INFINITARY PROPOSITIONAL RELEVANT LANGUAGES
WITH ABSURDITY
GUILLERMO BADIA
Department of Knowledge-Based Mathematical Systems Johannes Kepler Universität Linz
Abstract. Analogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logicL∞ωholds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic rela-tion of relevant directed bisimularela-tion as well as a Beth definability property.
§1. Introduction. In these pages we explore the model theory of a twofold nonclas-sical logic: infinitary relevant propositional logic. By extending the language of relevant logic by adding infinitary conjunctions and disjunctions, we naturally gain some expressive power. Such extensions have been toyed with from time to time in the context of relevant logic in an unsystematic and informal way (cf. [15, 16, 25]). In [25] (p. 336), Routley reports some unpublished (and, according to him, not overly successful) attempts to study infinitary relevant logic.
We will be working in the well-known Routley–Meyer semantics [13, 26–29]. This is the more or less standard nonalgebraic semantics for relevant logic ( [24, 30] are examples of quite recent applications). The reader can find a survey of the alternatives in [13], though.1
Though the heyday of infinitary logic seems to be long gone, important results remain. In the next sections, we will obtain relevant analogues of some of them such as Karp’s theorem or Scott’s isomorphism theorem. Karp’s theorem (Corollary 3.5.3 in [17]) is the claim that for any two models, L∞ω-equivalence is the same as the existence of a family of partial isomorphisms with the back and forth properties. Scott’s isomorphism theorem (Corollary 3.5.4 in [17]) says that, for denumerable models, making a single special for-mula true suffices to characterize a structure up to isomorphism.
The main problem we will solve here, though, is that of characterizing the expressive power of infinitary relevant logic. This will be accomplished by establishing a generalized interpolation result for the classical infinitary logic L∞ω, from which the desired char-acterization will follow in the form of a preservation theorem involving relevant directed bisimulations. On a historical note, directed bisimulations were introduced in [20] and
Received: January 26, 2016.
2010Mathematics Subject Classification: 03B47, 03C75.
Key words and phrases: relevant logic, model theory, infinitary logic, interpolation, Routley– Meyer semantics.
1 [21] is a recent contender for the quantificational case, where incompleteness had been found by
Fine [16].
though it was hinted there, it seems like [22] is the first time they were applied to the study of substructural logics in print. Recently, they have been shown to have a fundamental place in the model theory of relevant logic in the Routley–Meyer semantic framework (cf. [2], where the finitary case has been studied) analogous to bisimulations in the Kripke semantics for modal logic.
The results on expressive power in this paper can be seen as a continuation of the work in [2], turning our attention this time to the realm of infinitary languages. There are certain differences in method worth mentioning, though. In [2], there was an appeal to the machin-ery of saturated models in order to establish a preservation theorem characterizing relevant formulas as a fragment of first order logic. This was, in fact, unnecessary for a much more direct proof through a simple application of the compactness theorem of first order logic was possible. It would have simply required the introduction of the notion of a rele-vant directedn-bisimulation, a finite approximation of a relevant directed bisimulation.2 This approach is so basic that it generalizes to logics having some minimal forms of compactness such asL∞ω. That is the main motivation behind our introduction of relevant directedα-bisimulations in Definition 4.3.
In §2, we introduce the Routley–Meyer semantics for infinitary propositional relevant languages with absurdity. In §3, we show that infinitary relevant languages with absurdity are, in general, lacking compactness and most reasonable formal systems based on them are not strongly complete. In §4, we define relevant directed bisimulations establishing some basic propositions, including a relevant Karp theorem while in §5, we prove a relevant analogue of Scott’s isomorphism theorem. In §6, we prove an interpolation theorem for the infinitary quantificational boolean logic L∞ω which implies a preservation theorem say-ing that the formulas of L∞ωpreserved under relevant directed bisimulations are exactly infinitary relevant formulas, as well as a Beth definability result. Finally, in §7 we briefly summarize our work.
§2. Routley–Meyer semantics. In this section, we will review the Routley–Meyer semantics for propositional infinitary relevant languages with absurdity and their embed-dability in more traditional infinitary languages.
