2.12.
Suppose the root of ϕ is a cycle vertex and p does not occur in ϕ .Suppose p is modalised in ψ . Then,i. c ( ϕ p ) < c ( ϕ ) .ii. ̥ p. ( ϕ p ) ∼ = ϕ .iii. ( ϕ ϕ ) ≃ ϕ .iv. ̥ p.ψ ≃ ψ [ p : ̥ p.ψ ] . Remark 2.13.
The operation ( · · ) has to be treated with great care since wemay have ϕ ≃ ϕ ′ but ( ϕ p ) ( ϕ ′ p ). This can be easily seen from the fact thatwe can always unravel a formula a bit to a bisimilar one of which the root is noton a cycle. ❍ YCLIC HENKIN LOGIC 7
Interestingly, there is something like uniqueness of fixed points modulo bisimu-lation, which gives an intriguing analogy with the de Jongh-Sambin-Bernardi The-orem.
Theorem 2.14.
Suppose p is modalised in ϕ and ψ ≃ ϕ [ p : ψ ] . Then ψ ≃ ̥ p.ϕ .Proof. Let E be the embedding of the vertices of ϕ that are not occurrences of p into ϕ [ p : ψ ]. Let F be the embedding of the vertices of ψ into the vertices ofthe substituted copy of ψ in ϕ [ p : ψ ]. We note that F is a (non-root-preserving)bisimulation between ψ and ϕ [ p : ψ ].Let R be the maximal bisimulation between ϕ [ p : ψ ] and ψ . We claim that R ⋆ := E ; R is a root-preserving bisimulation between ̥ p.ϕ and ψ . (Here ‘;’ iscomposition in the order of reading.)Clearly R ⋆ connects the roots of ̥ p.ϕ and ψ (since the root of ϕ cannot be a p -occurrence). Consider any node a of ̥ p.ϕ . Then a is, by definition, a non- p -occurrence in ϕ . Suppose aR ⋆ b , say aEcRb . Let a ′ := S ̥ p.ϕ,i a and b ′ := S ϕ [ p : ψ ] ,i b , c ′ := S ψ,i c , d ′ := S ϕ,i a Case 1: Suppose d ′ is not a p -occurrence in ϕ . In this case a ′ = d ′ Ec ′ . Finally,because cRb it follows that c ′ Rb ′ and, hence, a ′ R ⋆ b ′ .Case 2: Suppose d ′ is a p -occurrence in ϕ . In this case a ′ = r ϕ Er ϕ [ p : ψ ] . Moreover, c ′ is the root of the substituted copy of ψ in ϕ [ p : ψ ] and c ′ Rb ′ . We have r ϕ [ p : ψ ] Rr ψ F c ′ Rb ′ . Since, R ; F ; R is a bisumulation, it is contained in R , so r ϕ [ p : ψ ] Rb ′ . It follows that a ′ Er ϕ [ p : ψ ] Rb ′ , i.o.w., a ′ R ⋆ b ′ .We may conclude that ̥ p.ϕ ≃ ψ . ❑ We note that the above proof does not use the full guard condition. It just usesthat the root is not a p -occurrence.The fixed point theorem as given in Lemma 2.12 has a easy generalisation tosystems of equations. Suppose E is a system of equations, i.e., a function froma finite set Q of variables to formulas ϕ q . We form a directed (unlabeled) graph(without ordered successors) G E with domain Q where we have an arrow from q to q ′ precisely if ϕ q is not modalised in q ′ . We say that E is modalised if G E is acyclic.This condition generalises the usual one: ϕ is modalised in p iff the equation p ϕ is modalised.We want to solve E . This means that we want to find a function F : q ψ q on Q , such that ψ q ≃ ϕ q F , for all q ∈ Q , where in the right-hand-side we view F as asubstitution. We demand that the q ′ ∈ Q do not occur in the ψ q .We define a new unrooted graph Ψ as follows. We define a sub-graph G ◦E of G E as follows. We have an arrow from q to q ′ iff λ ϕ q ( r ϕ q ) = q ′ . This new graph isclearly non-cyclic and every q in Q has at most one outgoing arrow. Let end ( q ) bethe variable at the end of the unique outgoing path from q .We have the following definitions. • V Ψ is the set of all pairs h ϕ q , a i , where q ∈ Q and a ∈ V ϕ q and λ ϕ q ( a ) Q . • λ Ψ ( h ϕ q , a i ) := λ ϕ q ( a ). • Suppose λ ϕ q ( a ) = q ′ ∈ Q . Then, idfy ( h ϕ q , a i ) := h ϕ end ( q ′ ) , r ϕ end ( q ′ ) i . In allother cases, idfy ( h ϕ q , a i ) := h ϕ q , a i . We note that the label of a value of idfy cannot be in Q . • Suppose S ϕ q a = h b , . . . , b n − i . Then, S Ψ h ϕ q , a i := h idfy ( h ϕ q , b i ) , . . . , idfy ( h ϕ q , b n − i ) i . ALBERT VISSER • r q := idfy ( h ϕ q , r ϕ q i ). In a sense the r q are multiple roots of Ψ. • ̥ E := F , where q F := ψ q := Ψ ↓ r q . We write ̥ q E for q ̥ E . Theorem 2.15. ̥ E is a solution of E .Proof. Let ϕ q , ψ q and F be as above. Without loss of generality we may assumethat the vertices of Ψ are disjoint from the vertices of the ϕ q . We define a bisimu-lation between ψ q and ϕ q F as follows: αRβ iff α = β or ( α = h ϕ q , a i and β = a ).We note that, in case ϕ q = q ′ ∈ Q , the second disjunct cannot become active.In this case, ψ q and ϕ q F will be identical.Suppose that ϕ q is not in Q . We consider the case where α = h ϕ q , a i and β = a .We need an auxiliary definition. Suppose λ ϕ q ( a ) = q ′ ∈ Q . Then, idfy ( a ) := h ϕ end ( q ′ ) , r ϕ end ( q ′ ) i . In all other cases, idfy ( a ) := a . We lift the relation R tosequences in the obvious way.Let S ϕ q a = h b , . . . , b n − i . Then, S ψ q ( α ) = h idfy ( h ϕ q , b i ) , . . . , idfy ( h ϕ q , b n − i ) i R h idfy ( b ) , . . . , idfy ( b n − ) i = S ϕ q F a ❑ Cyclic Henkin Logic
In this section we develop the logic
CHL . We choose to develop it for itself and notas part of a wider class of logics, even if, from a systematic standpoint, that would bebetter. The reason is simply the desire not to overburden the presentation. We willtouch on the broader perspective in Section 4 and, specifically, in Subsection 4.1.Subsection 3.1 provides the basic development of
CHL . Curiously, the central re-sult of the subsection is inter-substitution of equivalents, Theorem 3.6. All furtherdevelopment rests on this central result. In Subsection 3.2, we consider some alter-native axiomatisations. Finally, in Subsection 3.3, we provide the Kripke semanticsfor
CHL .3.1.
Basic Development.
Cyclic Henkin Logic or
CHL is axiomatised as follows. chl
1. If ⊢ ϕ and ⊢ ϕ → ψ , then ⊢ ψ . chl
2. If ⊢ ϕ , then ⊢ ϕ . chl
3. We have all substitution instances of propositional tautologies. Here we thinkof these tautologies as given by the usual parse trees. chl ⊢ ( ϕ → ψ ) → ( ϕ → ψ ). chl
5. Suppose ϕ ≃ ψ . Then ⊢ ϕ ↔ ψ . chl
6. If ⊢ ϕ → ϕ , then ⊢ ϕ .We note that if we think of our formulas as the result of dividing out bisimularity,then Axiom Scheme chl K . L¨ob’s Rule provides thissystem with bite. We remind the reader that ordinary K is closed under L¨ob’s Rule,so it is the presence of circularity that makes the rule powerful.We will use the notation Γ ⊢ CHL ϕ , where Γ is a finite set of formulas for: CHL ⊢ V Γ → ϕ . Trivially, we have the deduction rule for this notion.The following theorem tells us that CHL is indeed a logic.
YCLIC HENKIN LOGIC 9
Theorem 3.1. If CHL ⊢ ϕ , then CHL ⊢ ϕ [ p : ψ , . . . , p k − : ψ k − ] .Proof. By a simple induction on proofs. In the case of chl
5, this uses the safety ofsubstitution. ❑ Theorem 3.2. i. CHL ⊢ ϕ ↔ ( ϕ ϕ ) .ii. CHL ⊢ ̥ p.ψ ↔ ψ [ p : ̥ p.ψ ] , assuming that ψ is modalised in p .Proof. This is immediate by the fact that ϕ ≃ ( ϕ ϕ ) and ̥ p.ψ ≃ ϕ [ p : ̥ p.ψ ]. ❑ We define • ϕ := ̥ p. ( ϕ ∧ p ), where p does not occur in ϕ . Theorem 3.3.
We have:i. If
CHL ⊢ ϕ , then CHL ⊢ • ϕ .ii. If CHL ⊢ ϕ , then CHL ⊢ • ϕ . ( Necessitation for • ) iii. If CHL ⊢ • ϕ → ϕ , then CHL ⊢ ϕ . ( LR for • ) Proof.