Let κ be some infinite cardinal. An infinitary relevant language with absurdity L→κω contains a possibly finite listPROPof propositional variables p,q,r. . . and the logical symbols⊥(an absurdity constant),∼(negation),(conjunction),(disjunction), and →(implication). Formulas are constructed as expected:
φ::= p⊥ ∼φ i∈Iφi i∈Iφi φ→ψ,
wherep∈PROPand|I| < κ. The infinitary relevant language with absurdityL→∞ωcomes from letting the index I of a disjunction or a conjunction take any cardinality whatsoever. L→ωωis just an ordinary finitary relevant language.
A comment on the presence of ⊥in our languages is in place here, given that ⊥is not standardly part of the languages of relevant logic (cf. [1]). The results in these pages cannot dispense with⊥, since languages without⊥have no reasonable model-theoretic characterization. The interested reader is advised to consult §4 in [2].
Note that implications are still finitary in the sense that we can only build formulas of the form
φ0→(φ1→(φ2→(· · · →φλ) . . . )
2 Incidentally, this is how the main result of [3] characterizing the expressivity of propositional
whenλis finite. This is the reason for writingωinL→κω, it basically bounds the possible number of iterations of a→ symbol in a formula. This notation should not be confused with the classical notation where the second subscript is used to bound the possible length of a string of quantifiers.3
An example of a connective definable in L→∞ω (but not in L→ωω) is −→ω (iterated entailment), which was introduced by Humberstone (see [10], p. 36). The formulaφ−→ω ψ means that for some natural numbern >1,
φ→(φ→(. . . (φ
n−φs
→ψ) . . . ))
holds. This, of course, boils down to an infinitary disjunction of finitary implications:
n>1φ →(φ→(. . . (φ n−φs
→ψ) . . . )).
As we announced in §1, we will be working in the Routley–Meyer semantic framework. In this setting, amodelfor L→κω will be a structure M = W,R,∗,T,V, whereW is a nonempty set, T ∈ W,∗is an operation∗ : W −→ W (the so called Routley star), R ⊆W×W×W andVis a valuation functionV :PROP−→℘ (W). In what follows we frequently omit T from the presentation of our models since nothing essential hinges on that (given that we will not be considering any connectives involvingT in its semantics) and the reader can easily fill in the omitted details.
We define satisfaction atwinM recursively as follows:
M, w⊥ never
M, wp iff w∈V(p)
M, w(∼φ) iff M, w∗φ
M, w(i∈Iφi) iff M, wφi for everyi∈ I.
M, w(i∈Iφi) iff M, wφi for somei∈ I.
M, wφ→ψ iff for everya,bsuch thatRwab,
ifM,a φthenM,bψ.
Note that as⊥gives us a means to define the empty class of models, =d f (∼ ⊥)allows
defining the class of all models since it is invariably true (for recall that⊥invariably fails atw∗for anyw).
The basic semantic units in relevant logic are (as in modal logic)pointed models, that is, pairs(M, w)wherewis some distinguished element ofW. This is simply due to the fact that formulas are evaluated locally, at worlds.
3 It is opaque whether there is a connection here. For instance,φ→(φ→(. . . (φ
ω−φs
→ψ) . . . )) could be naïvely translated—without the intervention of infinitely long strings of quantifiers— into a “classical infinitary” language with the appropriate signature, using the translation function given below, as ∀y0z0(Rx y0z0 ∧ Tx(φ)y0/x ⊃ ∀y1,z1(Rz0y1z1 ∧ Tx(φ)y1/x ⊃
(. . .∀vu(Rzωvu∧Tx(φ)v/x ⊃ Tx(ψ)u/x) . . . ))). The problem is that this is not a formula of
any classical infinitary languageLκλ. The reason is that it violates the well-foundedness of the subformula relation (Lemma 1.3.3 from [12]). To see this note that the collection of formulas ∀yizi(Rzi−1yizi∧Tx(φ)yi/x⊃ ∀yi+1,zi+1(Rziyi+1zi+1∧Tx(φ)yi+1/x⊃(. . .∀vu(Rzωvu∧
Tx(φ)v/x ⊃Tx(ψ)u/x) . . . )))(0<i< ω) has no minimal element according to the subformula
By considering restricted classes of Routley–Meyer structures where the relationRhas certain properties and only some valuations are admitted, we can get classes of models corresponding to a number of formal systems of relevant logic likeB,T, orR. Next we will consider some famous examples from [26].
Consider a relevant language with absurdityL. A structureW,R,∗,T,Vis called a
B-modelif for anyx,y,z, v∈W:
(i) RT x x.
(ii) RT xvandRvyzimplies thatRx yz. (iii) x =x∗∗.
(iv) RT x yonly ifRT y∗x∗.
(v) x ∈V(p)andRT x yimplies thaty∈V(p).
AnR-modelis aB-model where condition (iv) is strengthened to
(iv) Rzx yonly ifRzy∗x∗,
and, furthermore (abbreviating the claim that there is a u such that Rx yu and Ruzv as R2(x y)zv, and the claim that there is anusuch thatRxuvandR yzuasR2x(yz)v), for any x,y,z, v∈W:
(v) R2(x y)zvonly if R2x(yz)v.
(vi) Rx x x.
(vii) Rx yzonly if R yx z.
AnRM-modelis anR-model such that and for anyx,y,z∈W:
(v) Rx yzonly if eitherRT x zorRT yz.
Whenis the deducibility relation of some formal systemSof relevant logic, a syntactic claim of the formφ ψis to be interpreted on the class of corresponding modelsVS as saying that M,T φonly if M,T ψfor every model M ∈ VS. In what follows we will use the symbolVSas a variable for the class of models corresponding to any systemS described in [26] betweenBandRM.4
Next we give an example of the increased expressive power of infinitary relevant
lan-guages. Suppose and are sets of formulas. We speak of the pair ( , )as being
satisfiableorhaving a modelin a classKof pointed models if there is a model(M, w)∈K
such that M, w φfor eachφ ∈ andM, w ψ for everyψ ∈ . These pairs are
calledtableauxin [11] (pp. 37–38).5LetV be a class of pointed models. A class of pointed models K ⊆ V is said to beaxiomatizableinL→ωω with respect toV if there is a set of
formulasof L→ωω such thatK = Mod()—whereMod()the class of pointed model
satisfying. Let(M, w)be a model for L→ωω. We say that(M, w) is inconsistent if for somep∈PROP,M, w(p∧(∼p)).
4 A caveat is in place here. Thevariable sharing propertyis a folklore requirement from any formal
system of relevant logic. The property states that wheneverφ→ψ is a theorem thenφandψ must share some propositional variable in common. When our language has ⊥, the principle fails quite easily since⊥ → θ (for arbitraryθ) would be a theorem, tempting one to claim that no system involving ⊥should qualify as a system of relevant logic. However, Yang [31] has suggested recently thestrong implicit relevance propertyas a nice substitute of the variable sharing property that would allow for systems containing⊥.
Inconsistency is definable by a sentence of a propositional relevant language with absur-dity L→ωωifPROPis finite, for in this casep∈PROP(p∧(∼ p))expresses that a model is inconsistent. If the signature is not finite, inconsistency is not in general a property axiomatizable inL→ωω. This has been pointed out forL P essentially in [14] with an argu-ment using a version of Ło´s’s theorem on ultraproducts.
PROPOSITION2.1. If|PROP|ω, inconsistency is not a property of models
axiomati-zable in L→ωωwith respect to any VS.
Proof. Suppose it were. Say the theoryaxiomatizes the class of inconsistent models.
Now, the pair(, )where = {(p∧(∼ p)): p ∈PROP}is finitely satisfiable inVS. To see this take a finite subset{p0, . . . ,pn} ⊂PROP. Consider the modelW,R,∗,V,T
(inVSsince it is inVRM) such that
W = {t,s}.
∗ = {t,s,s,t}.
R= {t,t,t,t,s,t,t,s,s,s,t,t,s,t,s,s,s,t,s,s,s}.
T =t.
V(pi)=W (fori=0, . . . ,n).
V(q)= {t}(forq ∈PROP, andq =pi fori =0, . . . ,n).
We see that ifq ∈PROPbutq = pi fori =0, . . . ,n, thenM,t (q∧(∼q)). On the
other hand,M,t (pi∧(∼ pi)) (i =0, . . . ,n)sincet∗=s∈V(pi), which means that
M,t (∼ pi).