We verify (i). Suppose
CHL ⊢ ϕ . We have: • ϕ ⊢ CHL ( ϕ ∧ • ϕ ) ⊢ CHL • ϕ Hence, by L¨ob’s rule, ⊢ CHL • ϕ .(ii) is immediate from (i).We verify (iii). Suppose • ϕ ⊢ CHL ϕ . Then, ( ϕ ∧ • ϕ ) ⊢ CHL ϕ ∧ • ϕ . By LR ,we find CHL ⊢ ϕ . ❑ Theorem 3.4. • satisfies L¨ob’s Logic GL over CHL .Proof.
We have L1 , i.e. necessitation, by Theorem 3.3(ii). We verify L2 . ( • ( ϕ → ψ ) → ( • ϕ → • ψ )) ⊢ CHL • ( ϕ → ψ ) → (( ϕ → ψ ) ∧ • ( ϕ → ψ )) → (( ϕ → ψ ) ∧ ( • ϕ → • ψ )) → (( ϕ ∧ • ϕ ) → ( ψ ∧ • ψ )) → ( ( ϕ ∧ • ϕ ) → ( ψ ∧ • ψ )) → ( • ϕ → • ψ ) So, by L¨ob’s Rule, we are done. We verify L3 .( • ϕ → • • ϕ ) ⊢ CHL • ϕ → • ϕ → ( • ϕ ∧ • • ϕ ) → • • ϕ By L¨ob’s Rule we are done. We verify L¨ob’s Principle L4 . We have: • ( • ( • ϕ → ϕ ) → • ϕ ) ⊢ CHL • ( • ϕ → ϕ ) → • • ( • ϕ → ϕ ) → • • ϕ → • ϕ By LR for • (Theorem 3.3(iii)), we are done. ❑ We write ϕ for ϕ ∧ ϕ and, similarly for • . Corollary 3.5.
We have strengthened L¨ob’s Rule for • over CHL , i.e.,if ^ i Proof. Suppose V i Theorem 3.6. Suppose that the variables s i , for i < n and r j for j < m arepairwise distinct; and that ϕ is modalised in the r i . Then, we have: ^ i Suppose a formula ψ and substitutions σ and τ on q , . . . , q k − are given.We consider a conjunction α = V i Suppose a is a box-occurrence on a cycle, i.e., a is in the chosen guard. Let ϕ be ϕ ↓ a . Suppose α is acceptable for ϕ , σ, τ . We choose p not in ϕ , nor in the ψ i , χ i , θ j and ρ j and distinct from the q i and r j . We have: ϕ σ ≃ ( ϕ p )[ p : ϕ ] σ ≃ ( ϕ p )( σ ⋆ [ p : ϕ σ ]) . Similarly, ϕ τ ≃ ( ϕ p )( τ ⋆ [ p : ϕ τ ]).Since c ( ϕ p ) < c ( ϕ ) ≤ c ( ϕ ), we may apply the main induction hypothesis.Since ( ϕ p ) is modalised in p , we find that α ∧ • ( ϕ σ ↔ ϕ τ ) is acceptable for( ϕ p ) , σ ⋆ [ p : ϕ σ ] , τ ⋆ [ p : ϕ τ ]. Thus, by the main induction hypothesis: α, • ( ϕ σ ↔ ϕ τ ) ⊢ CHL ϕ σ ↔ ( ϕ p )( σ ⋆ [ p : ϕ σ ]) ↔ ( ϕ p )( τ ⋆ [ p : ϕ τ ]) ↔ ϕ τ By the strengthend L¨ob’s Rule, we may omit the assumption • ( ϕ σ ↔ ϕ τ ) andwe are done. ❑ Theorem 3.7. Suppose ϕ and ψ are modalised in p and CHL ⊢ ϕ ↔ ψ , then CHL ⊢ ̥ p.ϕ ↔ ̥ p.ψ .Proof. Suppose CHL ⊢ ϕ ↔ ψ . Then, • ( ̥ p.ϕ ↔ ̥ p.ψ ) ⊢ CHL ̥ p.ϕ ↔ ϕ [ p : ̥ p.ϕ ] ↔ ψ [ p : ̥ p.ϕ ] ↔ ψ [ p : ̥ p.ψ ] ↔ ̥ p.ψ So, by L¨ob’s rule, we are done. ❑ We prove the de Jong-Sambin-Bernardi Theorem about the uniqueness of fixedpoints. Theorem 3.8. Suppose ϕ is modalised in p . Then CHL ⊢ • ( p ↔ ϕ ) → ( p ↔ ̥ p.ϕ ) . Proof. We have: • ( p ↔ ϕ ) ⊢ CHL • ( p ↔ ̥ p.ϕ ) → ( p ↔ ϕ ↔ ϕ [ p : ̥ p.ϕ ] ↔ ̥ p.ϕ )By the Strengthened L¨ob’s Rule, we may omit the assumption • ( p ↔ ̥ p.ϕ ). ❑ We generalise the de Jong-Sambin-Bernardi Theorem to systems of equations asfollows. Theorem 3.9. Suppose E is modalised. Then, ^ q ∈ Q • ( q ↔ q E ) ⊢ CHL ^ q ∈ Q ( q ↔ ̥ q E ) . Proof. We write ϕ q =: q E and ψ q =: ̥ q E and F := ̥ E . Let χ := V q ∈ Q • ( q ↔ ϕ q )and ρ := • V q ∈ Q ( q ↔ ψ q ). We prove χ, ρ ⊢ q ↔ ψ q , by induction on G E .Let Q q be the set of all q ′ that can be reached from q in G E via a non-emptypath. We suppose we have the desired result for all q ′ in Q q . Let F q be restrictionof F to the Q q . It follows that we have: χ, ρ ⊢ CHL q ↔ ϕ q ↔ ϕ q F q ↔ ϕ q F q F↔ ϕ q F↔ ψ q The first equivalence is by χ . The second equivalence follows by the combination of ρ and the induction hypothesis. The third equivalence follows by ρ in combinationwith the fact that all variables from Q in ϕ q F q are guarded: the only variables that ϕ q can ‘see’ in G E have been removed by the substitution. Moreover, no variablesfrom Q occur in the ψ q ′ that are substituted. The fourth and fifth equivalence areimmediate.We have shown that: χ, • ^ q ∈ Q ( q ↔ ψ q ) ⊢ CHL ^ q ∈ Q ( q ↔ ψ q ) . So, by the Strengthed L¨ob’s Rule, we are done. ❑ Consider a formula ϕ . We assign to each -occurrence a a propositional variable q a , where the q a are pairwise distinct and also distinct from the propositionalvariables of ϕ . We map the nodes of ϕ to formulas of the language of ordinarymodal logic, i.e., the cycle-free formulas, as follows: • E ϕ ( a ) := q a , if a is a -occurrence. • E ϕ ( a ) := ( E ϕ ( S a ) ∧ E ϕ ( S a )), if a is a ∧ -occurrence. Similarly, for theother connectives and for the propositional variable-occurrences in ϕ .Our definition is correct by guard-recursion. We write ψ a for E ϕ ( a ). Theorem 3.10. Let G be the set of -occurrences of ϕ . We have: ^ a ∈ G • ( q a ↔ ψ S a ) ⊢ CHL ψ r ↔ ϕ. Proof. We show by guard induction with bo ( ϕ ) as guard that, for all nodes b , wehave: • ^ e ∈ V ( ψ e ↔ ϕ ↓ e ) , ^ a ∈ G • ( q a ↔ ψ S a ) ⊢ ψ b ↔ ϕ ↓ b. We first treat the case of conjunction. Suppose b is a ∧ -occurrence. We assume ourdesired conclusion for S b and S b . We have: • ^ e ∈ V ( ψ e ↔ ϕ ↓ e ) , ^ a ∈ G • ( q a ↔ ψ S a ) ⊢ CHL ψ b ↔ ( ψ S b ∧ ψ S b ) ↔ (( ϕ ↓ S b ) ∧ ( ϕ ↓ S b )) ↔ ϕ ↓ b YCLIC HENKIN LOGIC 13 The reasoning for the propositional variable-occurrences of ϕ and for the othernon-box connectives is similar. Suppose b is a -occurrence. We have: • ^ e ∈ V ( ψ e ↔ ϕ ↓ e ) , ^ a ∈ G • ( q a ↔ ψ S a ) ⊢ CHL q b ↔ ψ S b ↔ ( ϕ ↓ S b ) ↔ ϕ ↓ b So we find: • ^ e ∈ V ( ψ ( e ) ↔ ϕ ↓ e ) , ^ a ∈ G • ( q a ↔ ψ S a ) ⊢ CHL ^ e ∈ V ( ψ e ↔ ϕ ↓ e )We apply the strengthened version of L¨ob’s Rule to obtain: ^ a ∈ G • ( q a ↔ ψ S a ) ⊢ CHL ^ e ∈ V ( ψ e ↔ ϕ ↓ e )For this last insight, the desired conclusion is immediate. ❑ Remark 3.11. One would hope that the same kind of treatment we give here for CHL would be also possible for the µ -calculus. Formulas are defined in the sameway, only the guarding constraint is replaced by the following constraint. Considerany directive cycle γ : C → ω in ϕ . Consider the set X consisting of all occurrences a of ¬ in C plus all occurrences b of → in C such that γ ( b ) = 0. We demand that X has an even number of elements. In stead of L¨ob’s Rule one would have theMinimality Rule: if ⊢ ( ϕ α ) → α , then ⊢ ϕ → α . I have no idea how the detailsof this will work out. What replaces guard-induction and recursion? ❍ Alternative Axiomatisations. We provide some alternative axiomatisationsfor CHL . We consider the following axioms an rules. IPE: Suppose ϕ and ψ are modalised in p and ⊢ ϕ ↔ ψ , then ⊢ ̥ p.ϕ ↔ ̥ p.ψ (Intersubstitutivity of Provable Equivalents). N • : If ⊢ ϕ , then ⊢ • ϕ . H: ⊢ H , where H = ̥ p. p .We define the following theories: • K ◦− is given by chl • K ◦ is K ◦− plus IPE . • CHL is K ◦ plus H . • CHL is K ◦− plus N • .Here K ◦ is the reasonable circular version of K . We will show that over this theoryL¨ob’s rule is equivalent with the Henkin sentence (as axiom). Theorem 3.12. CHL and CHL prove the same theorems as CHL .Proof. To prove inclusion of CHL in CHL , we show that CHL is closed under N • Suppose CHL ⊢ ϕ . It follows that CHL ⊢ ( p ∧ ϕ ) ↔ p . Hence, by IPE , we have CHL ⊢ ̥ p. ( ϕ ∧ p ) ↔ ̥ p. p , i.e., CHL ⊢ • ϕ ↔ H . So, by axiom H , we find CHL ⊢ • ϕ .To prove inclusion of CHL in CHL it is sufficient to show that CHL is closedunder L¨ob’s Rule. This uses a well-known proof of L¨ob’s Rule. Suppose CHL ⊢ ϕ → ϕ . Let ν := ̥ q. • ( q → ϕ ), where q does not occur in ϕ . We have: CHL ⊢ ν → • ( ν → ϕ ) → (( ν → ϕ ) ∧ • ( ν → ϕ )) → (( ν → ϕ ) ∧ ν ) → ϕ → ϕ So, we have (a) CHL ⊢ ν → ϕ . Hence, by N • , we find CHL ⊢ • ( ν → ϕ ). Itfollows that (b) CHL ⊢ ν . Combining (a) and (b), we find: ⊢ ϕ .The inclusion of CHL in CHL follows from our previous results. ❑ Kripke Semantics. A Kripke model for CHL is given by a triple h W, < , f i ,where W is a non-empty set of worlds, < is an acyclic binary relation on W and f : W × Prop → { , } , where Prop is the set of propositional variables.Consider any formula ϕ . We define Ev ϕ : W × V → { , } as follows. We define