Finally, by Proposition 2.5 of [2], the pair(, )is satisfiable inVS, which is a
contra-diction since by definitionsays that at least one ofφ∈ must hold.
When|PROP| ω, inconsistency is expressible by a single formula in the extension L→|PROP|+ω of L→ωω. Again,p∈PROP(p ∧(∼ p))expresses that a model is inconsistent.
This fact shows thatL→|PROP|+ωis a proper expressive extension ofL→ωω.
Consider an infinitary language with equality and boolean negation admitting conjunc-tions and disjuncconjunc-tions of size at most κ (the standard reference for the study of such languages is [12]) and quantifications over at most finitely many variables that comes with an individual constant symbolT, one function symbol∗, a distinguished three place relation symbol R, and a unary predicate P for each p ∈ PROP. Following the tradition in modal logic, we might call this a correspondence language Lcorrκω for L→κω (cf. [9]). Now we can read a model M as a classical model for Lcorrκω in a straightforward way: W is taken as the domain of the structure, the constant T denotes the obvious distin-guished world, V specifies the denotation of each of the predicates P,Q, . . ., while ∗ is the denotation of the function symbol∗of Lcorr, and R the denotation of the relation
RofLcorr
κω .
Wheretis a term in the correspondence language, we writeφt/x for the result of replac-ingx witht everywhere in the formulaφ. As expected, it is easy to specify a translation from the formulas of the basic relevant language with absurdity to the correspondence language as follows: Tx(⊥) = ¬Rx x x∧Rx x x. Tx(p) = P x. Tx(∼φ) = ¬Tx(φ)x∗/x. Tx( i∈Iφi) = i∈ITx(φi). Tx( i∈Iφi) = i∈ITx(φi). Tx(φ→ψ) = ∀y,z(Rx yz∧Tx(φ)y/x ⊃Tx(ψ)z/x).
The symbols ¬and⊃appear here representing boolean negation and material implica-tion in quantificaimplica-tional infinitary logic (which should not be confused with the relevant
∼and→).
The following proposition gives a bridge between the satisfaction relationfor rele-vant propositional languages we just defined and the standard satisfaction relationfrom classical logic (where whenφis a classical formula, we writeM φ[w] to mean that the objectwsatisfiesφin the usual Tarskian sense).
PROPOSITION2.2. For anyw, M, wφif and only if M Tx(φ)[w].
Proof. We simply need to note that, according to the Routley–Meyer semantics, each
propositional relevant formula φ says the same aboutw as Tx(φ)does in the Tarskian
semantics.
The existence of a satisfaction preserving translation function allows us to study relevant languages with absurdity as fragments of model-theoretically better understood creatures.
§3. Failure of compactness and strong completeness. In this section, we study briefly a phenomenon pervasive in infinitary logic even at the propositional level: the loss of compactness. This quickly leads to a loss of strong completeness for any reasonable in-finitary formal system (cf. [18]). Such seems to be the price to pay for having infinitely long conjunctions and disjunctions around. Here we will focus our attention on specific classes of models since we will be discussing questions sensitive to the choice of formal system such as incompleteness.
DEFINITION3.1. Let L→κω be a relevant language with absurdity, K a class of Routley–
Meyer structures for it and( , )a pair of collections of relevant formulas. L→κωis said to
beλ-compact with respect to Kif for every 0⊆ and0⊆such that| 0|,|0| < λ,
the pair( 0, 0)has a model in K only if( , )has a model in K .
PROPOSITION3.2. Let|PROP|κ. L→κωisκ-compact with respect to some VSonly if
κis a regular limit cardinal.
Proof. Supposeκis a sucessor cardinalξ+1. Without loss of generality, assumePROP
is composed of double indexed propositional variablespλγ (λ < ξ+1, γ < ξ). Consider the set of formulas
= γ < ξpλγ :λ < ξ+1∪ {pλγ∧ pμγ → ⊥:μ=λ, μ, λ < ξ+1, γ < ξ}.
Take any 0 ⊂ such that |0| ξ. By the axiom of choice, there is a one-to-one
mapping f from the set of allλsuch thatpλγ for someγappears in a formula of0intoξ.