Ev ϕ ( w, · ) assuming that we already have defined Ev ϕ ( w ′ , · ) for all w ′ = w . We useguard-recursion w.r.t. bo ( ϕ ) as guard. • If a is an occurrence of p , then Ev ϕ ( w, a ) = f ( w, p ). • If a is an occurrence of ∧ , then Ev ϕ ( w, a ) = min ( Ev ϕ ( w, S a ) , Ev ϕ ( w, S a )) , and, similarly for the other non-box connectives. • If a is a -occurrence, we set Ev ϕ ( w, a ) = 1 iff, for all w ′ = w , we have Ev ϕ ( w ′ , S a ) = 1. We take Ev ϕ ( w, a ) = 0, otherwise.We note that this definition works since in the clause for box do not call on valuesfor w but on the previously defined values for w ′ = w .We define J ϕ K ( w ) := Ev ϕ ( w, r ϕ ) and we write w (cid:13) ϕ for J ϕ K ( w ) = 1. Since ourguard is preserved by bisimulation, we find that if ϕ ≃ ϕ ′ , then J ϕ K ( w ) = J ϕ ′ K ( w ).We easily verify the usual clauses like w (cid:13) ϕ ∧ ψ iff w (cid:13) ϕ and w (cid:13) ψ . Usingthis the validity of CHL is immediate. Remark 3.13. We can derive the Kripke completeness theorem for CHL if finiteacyclic models in two ways. The first is using the synonymy of CHL and theHenkin Calculus HC , which is essentially CHL on standard syntax using a fixed pointoperator. We can prove the Kripke completeness for HC , for example, by showingthat HC is synonymous to the well-founded part of the µ -calculus and invokingthe Kripke compleness theorem for the µ -calculus. We will give the details of thisargument in a sequel to this paper by Tadeusz Litak and myself. Alternatively,we can use the Kripke Completeness of WfL and use the reduction of CHL to thattheory: see Subsection 5.2 for more detail. Of course, it would be much nicer togive a direct proof of the completeness theorem in terms of the circular syntax. Todo this remains open. ❍ L¨ob’s Logic meets Cyclic Syntax The counterpart of CHL in cycle-free language is the Henkin Calculus HC . Thiscalculus employs a variable-binding fixed point operator in the object language. Thelogics CHL and HC are synonymous/definitionally equivalent. We will establish thisfact in a later paper in which HC is developed. Here we will treat the simpler case YCLIC HENKIN LOGIC 15 of the synonymy of GL ◦ , the extension of CHL with the transitivity axiom scheme L3 , aka 4, and L¨ob’s Logic GL .In Subsection 4.1, we set up a modest framework for a province in which boththeories live and some basics of comparing theories living in that province. InSubsection 4.2, the logic GL ◦ is introduced and in Subsection 4.3 we do the samefor ordinary GL (as it appears in our framework). In Subsection 4.4, we developthe de Jongh-Sambin algorithm as one half of the witness of a synonymy between GL ◦ and GL .4.1. Languages, Translations and Interpretations. In this section we look ata restricted class of logics and employ a very restricted framework of interpretations.The interpretations are something like -preserving logic-interpretations . However,in this section we will call them ℓ -interpretations .A language L in this section will simply be a sub-set of our full language L ◦ thatis closed under (i) bisimilarity, (ii) the propositional variables and the syntacticoperations associated with the connectives, (iii) subformulas and (iv) substitution.The minimal language is L , the set of all acyclic formulas. We may view L as theordinary language of modal logic since each bisimulation equivalence class of anacyclic formula has a unique finite tree (modulo isomorphism) as normal form.We define HL ( L ) as the logic in the language L that is axiomatised by chl CHL = HL ( L ◦ ) andthat K proves the same theorems as HL ( L ). A logic Λ will be a rule-preservingextension of one of the HL ( L ) in the same language by schematic rules.A salient language is L • . This is the language generated by the variables, thelogical connectives including and • . We define WfL := HL ( L • ).Here is an important observation. Observation 4.1. Suppose L is a language that extends L • . Then Theorems 3.1,3.2 ( i ) , 3.3, 3.4, 3.6 and Corollary 3.5 remain valid when we replace CHL by HL ( L ) . We can simply check that these results do not use closure under the operation ̥ p. ( · ).A local translation T of a formula ϕ into a logic Λ is a mapping from V ϕ tothe language of Λ, such that if a is $-occurrence for a connective $ of arity n or avariable treated as having arity 0, whereΛ ⊢ T ( a ) ↔ $( T ( S a ) , . . . , T ( S n − a )) . Strictly speaking the local translation is given as the triple of the formula, thefunction and the logic. Theorem 4.2. Consider any logic Λ . Suppose T is a local translation of of ϕ in Λ and T ′ is a local translation of of ϕ ′ in Λ . Suppose further that R is a bisimulationbetween ϕ and ϕ ′ , where R does not need to be root-preserving. Suppose aRa ′ .Then, Λ ⊢ T ( a ) ↔ T ′ ( a ′ ) .Proof. Let χ := V bRb ′ ( T ( b ) ↔ T ′ ( b ′ )). We prove by guard-induction on -occurrences d in ϕ that, if dRd ′ , then χ ⊢ Λ T ( d ) ↔ T ′ ( d ′ ). The cases of thenon-box occurrences are trivial. In case d is a -occurrence, we have: χ ⊢ T ( d ) ↔ T ( S d ) ↔ T ′ ( S ′ d ′ ) ↔ T ′ ( d ′ ) It follows that χ ⊢ Λ χ . So, by L¨ob’s rule, we are done. ❑ A global translation G of a language L into a logic Λ is a function from L to thelanguage of Λ that commutes modulo Λ-provable equivalence with the propositionalvariables and the connectives. Strictly speaking a global translation is given asthe triple of language, function and logic. We usually omit the ‘global’ of ‘globaltranslation’.We collect some trivial insights. Theorem 4.3. i. Suppose G is a global translation of L into Λ . Let ϕ be an L -formula. For a ∈ V ϕ , we define T ( a ) := G ( ϕ ↓ a ) . Then, T is a localtranslation of ϕ in Λ .ii. Suppose every ϕ in L has a local translation T ϕ in Λ . Then G with G ( ϕ ) := T ϕ ( r ϕ ) is a global translation of L in Λ .iii. If G and G ′ are global translations of L in Λ . Then, for each ϕ , we have Λ ⊢ G ( ϕ ) ↔ G ′ ( ϕ ) . A translation G is an ℓ -translation if it commutes with substitution. This meansthat, for all formulas ϕ and all L -substitutions σ , we have Λ ⊢ G ( ϕσ ) ↔ G ( ϕ )( G ◦ σ ).A interpretation K : Λ → Λ ′ is given as a triple of Λ, a function G , and Λ ′ ,where G is a translation from the language of Λ into Λ ′ . We demand that if Λ ⊢ ϕ ,then Λ ′ ⊢ G ( ϕ ). Usually, we will confuse K with its underlying translation, writinge.g. K ( ϕ ).An interpretation is an ℓ -interpretation iff the underlying translation is an ℓ -translation.We count two interpretations as the same if their values are provably equivalentin the target logic. The identity interpretation ID Λ on Λ is the interpretation basedon the identity function Id L , where L is the language of Λ. We define compositionof interpretations in the expected way. We leave the simple verification that inter-pretations are closed under composition to the reader. It is also easy to see thatthe composition of ℓ -interpretations delivers a ℓ -interpretation.We note that the fact that we have at most one interpretation (from our restrictedclass) between two logics implies that, whenever we have mutual interpretability,we have synonymy: the composition of an interpretation from Λ to Λ ′ an an inter-pretation from Λ ′ to Λ will be an interpretation from Λ to Λ and this must be theidentity.4.2. L¨ob’s Logic with Cycles. We define GL ◦ as CHL plus L3 or 4, to wit ⊢ ϕ → ϕ . Here the intended notion of extension is one that preserves the rules.The theory GL ◦ is extensionally the same as the theory K ◦− plus L3 . This isbecause in the presence of the fixed points we may prove L¨ob’s Principle in theusual way. Then, the desired closure under L¨ob’s rule follows, by the usual proof ofL¨ob’s Rule from L¨ob’s Principle. We will write ≡ ◦ for provable equivalence in GL ◦ .We show that in GL ◦ the modalities and • coincide. Theorem 4.4. GL ◦ ⊢ • ϕ ↔ ϕ .Proof. We have: ( • ϕ ↔ ϕ ) ⊢ GL ◦ • ϕ ↔ ( ϕ ∧ • ϕ ) ↔ ( ϕ ∧ ϕ ) ↔ ϕ ❑ YCLIC HENKIN LOGIC 17 Thus, all insights that we accumulated for • in CHL transfer to in GL ◦ .Here is a GL ◦ reformulation of a well-known insight due to Dick de Jongh. Lemma 4.5. Suppose the root of ψ is an -occurrence. Then, we have ψ ≡ ◦ ( ψ ⊤ ) .Proof. If the root of ψ is not on a cycle, this is trivial. So, suppose the root is ona cycle. Let χ := su ( ψ ). We have ψ ≡ ◦ χ , and hence GL ◦ ⊢ ψ → ψ . Similarly,for ( ψ ⊤ ). Suppose p does not occur in ψ . We have: ψ ⊢ GL ◦ ψ ∧ ( ψ p )[ p : ψ ] ⊢ GL ◦ ( ψ ↔ ⊤ ) ∧ ( ψ p )[ p : ψ ] ⊢ GL ◦ ( ψ p )[ p : ⊤ ] ⊢ GL ◦ ( ψ ⊤ )We also have: ψ, ( ψ ⊤ ) ⊢ GL ◦ ( ψ ↔ ⊤ ) ∧ ( ψ p )[ p : ⊤ ] ⊢ GL ◦ ( ψ p )[ p : ψ ] ⊢ GL ◦ ψ By the Strengthed L¨ob’s rule, we find ( ψ ⊤ ) ⊢ GL ◦ ψ . ❑ L¨ob’s Logic without Cycles. In our context we may define L¨ob’s Logic GL simply as HL ( L ) plus L 3. We note that this is K plus L¨ob’s Rule plus L 3. We willemploy the usual facts about GL and especially the de Jongh result: Lemma 4.6. GL ⊢ ( ϕ )[ p : ⊤ ] ↔ ( ϕ )[ p : ( ϕ )[ p : ⊤ ]] . The de Jongh-Sambin Interpretation. We define functions js and js ⋆ . Ouraim is to show that js ⋆ ( ϕ, · ) is a local ℓ -translation of ϕ to GL . • Suppose a is not in bo ◦ ( ϕ ). We treat the case of a ∧ -occurrence, the othercases being similar. We take js ⋆ ( ϕ, a ) := ( js ⋆ ( ϕ, S a ) ∧ js ⋆ ( ϕ, S a )). • Suppose a is in bo ◦ ( ϕ ). Then, js ⋆ ( ϕ, a ) := js ⋆ ((( ϕ ↓ a ) ⊤ ) , a ). • js ( ϕ ) := js ⋆ ( ϕ, r ϕ ).We have: Lemma 4.7. js ⋆ ( ϕ, a ) is in L . Lemma 4.8. js ⋆ ( ϕ, a ) = js ( ϕ ↓ a ) . The desired results follow trivially by course of values induction on c ( ϕ ) and byguard induction on bo ◦ ( ϕ ).We prove a result on commutation with substitution. Lemma 4.9. Let σ be a substitution on Q . Then, for a ∈ V ϕ , we have js ⋆ ( ϕσ, a ) ≃ js ⋆ ( ϕ, a )( js ◦ σ ) .Proof. The proof is by course of values induction on c ( ϕ ) and by guard inductionon bo ◦ ( ϕ ).Suppose a bo ◦ ( ϕ ). Suppose, e.g., a is a ∧ -occurrence. We have: js ⋆ ( ϕσ, a ) = ( js ⋆ ( ϕσ, S a ) ∧ js ⋆ ( ϕσ, S a )) ≃ ( js ⋆ ( ϕ, S a )( js ◦ σ ) ∧ js ⋆ ( ϕ, S a )( js ◦ σ )) ≃ js ⋆ ( ϕ, a )( js ◦ σ ) The cases where a is an occurrence of a variable not in Q or where a is an occurrenceof another connective are similar. Suppose a is an occurrence of q ∈ Q . We have: js ⋆ ( ϕσ, a ) ≃ js ( ϕσ ↓ a )= js ( qσ )= q ( js ◦ σ )= js ( q )( js ◦ σ )= js ( ϕ ↓ a )( js ◦ σ ) ≃ js ⋆ ( ϕ, a )( js ◦ σ )Suppose a ∈ bo ◦ ( ϕ ). We have: js ⋆ ( ϕσ, a ) = js ⋆ ((( ϕσ ↓ a ) ⊤ ) , a )= js ⋆ ((( ϕ ↓ a ) ⊤ ) σ, a ) ≃ js ⋆ ((( ϕ ↓ a ) ⊤ ) , a )( js ◦ σ )= js ⋆ ( ϕ, a )( js ◦ σ ) ❑ We write ≡ ◦ for provable equivalence in GL ◦ and ≡ for provable equivalence in GL .For the proof of our main insight, Theorem 4.11, we need a lemma, that is astrengthening of Theorem 4.2. Consider formulas ϕ and ϕ ′ . Let F : V ϕ → L and let F ′ : V ϕ ′ → L . We define C ϕ ( F ) as the conjunction of formulas F ( a ) ↔ $( F ( S a ) , . . . , F ( S n − a )), where $ is the label of a in ϕ and where $ is n -ary. Wetreat the variable as a 0-ary operation here. Lemma 4.10. Suppose R is a bisimulation between ϕ and ϕ ′ and aRa ′ . Then,i. C ϕ ( F ) ∧ C ϕ ′ ( F ′ ) ⊢ GL F ( a ) ↔ F ( a ′ ) .ii. C ϕ ( F ) ∧ C ϕ ′ ( F ′ ) ⊢ GL ( F ( a ) ↔ F ( a ′ )) .Proof. The proof of (i) is an immediate adaptation of the proof of Theorem 4.2,replacing L¨ob’s Rule by the Strengthened L¨ob’s Rule. Item (ii) is immediate from(i). ❑ Theorem 4.11. js ⋆ ( ϕ, · ) is a local translation of ϕ into GL .Proof. We employ course of values induction on the number of cycles in ϕ and,then, guard induction on bo ◦ ( ϕ ). The only non-trivial case is where a ∈ bo ◦ ( ϕ ).So, suppose a ∈ bo ◦ ( ϕ ). We have js ⋆ ( ϕ, a ) = js ⋆ ((( ϕ ↓ a ) ⊤ ) , a ). Let us write ψ b := js ⋆ ( ϕ, b ), for b ∈ V ϕ and ψ ′ c := js ⋆ ((( ϕ ↓ a ) p ) , c ), for c ∈ V (( ϕ ↓ a ) p ) , where p is a fresh variable. Let S ′ be the successor function of (( ϕ ↓ a ) p ).We have ( † ): ψ a = js ⋆ ((( ϕ ↓ a ) p )[ p : ⊤ ] , a )= ψ ′ a [ p : ⊤ ]= ( ψ ′ S ′ a )[ p : ⊤ ] ≡ ( ψ ′ S a )[ p : ( ψ ′ S ′ a )[ p : ⊤ ]]= ( ψ ′ S ′ a )[ p : ψ a ]= ( ψ ′ S ′ a [ p : ψ a ]) YCLIC HENKIN LOGIC 19 Here the second step is by Lemma 4.9. The third step uses the fact that a is noton a cycle in (( ϕ ↓ a ) p )[ p : ⊤ ]. The fourth step uses Lemma 4.6.We now define F from V ϕ ↓ a to L as follows. F ( a ) := ψ a and F ( b ) := ψ ′ b [ p : ψ a ] if a = b . We prove that F is a local translation from ϕ ↓ a to GL . The case of a is by( † ). Suppose b is a $-occurrence unequal to a , where $ is n -ary. By the inductionhypothesis, we have GL ⊢ ψ ′ b ↔ $( ψ ′ S ′ b , . . . , ψ ′ S ′ n − b ) and hence GL ⊢ ψ ′ b [ p : ψ a ] ↔ $( ψ ′ S ′ b [ p : ψ a ] , . . . , ψ ′ S ′ n − b [ p : ψ a ]). We note that:i. ψ ′ b [ p : ψ a ] = F ( b );ii. if S ′ i b is not a p -occurence, then ψ ′ S ′ i b [ p : ψ a ] = ψ ′ S i b [ p : ψ a ] = F ( S i b );iii. if S ′ i b is a p -occurrence, then ψ ′ S ′ i b [ p : ψ a ] = ψ a = F ( a ) = F ( S i b ).Thus, we have GL ⊢ C ϕ ↓ a ( F ) and, hence, ( ‡ ) GL ⊢ C ϕ ↓ a ( F )We define G from V ϕ to the acyclic formulas by G ( b ) := js ⋆ ( ϕ, b ). It follows that: C ϕ ( G ) ⊢ GL C ϕ ↓ a ( F ) ∧ C ϕ ( G ) ⊢ GL ( F ( S a ) ↔ G ( S a )) ⊢ GL F ( S a ) ↔ G ( S a ) ⊢ GL G ( a ) ↔ G ( S a )Here the first step is by ( ‡ ), the second step is by Lemma 4.10(ii) and the fourthstep is by ( † ).Since a was an arbitrary -occurrence on a cycle, we find C ϕ ( G ) ⊢ GL C ϕ ( G ).Ergo, GL ⊢ C ( G ). Thus, G = js ⋆ ( ϕ, · ) is a local translation of ϕ to GL . ❑ Theorem 4.12. js carries an ℓ -interpretation JS of GL ◦ in GL .Proof. By Theorem 4.11, js ⋆ ( ϕ, · ) is a local translation of ϕ in GL . It follows that js is a translation of L ◦ into GL . Our translation is an ℓ -translation by Lemma 4.9.So we need just to verify the translations of the axioms and rules of GL ◦ in GL . However, the translations of the axioms and rules of GL ◦ except chl 5, are allinstances of the same axioms and rules of GL , modulo GL -provable equivalence. Theaxiom chl ❑ The identical translation emb of L into GL ◦ clearly carries an ℓ -interpretationof GL in GL ◦ . So, the pair JS , Emb forms an ℓ -synonymy. Since the arrows of asynonymy are faithful, it follows that GL ◦ is conservative over GL . In other words, GL is the acyclic fragment of GL ◦ .Consider any acyclic ϕ . Clearly, GL ⊢ js ( ϕ ) ↔ ϕ . (Inspection shows that we evenhave dj ( ϕ ) ≃ ϕ .) Suppose p is modalised in ϕ . We have: GL ◦ ⊢ ̥ p.ϕ ↔ ϕ [ p : ̥ p.ϕ ].It follows that GL ⊢ js ( ̥ p.ϕ ) ↔ ϕ [ p : js ( ̥ p.ϕ )]. Thus js ( ̥ p.ϕ ) is a de Jongh-Sambinexplicit fixed point of ϕ . Remark 4.13. Of course, we could also develop the synonymy by using the knownde Jongh-Sambin Theorem for GL . However, the advantage of the present set-up isthat we can see the algorithm stated using guard recursion. ❍ Further Inter-theory Relations We develop the relations of CHL to two other theories, to wit, Multiple FixedPoint Theory MFT and Well-foundedness Logic WfL (or: HL ( L • )). In Subsec-tion 5.1, we discuss MFT . Our result on MFT will play a role in the definition ofarithmetical interpretations. In Subsection 5.2, we address WfL . Our result in thatsubsection is an ingredient of one possible Kripke completeness proof for CHL .5.1. Multiple Fixed Point Theory. We define the following theory MFT in L extended with fresh constants. We have K plus L¨ob’s Rule plus, for every systemof equations E on a set of fixed-point variables Q for the modal language withoutconstants, constants c E ,q and axioms stating that these constants solve E . Let ussay that an extension Θ of MFT (in the same language) is a strong extension if itis closed under necessitation and L¨ob’s Rule. Here we do not demand that Θ isclosed under substitution: it is a theory not a logic.We adapt the notion of local translation to the new setting in the following way.In the CHL - MFT direction we allow formulas containing the constants. In the MFT - CHL connection we allow the constants to be interpreted by formulas. Similarly,for global translations.We have an immediate adaptation of Theorem 4.2 to the slightly modified settingthat we formulate here for completeness. Theorem 5.1. Suppose R is a bisimulation between ϕ and ϕ ′ and suppose T isa local translation of ϕ in Θ and T ′ is a local interpretation of ϕ ′ in Θ . Supposefurther that, for any variable p , if a and a ′ are occurrences of p , then Θ ⊢ T ( a ) ↔ T ′ ( a ′ ) . Then, whenever aRa ′ , we have Θ ⊢ T ( a ) ↔ T ′ ( a ′ ) . Our next order of business is to prove the existence of a local translation of ϕ . InSection 3, we introduced the mapping E . For convenience, we repeat the definitionhere. Consider a formula ϕ . We assign to each -occurrence a a propositional vari-able q a , where the q a are pairwise distinct and also distinct from the propositionalvariables of ϕ . We map the nodes of ϕ to formulas of the language of ordinarymodal logic as follows: • E ϕ ( a ) := q a , if a is a -occurrence. • E ϕ ( a ) := ( E ϕ ( S a ) ∧ E ϕ ( S a )), if a is a ∧ -occurrence. Similarly, for theother connectives and for the propositional variable-occurrences in ϕ .We write ψ a for E ϕ ( a ). We note that ψ a is cycle-free.Let E be defined by q a E := E ϕ ( S a ), where a is an -occurrence in ϕ . In MFT this system of equations has a solution, say q a F := c a . Finally we define, for every b ∈ V ϕ , the mapping cyco ⋆ϕ by cyco ⋆ϕ ( b ) : ψ b F . Lemma 5.2. cyco ⋆ϕ is a local translation of ϕ in MFT .Proof. The cases where b is not a -occurrence are simple. Suppose b is an -occurrence. We need that MFT ⊢ ψ b F ↔ ψ S b F . However, this is precisely MFT ⊢ c b ↔ ψ S b F , the promised solution of E . ❑ It follows that cyco defined by cyco ( ϕ ) := cyco ⋆ϕ ( r ϕ ) is a global translation of L ◦ in MFT . Using Theorem 5.1, we now find: Theorem 5.3. There is an interpretation CyCo based on cyco of CHL in MFT . YCLIC HENKIN LOGIC 21 In the other direction we define a translation cocy that commutes with propo-sitional variables and connectives and that sends a constant c q in introduced for asystem of equations E to ̥ q E , the solution of E for q as guaranteed by Theorem 2.15.We find: Theorem 5.4. There is an interpretation CoCy based on cocy of MFT in CHL .Moreover, this interpretation is unique.Proof. The verification that cocy carries an interpretation is entirely as expected.For the uniqueness we use Theorem 3.9. ❑ Using the analogue of Theorem 3.9 in MFT , we find: Theorem 5.5. CyCo and CoCy form a synonymy. We note that the synonymy we are looking at here is a synonymy of theories notlogics. It is a form of sameness weaker than the sameness of GL ◦ and GL .5.2. Well-Foundedness Logic. It is interesting to note that the global transla-tions JS and Emb still carry a synonymy between WfL plus L GL when werestrict JS to L • . In this case we simply have JS ( • ϕ ) = ( JS ( ϕ ) ∧ ⊤ ).We borrow a result form a forthcoming paper with Tadeusz Litak: we have thecompleteness theorem for WfL in finite acyclic Kripke models.We remind the reader that bo ( ϕ ) is the set of -occurrences in ϕ . Modulo theannounced result, we have: Announced Theorem 5.6. Consider any L -formula ϕ . We use ψ a := E ϕ ( a ) asin Subsection 5.1. We have: CHL ⊢ ϕ iff ^ a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) ⊢ WfL ψ r . Proof. Suppose V a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) ⊢ WfL ψ r . Then, a fortiory , ^ a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) ⊢ CHL ψ r . Then, by Theorem 3.10, we have: ( † ) V a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) ⊢ CHL ϕ . Wenote that ϕ does not contain occurrences of the q a . By the Simultaneous FixedPoint Theorem 2.15, we have CHL -verifiable solutions χ a of the equations in theantecedent conjunction. Substituting these for the q i , we find CHL ⊢ ϕ .Conversely, suppose V a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) WfL ψ r . By the announced Com-pleteness Theorem for WfL , we have an acyclic finite Kripke model K with root r such that r (cid:13) V a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) and r ψ r . Clearly the forcing relationfor L • is the restriction of the forcing relation on L ◦ to L • . Since K forces CHL itfollows, by Theorem 3.10, that r ϕ . So, CHL ϕ . ❑ Announced Theorem 5.7. We have completeness for CHL in finite acyclic Kripkemodels.Proof. Suppose CHL ϕ . Then, V a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) WfL ψ r . So, we have,by the announced completeness theorem for WfL , a Kripke model K with root r such that r (cid:13) V a ∈ bo ( ϕ ) • ( q a ↔ ψ S a ) and r ψ r . It follows by Theorem 3.10,that r ϕ . ❑ Arithmetical interpretations In this section, we introduce the notion of arithmetical interpretation and verifyits basic properties.We fix a theory U , an interpretation N of S in U . We suppose that we have aprovability predicate bew for numerals in N that satisfies L A for bew ( p A q ). Remark 6.1. In our paper [Vis19], we show that if we have a predicate thatuniformly semi-represents a given axiom set of U (w.r.t. N ) in U . The predicate prov N [ α ] , where prov is a standard provability predicate, has the desired propertieswe ask of bew . We discuss these ideas in some detail in Appendix A.5. ❍ A translation of ϕ in U (for bew ) is a mapping T from V ϕ to U -sentences, suchthat: • If a is an occurrence of ∧ then U ⊢ T ( a ) ↔ ( T ( S a ) ∧ T ( S a )). Similarlyfor the other non-box connectives. • If a is an occurrence of and Sa = h b i , then U ⊢ T ( a ) ↔ T ( S a ). • If a and a ′ are occurrences of p , then U ⊢ T ( a ) ↔ T ( a ′ ).An interpretation of L ◦ in U is a mapping U from L ◦ to U -sentences that commutesmodulo U -provability with the propositional connectives and the commutes within the sense that U ⊢ U ( ϕ ) ↔ U ( ϕ ).Modulo U -provable equivalence, arithmetical translations are preserved underbisimulation. Theorem 6.2. Suppose R is a bisimulation between ϕ and ϕ ′ and suppose T is atranslation of ϕ in U and T ′ is a translation of ϕ ′ in U . Suppose further that, forany variable p , if a and a ′ are occurrences of p , then U ⊢ T ( a ) ↔ T ′ ( a ′ ) . Then,whenever aRa ′ , we have U ⊢ T ( a ) ↔ T ′ ( a ′ ) . The proof is just a minor variation of the proof of Theorem 4.2. We note that itfollows that, if we have translations T ϕ for each ϕ ∈ L ◦ , we can base an interpre-tation on them by defining U ( ϕ ) := T ϕ ( r ϕ ). The Uniqueness Theorem guaranteesthat the local pieces add up to a coherent whole.As a preparation of the theorem concerning the existence of arithmetical transla-tions, we first remind the reader of the simultaneous fixed point lemma and providetwo proofs of it. Lemma 6.3. Suppose N interprets S in U . Consider U -formulas A i ( x , . . . x n − , ~y ) for i < n . Here the variables x i range over the domain of N . Then, there are for-mulas B ( ~y ) , . . . , B n − ( ~y ) such that U ⊢ B i ( ~y ) ↔ A i ( p B ( ˙ ~y ) q , . . . , p B n − ( ˙ ~y ) q , ~y ) , for i < n . First Proof. The proof is by induction on n . By the usual fixed point lemma withparameters we find a C ( x , . . . , x n − , ~y ) such that U ⊢ C ( x , . . . , x n − , ~y ) ↔ A ( p C ( ˙ x , . . . , ˙ x n − , ˙ ~y ) q , x , . . . , x n − , ~y ) . We assume here that the variables of a formula or term are among the variables that aredisplayed. We employ the Smory´nsky dot notation. YCLIC HENKIN LOGIC 23 Now we define: C i +1 ( x , . . . , x n − , ~y ) := A i +1 ( p C ( ˙ x , . . . , ˙ x n − , ˙ ~y ) q , x , . . . , x n − , ~y ) . We solve the system C , . . . C n − as is guaranteed by the induction hypothesis,resulting in B , . . . , B n − . Finally we set: B ( ~y ) := C ( p B ( ˙ ~y ) q , . . . , p B n − ( ˙ ~y ) q , ~y ) . It is now easy to see that the B i form the desired simultaneous fixed point. ❑ Second Proof. Let us write x [ w ] for the term representing the result of substitutingthe numeral of w for the variable z in the formula represented by x . We define D ( z, x, ~y ) := W i MFT in U is the obvious one, since MFT hasacyclic syntax. Using the simultaneous fixed point lemma, we find: Theorem 6.4. For every f from the propositional variables U -sentences, there isan interpretation FA f based on fa f of MFT in U such that fa f ( p ) = f ( p ) . Theorem 6.5. For every f from the propositional variables U -sentences, there isan interpretation HA f based on a translation ha f of CHL in U such that fa f ( p ) = f ( p ) . This interpretation is unique modulo provable equivalence.Proof. We take ha f := fa f ◦ cyco . For the verification of chl ❑ We note that we can view ha f as a mapping of formulas modulo bisimulation toelements of the diagonalised algebra of U .If U satisfies some further schematic logical principle, it is easy to see that ha f carries an interpretation H of CHL plus that logical principle. So, e.g., if satisfies L U , we have in interpretation H based on ha f of GL ◦ in U , etcetera. Remark 6.6. Clearly, we can use Theorem 4.12 to prove arithmetical complete-ness for GL ◦ in a Σ -sound extension U of Elementary Arithmetic for Fefermanianprovability with respect to an elementary representation α of the axiom set. Wecan do it directly. It is somewhat remarkable that (a lifted version of) Theorem 6.5is not needed in the proof. The Solovay construction delivers translations with thedesired properties directly.Here is a sketch of how this works. Consider any tail-model K for GL / GL ◦ . See[Vis84] for this notion. Let X be a finite or cofinite set of nodes. We write [ X ] for (anappropriate paraphrase of) ∃ x ∈ X ℓ = x , where ℓ is the limit statement constructedby Solovay (for K and for α ). We can now show: U ⊢ [ X ∩ Y ] ↔ [ X ] ∧ [ Y ] and,similarly, for the other non-modal connectives. We write X := { y | ∀ x ( y < x ⇒ x ∈ X ) } . We have: U ⊢ [ X ] ↔ α [ X ].Consider any ϕ ∈ L ◦ . We define S ⋆ K ,ϕ ( a ) := [ { x | x (cid:13) ϕ ↓ a } ]. (Here we needto check that { x | x (cid:13) ψ } is always finite or co-finite not just for L as is provenin [Vis84], but also for L ◦ .) It is easy to see that we can base an interpretations S K on the S ⋆ K ,ϕ and that these interpretations witness the desired arithmeticalcompleteness for GL ◦ .Thus, in this proof we use the Magari Algebra of K to replace the use of MFT in the proof of Theorem 6.5. We interpret GL ◦ in the Magari Algebra of K via ourresult on evaluation in Kripke models and we interpret K in U via the standardSolovay argument as applied to tail models. ❍ References [AB04] S.N. Artemov and L.D. Beklemishev. Provability logic. In D. Gabbay and F. Guenthner,editors, Handbook of Philosophical Logic, 2nd ed. , volume 13, pages 229–403. Springer,Dordrecht, 2004.[Acz88] P. Aczel. Non-well-founded sets , volume 14 of CSLI Lecture Notes . CSLI, Stanford, 1988.[BE87] J. Barwise and J. Etchemendy. The Liar, an essay in truth and circularity . OxfordUniversity Press, New York, Oxford, 1987.[Boo93] G. Boolos. The logic of provability . Cambridge University Press, Cambridge, 1993.[JdJ98] G. Japaridze and D. de Jongh. The logic of provability. In S. Buss, editor, Handbook ofproof theory , pages 475–546. North-Holland Publishing Co., Amsterdam, 1998.[Lin96] P. Lindstr¨om. Provability logic – a short introduction. Theoria , 62(1-2):19–61, 1996.[Sha14] D. S. Shamkanov. Circular proofs for the G¨odel-L¨ob provability logic. MathematicalNotes , 96(3-4):575–585, 2014.[Sha20] D. S. Shamkanov. Non-well-founded derivations in the G¨odel-L¨ob provability logic. TheReview of Symbolic Logic , 13(4):776–796, 2020.[Smo85] C. Smory´nski. Self-Reference and Modal Logic . Universitext. Springer, New York, 1985.[ˇSve00] V. ˇSvejdar. On provability logic. Nordic Journal of Philosophical Logic , 4(2):95–116,2000.[VB06] J. Van Benthem. Modal frame correspondences and fixed-points. Studia Logica , 83(1-3):133–155, 2006.[Vis84] A. Visser. The provability logics of recursively enumerable theories extending PeanoArithmetic at arbitrary theories extending Peano Arithmetic. Journal of PhilosophicalLogic , 13:97–113, 1984.[Vis05] A. Visser. L¨ob’s Logic Meets the µ -Calculus. In Aart Middeldorp, Vincent van Oostrom,Femke van Raamsdonk, and Roel de Vrijer, editors, Processes, Terms and Cycles, Stepson the Road to Infinity, Essays Dedicated to Jan Willem Klop on the Occasion of His60th Birthday , LNCS 3838, pages 14–25. Springer, Berlin, 2005.[Vis19] A. Visser. Another look at the second incompleteness theorem. The Review of SymbolicLogic , pages 1–27, 2019. Appendix A. Arithmetical Matters In this appendix, we summarise the relevant ideas from our paper [Vis19] andprove some additional results that connect that paper to the present one.In Subsection A.1, we provide some basic insights. In Subsection A.2, we reprovethe version of the Second Incompleteness Theorem from [Vis19] using the notationsof the present paper. In Subsection A.3, we revisit the transitive closure modality • and show that some of its properties also hold globally. Moreover, we show that,under certain conditions, the Fefermanianness of the input modality is preserved toits transitive closure. In Subsection A.4, we discuss the operations of Craigificationand Smoothening. Finally, we consider an example with some remarkable propertiesin Subsection A.5.A.1. Preliminaries. We consider theories T in predicate logic of finite signature.We allow T to have any complexity. We always assume that T is equipped withan interpretation N : S → T . Unless stated otherwise, displayed variables rangeover the domain of N . In other words, we pretend that T is an arithmetical theory. YCLIC HENKIN LOGIC 25 Moreover, we will use single variables to range over the domain of N , even if N might be multi-dimensional.Our treatment is, at places, somewhat dependent on details of the coding. Let ussimply assume that we base our coding on an alphabet that contains (at least) thesigns of the language plus some extra brackets “[” and “]”. Our G¨odel numbering isgiven by the length-first ordering of strings from this alphabet. We code finite setsof expressions as (the G¨odel numbers of) strings of expressions, so p A A . . . A k − q and finite sets of numbers as the finite sets of their efficient numerals. We representproofs with assumptions as strings of formulas where some formulas, the assump-tions, are enclosed between square brackets. We code finite functions B i n i fromsentences to numbers as strings of the form B n . . . B m − n m − . Etcetera. Ofcourse, most of these details are immaterial. The main things we need are proper-ties like the following: the function that sends n to the G¨odel number of its efficientnumeral, p n q , is p-time; the assumption set of a proof can be efficiently extractedfrom the proof; etcetera.Consider a T -predicate bew . We write A for bew ( p A q ). Here the numerals aredefined w.r.t. N . We say that bew is a provability predicate if it satisfies the firsttwo L¨ob conditions w.r.t. T . L 1. If T ⊢ A , then T ⊢ A . L T ⊢ ( A ∧ ( A → B )) → B .A special class of provability predicates are the Feferman-style predicates. Thisworks as follows. We fix arithmetisations of proof † ( p, y ) and ass ( p ), where proof † ( p, y )is a good arithmetisation of: p codes a proof in predicate logic of y (of the ambientsignature) from assumptions in ass ( p ). We write: • proof α ( p, y ) for ( proof † ( p, y ) ∧ ∀ z ∈ ass ( p ) α ( z )), • prov α ( y ) for ∃ p proof α ( p, y ), • α A for prov α ( p A q ).Now consider the theory T with axiom set X . We suppose we have an interpretation N of S in T . Suppose that α semi-numerates X in T , i.e., if A ∈ X , then T ⊢ α ( p A q ). In these circumstances, it is easy to see that prov Nα is a provabilitypredicate for T . We say that prov Nα is a Fefermanian provability predicate. We notethat α need not be of the form β N , where β is an arithmetical predicate.If X is a finite set of T -sentences, we write [ X ] for W B ∈ X x = p B q . We write[ A ] for [ { A } ]. We note that [ ∅ ] , [ ⊤ ] and ⊥ all represent provability in predicatelogic of the ambient signature.Here are some further definitions. • bew is an LR-provability predicate for T if, it satisfies ( L 2) and T is closedunder L¨ob’s rule LR for bew : if T ⊢ A → A , then T ⊢ A . • bew is a uniform provability predicate for T , if it satisfies the following threeprinciples L un 1. Whenever T ⊢ A , there is finite set of T -sentences X such that, X ⊢ A and, for each B ∈ X , we have T ⊢ B and X ⊢ B . L un T ⊢ ∀ b ( prov N [ ∅ ] ( b ) → bew ( b )). L un T ⊢ ∀ a ∀ b (( bew ( a ) ∧ bew ( imp ( a, b ))) → bew ( b )). • bew is a global provability predicate , if it satisfies L1 and L un L un Theorem A.1. Suppose bew is an LR-provability predicate for T . Then, bew is aprovability predicate for T .Proof. The proof is just a minor adaptation of an argument due to Dick de Jongh.Suppose is an LR-provability predicate for T . Suppose T ⊢ A . Then, by K -reasoning, T ⊢ ( A ∧ A ) → ( A ∧ A ). So, by L¨ob’s rule, we have T ⊢ A . ❑ Theorem A.2. If bew is a uniform provability predicate, it is a global provabilitypredicate. If bew is a global provability predicate, it is a provability predicate.Proof. We verify e.g. that a uniform provability predicate satisfies L1 . Suppose T ⊢ A . We find X = { B , . . . , B n − } as promised. We have X ⊢ A and, hence, T ⊢ N [ ∅ ] C , where C := ( B → ( B → . . . ( B n − → A ) . . . )). It follows that T ⊢ C . We also have T ⊢ B i for i < n , so by repeated application of L un 3, wefind T ⊢ A . ❑ The next theorem is obvious. Theorem A.3. Any Fefermanian provability predicate is global. A.2. A Version of the Second Incompleteness Theorem. We present therelevant result of [Vis19] in the terminology of the present paper. Theorem A.4. Suppose is a uniform provability predicate for T . Then, is anLR-provability predicate for T .Proof. Let be a uniform provability predicate for T . Suppose T ⊢ A → A . Let C be the conjunction of the following statements: • ( V S ) N (we assume that the axioms of S include the axioms of identity), • ∀ b ( prov N [ ∅ ] ( b ) → bew ( b )), • ∀ a ∀ b (( bew ( a ) ∧ bew ( imp ( a, b ))) → bew ( b )), • A → A .Let X be as promised for C in the definition of uniformity. We have X ⊢ C .Hence, X ⊢ N [ X ] A → N [ ∅ ] ( ^ X → A ) → ( ^ X → A ) → A → A So, we find that X ⊢ N [ X ] A → A . Since we have L¨ob’s Rule for finitely axioma-tised theories with standard axiomatisation (w.r.t. an interpretation N of S ), wefind X ⊢ A and, hence, T ⊢ A . ❑ Open Question A.5. It would be interesting to have an example of a provabilitypredicate for which we have the Second Incompleteness Theorem, but not L¨ob’sRule.It would be interesting to have an example of an LR-provability predicate thatis not uniform. ❍ YCLIC HENKIN LOGIC 27 A.3. Global Properties of the Transitive Closure Modality. We show thata number of desirable properties can be lifted from bew to bew • .Suppose bew is an LR-provability predicate for T w.r.t. N , where N is aninterpretation of S . Let a, b, c, . . . range over codes of sentences. We use conj , imp ,etcetera, for the arithmetisations of the obvious syntactical operations.By the fixed point lemma with parameters we find bew • , such that: • T ⊢ bew • ( a ) ↔ bew ( conj ( a, p bew • ( ˙ a ) q )).This is just our previous definition but now not locally for each sentence but for allsentences at once. The definition is unproblematic using the fixed point constructionwith parameters. We note that by the uniqueness result our global definition willlocally coincide with whatever way we implemented the local definitions. Theorem A.6. Suppose bew is a global LR-provability predicate for T w.r.t. N ,where N is an interpretation of S . Then, bew • is a global LR-provability predicatew.r.t. N .Proof. We assume the conditions of the theorem. We can copy the reasoning of theproof of Theorem 3.3, to show that • is closed under necessitation and L¨ob’s Rule.We verify L un bew • . Let A := ( V S ) N , where we assume S to include thetheory of identity. Reason in T . Suppose prov N [ ∅ ] ( b ). By the formalisation in S of global ∃ Σ b -completeness, we find prov [ ∅ ] ( imp ( p A q , p prov N [ ∅ ] (˙ b ) q )). It follows that bew ( imp ( p A q , p prov N [ ∅ ] (˙ b ) q )). Since, by necessitation, we also have bew ( p A q ). So,we find, by L un bew , that bew ( p prov N [ ∅ ] (˙ b ) q ). We leave T . We have shown: T ⊢ ∀ b ( prov N [ ∅ ] ( b ) → bew ( p prov N [ ∅ ] (˙ b ) q )) . It follows that:( ∀ a ( prov N [ ∅ ] ( a ) → bew • ( a ))) ⊢ T prov N [ ∅ ] ( b ) → ( bew ( b ) ∧ bew ( p prov [ ∅ ] N (˙ b ) q )) → ( bew ( b ) ∧ bew ( p bew • (˙ b ) q )) → bew ( conj ( b, p bew • (˙ b ) q )) → bew • ( b ) . We may conclude that ( ∀ a ( prov N [ ∅ ] ( a ) → bew • ( a ))) ⊢ T ∀ a ( prov N [ ∅ ] ( a ) → bew • ( a )),hence, by L¨ob’s Rule, T ⊢ ∀ a ( prov N [ ∅ ] ( a ) → bew • ( a )).We verify L un 3. Let B := ∀ a ∀ b (( bew • ( a ) ∧ bew • ( imp ( a, b ))) → bew • ( b )). Wehave: B ⊢ ( bew • ( a ) ∧ bew • ( imp ( a, b ))) → bew ( conj ( a, p bew • ( ˙ a ) q )) ∧ bew ( conj ( imp ( a, b ) , p bew • ( imp ( ˙ a, ˙ b )) q )) → bew ( a ) ∧ bew ( imp ( a, b )) ∧ bew ( conj ( bew • ( ˙ a ) , bew • ( imp ( ˙ a, ˙ b )))) → bew ( b ) ∧ bew ( bew • (˙ b )) → bew ( conj ( b, bew • (˙ b ))) → bew • ( b ) It follows that B ⊢ T B . So, by L¨ob’s Rule, we have T ⊢ B . ❑ We can now show that the ( · ) • -operation preserves Fefermanianness. Theorem A.7. Suppose prov Nα is an LR provability predicate. Then ( prov Nα ) • isalso a Fefermanian provability predicate.Proof. To simplify the presentation, we will omit the relativisation to N . We rea-son inside N but for the fact that α is not an internal N -formula. Inspectingthe fixed point construction, we see that prov • α ( a ) is of the form ( $ ) ∃ b ( S ( a, b ) ∧ prov α ( conj ( a, b ))), where S is an ∃ Σ b -formula that represents the relevant term,such that, T -verifiably, ∀ a ∀ b ( S ( a, b ) ↔ b = p prov • α ( ˙ a ) q ). We will treat the exis-tential quantifier of prov α in ( $ ) as giving the primary witness of prov • α ( a ).To define α • ( a ), we, sloppily, use meta-notations in the object language. Thus wewrite, for example, ( A ∧ B ) for conj ( a, b ). I think the gain in readability outweighsthe loss of precision. The reader just should remember that, locally, the romancapitals represent internal variables.We take α • ( A ) iff A is of the form ( p = p ∧ B ), where overlining gives us efficientnumerals and where p is a primary witness of • α B . In other words, p is a witnessof α ( B ∧ • α B ). The attentive reader will see that the definition of α • is anincarnation of Craig’s trick.We work in T . Suppose • α B . Let p be a primary witness. Then, ( p = p ∧ B ) isin α • . Hence trivially α • B .Now suppose α • B . Let q be a witnessing proof. Suppose ( p = p ∧ C ), . . . ,( p k − = p k − ∧ C k − ) are the (possibly non-standardly many) axioms used in q .We want to prove • α B . Let D := ( C → ( C → . . . ( C k − → B ) . . . )). We claimthat, inside α , we have:I. BII. • α C i , for i < k ,III. • α D ,IV. ∀ a, b (( prov • α ( a ) ∧ prov • α ( imp ( a, b ))) → prov • α ( b )).(I) follows from the fact that we can effectively transform q into an α -proof of B .We have (II) by the definition of the p i . We note that the finite set of witnessesof the α • α C i is not much larger than q . (III) follows from the fact that we cantransform q effectively into a witness of [ ∅ ] D . Since, by Theorem A.6, prov • α isglobal, it follows that • α D and, hence, α • α D . Finally, we have (IV) since α isglobal combined with necessitation for α .Now we combine (II), (III) and (IV) to effectively find a witness of α • α B . So,combining this with (I), we find α ( B ∧ • α B ). Hence, we have • α B . ❑ We have the following immediate corollary. Corollary A.8. Suppose prov Nα is a Fefermanian provability predicate. Then, prov Nα is an LR-predicate for T iff there is a Fefermanian predicate prov γ suchthat T ⊢ Nγ B → Nα B , for all T -sentences B , and prov Nγ satisfies L¨ob’s Logic over T . We also have: Corollary A.9. Suppose α is an LR-predicate for T . Then T interprets T + α ⊥ .Proof. Since we have the Interpretation Existence Lemma for Fefermanian prov-ability predicates, we have the result for prov α • . Moreover α • ⊥ is equivalent over T to • α ⊥ , which is again equivalent to α ⊥ . By the usual argument, we have T interprets T + α • ⊥ . Hence, T interprets T + α ⊥ . ❑ YCLIC HENKIN LOGIC 29 A.4. Craigification and Smoothening. To prepare the reader for the exampleof Section A.5, we briefly discuss Craigification and smoothening. The main pointis that smoothening does preserve the Feferman property of being a provabilitypredicate of an axiom class.In this subsection, we follow the sloppy ways of the proof of Theorem A.7 anduse meta-notations in the object language. We also will suppress the superscript N that signals relativisation to the chosen numbers.Consider α ( x ) of the form ∃ y α ( y, x ). We can transform α to its Craigification α cr as follows: • α cr ( A ) iff, for some n < A and B < A , we have A = ( n = n ∧ B ) and α ( n, B ).The smoothening prov sm α is defined as follows: • prov sm α ( A ) iff ∃ p ∃ f ( proof ( p, A ) ∧ ∀ B ∈ ass ( p ) α ( f ( B ) , B )).Here ‘ f ’ ranges over finite functions coded as numbers.We note that the smoothening takes the syntactic form of α as input, so the notationis a bit misleading. The basic insight on the relationship between Craigification andsmoothening is simply this: Theorem A.10. T ⊢ ∀ y ( prov sm α ( y ) ↔ prov α cr ( y )) . In other words, smoothening preserves the Fefermanian character of a provabilitypredicate.We will not give the proof here. The main thing is seeing that given an α -proof p and the finite function f , we can construct in p-time the corresponding α cr -proof p ∗ . Conversely, from p ∗ we can efficiently find both a p and and f . A.5. An Example. The reader does have to glance through Section 6.2 of [Vis19]to understand what is going on in this subsection.We use β as a standard representation of some single axiom that axiomatises EA .In the example we constructed an axiomatisation σ of EA with various properties.We will show that prov β and prov sm σ and prov • σ and prov σ are pairwise distinct over EA . Distinctness means that EA does not prove sameness.We will also show that the G¨odel sentence of the provability predicate defined isthe example has an explicit representation.A.5.1. Inclusions. We first note that since EA is finitely axiomatisable, prov σ is auniform provability predicate for EA and, hence, uniform. So, we do have L¨ob’sRule and hence all the desirable properties of • σ . We have:a. EA ⊢ ∀ x ( prov β ( x ) → prov sm σ ( x ))b. EA ⊢ ∀ x ( prov sm σ ( x ) → prov • σ ( x ))c. EA ⊢ ∀ x ( prov • σ ( x ) → prov σ ( x )) Proof. We prove (b). We note that prov sm σ ( x ) is Σ and that hence we have EA ⊢ ∀ x ( prov sm σ ( x ) → prov σ ( p prov sm σ ( ˙ x ) q )) . We choose our coding in such a way that these transformations are feasible. Note that if wehad e.g. the Ackermann coding for finite functions and a string style coding for sequences all thiswould get far less clear. Let A : ∀ x ( prov sm σ ( x ) → prov • σ ( x )). We have: EA + σ A ⊢ prov sm σ ( a ) → ( prov σ ( a ) ∧ prov σ ( p prov sm σ ( ˙ x ) q )) → ( prov σ ( a ) ∧ prov σ ( p prov • σ ( ˙ x ) q )) → prov • σ ( a )It follows that EA ⊢ σ A → A and, hence, by L¨ob’s rule, EA ⊢ A . ❑ A.5.2. Separations. We separate prov β from prov sm σ . Clearly, EA + S ⋆ ⊢ sm σ ℓ p = 1and EA + S ⋆ ⊢ β ℓ p = 1 → β ⊥ . However, as is shown in [Vis19], EA + S ⋆ β ⊥ .So, EA + S ⋆ β ℓ p = 1. We separate prov sm σ from prov • σ . In EA + S ⋆ , smoothening does allow us to usemore and more of the non-standardly finitely many axioms but never all in oneproof. It now follows by a minor adaptation of the argument for Lemma 6.13(b) of[Vis19] that, over EA + S ⋆ , we have that sm σ ⊥ is equivalent to β ⊥ . At the sametime, • σ ⊥ is equivalent to σ ⊥ . Moreover, by the results of [Vis19], we have theequivalence of σ ⊥ and β β ⊥ . By a model theoretic argument analogous to theargument in [Vis19], we find that EA + S ⋆ β β ⊥ → β ⊥ . So, EA + S ⋆ does notprove the equivalence of sm σ ⊥ and • σ ⊥ . We separate prov • σ from prov σ . Suppose we would have the equivalence of prov σ and prov • σ over EA . It would follow that we have L¨ob’s Logic for σ . But this wasrefuted in [Vis19].A.5.3. Explicit G¨odel Sentence. We have seen that the G¨odel sentence for σ isunique over EA . But what is it? The next theorem answers this question. Theorem A.11. The G¨odel sentence of σ is, modulo EA -provable equivalence, σ σ ⊥ .Proof. Since we already have uniqueness, it is sufficient to show that σ σ ⊥ is EA -provably equivalent to ¬ σ σ σ ⊥ . In other words, we want to show that σ σ ⊤ is EA -provably equivalent to σ σ σ ⊥ .We remind the reader of the following lemmas of [Vis19]. Lemma 6.9: EA + S ⋆ ⊢ β ¬ S ⋆ . Lemma 6.11: EA + S ⋆ ⊢ σ A ↔ β ( β ⊤ → A ). Lemma 6.14(a): EA + S ⋆ ⊢ σ σ ⊤ . Lemma 6.14(b): EA + ¬ S ⋆ ⊢ σ σ ⊤ ↔ β ⊥ .By Lemma 6.14(b), it follows that: EA + ¬ S ⋆ ⊢ σ σ σ ⊥ → σ σ ⊤→ β ⊥→ σ σ σ ⊥ So, in EA + ¬ S ⋆ we have the desired equivalence.We show we also have the equivalence in EA + S ⋆ . In the light of Lemma 6.14(a),it suffices to show EA + S ⋆ ⊢ σ σ σ ⊥ . We reason in EA + S ⋆ . By 6.11, σ σ σ ⊥ is equivalent to β ( β ⊤ → σ σ ⊥ ). By 6.9, we have (i) β ¬ S ⋆ . By applyingnecessitation to 6.14(b), we have(ii) β ( ¬ S ⋆ → ( σ σ ⊤ ↔ β ⊥ )) . YCLIC HENKIN LOGIC 31 By combining (i) and (ii), we find that σ σ σ ⊥ is equivalent to β ( β ⊤ → β ⊤ ), which in its turn is equivalent to ⊤ . So, we are done. ❑ Open Question A.12. Can we give an example of a modalised fixed point thathas no definable solution for the case of EA , σ ?Can we given an example of a pair EA , τ , where τ is Σ and τ is a provabilitypredicate for EA such that the G¨odel sentence is not explicitly definable? If not,what about the more general case? ❍ Open Question A.13. Consider an Σ -predicate τ that axiomatises EA in EA .The provability logic of • τ contains GL . However, prima facie , Solovay’s proof fails.Can we still prove that the logic is precisely GL . What about the logic for • σ forthe specific predicate σ studied above? ❍ Philosophy, Faculty of Humanities, Utrecht University, Janskerkhof 13, 3512BL Utrecht,The Netherlands Email address ::