We build the model where W = {t}, R = {t,t,t},∗ = {t,t}, and we define V as follows:V(pλf(λ))=W, andV(pλγ)= ∅ifγ = f(λ). It is clear thatM,tγ < ξ pλγ
for all disjunctions in0withγ < ξ+1. Now take anypλγ∧pμγ → ⊥ ∈0such that μ=λ, μ, λ < ξ+1,andγ < ξ. Since f is an injection we have that f(μ)= f(λ), so pλγ andpμγ will never hold simultaneously at any world inW by our definition ofV. Hence, M,t pλγ ∧ pμγ → ⊥by antecedent failure. However,itself has no model, contradictingκ-compactness.
Suppose on the other hand thatκis singular. In [12] (p. 85) it is noted that the infinitary languages Lκω where κ is singular are exactly as expressive as languages Lκ+ω. The argument holds forL→κωas well. Hence, without loss of generality, we can take
to be a perfectly good collection of formulas ofL→κω. As before every subset0⊂such
that|0| < κhas a model inVSbutdoes not.
A Hilbert-style formal systemHfor a languageL→κωwith respect to the class of models for a standard system for relevant logic will be formed by a set of formulas ofL→κωtaken as the collection of axioms and a collection of rules of inference each with less than κ premises. Ifis a collection of formulas of L→κω andφa formula of L→κω, we will write
H φif there is a sequence of formulasSof length less thanκ such that every formula
inSis either an axiom, one of the formulas inor it follows from previous formulas inS using one of the inference rules.
PROPOSITION3.3. Let|PROP| κ+. Let H be a formal system for L→κ+ω sound with
respect to some VS. Then H is not strongly complete.
Proof. Takein the proof of Proposition 3.2. Since every0⊆with|0| < κhas
a model inVS, by the soundness ofH, we see that0H ⊥, but that means thatH ⊥.
Howeversemantically implies⊥overVS, since it has no model.
§4. Relevant directed bisimulations and Karp’s Theorem. In this section, we in-troduce relevant directed bisimulations, establish some basic facts that will be needed in §6 and prove the relevant analogue of Karp’s theorem. The present section as well as §6 focuses on the infinitary relevant language with absurdityL→∞ω.
DEFINITION4.1. The degreeof an infinitary relevant formulaφ, in symbols, dg(φ), is
defined inductively in the following way:
dg(⊥) = 0, dg(p) = 0, dg(i∈Iφi) = sup{dg(φi):i ∈ I}, dg(i∈Iφi) = sup{dg(φi):i ∈ I}, dg(∼φ) = dg(φ), dg(φ→ψ) = sup{dg(φ),dg(ψ)} +1.
We will say that two formulasφandψareequivalentif for any model(M, w),M, w
φiffM, wψ.
PROPOSITION4.2. For each ordinalα, there are only set-many nonequivalent formulas
of L→∞ωwith degreeα.
Proof. Consider firstLcorr∞ω. Define the quantifier rank of a formula ofLcorr∞ω following [4]
(Definition 10. 4) which deals appropriately with the presence of functions in the language. According to Corollary 10.9 in [4], forκsome fixed point of the functionwith cardinality bigger than the cardinality of the signature ofLcorr∞ω (there is always some suchκgiven that
is normal), every formula ofLcorr∞ω with quantifier rankαis equivalent to a disjunction of size smaller than κ of formulas of a certain classwith fewer thanκ nonequivalent members. Clearly, there are only set-many nonequivalent such disjunctions. Hence, there are only set-many nonequivalent formulas ofLcorr∞ω with quantifier rankα.
Finally since relevant formulas of degreeαcan be seen via the translation as formulas ofLcorr
∞ω with quantifier rankβ for sufficiently bigβ, we have established the result.
Relevant directed bisimulations—as bisimulations in modal logic—are “nonclassical” analogues of back and forth games from classical model theory. In this sense, the next definition introduces the analogue of Definition 5.3.3 from [12].
DEFINITION4.3. Let M1= W1,R1,∗1,V1and M2= W2,R2,∗2,V2be two models.
Arelevant directedα-bisimulationforPROPbetween M1and M2is a system of pairs of
nonempty relationsZ01,Z02, . . . ,Zα1,Zα2where
Zβ1⊆W1×W2and Zβ2⊆W2×W1 (0βα)
such that
Zα1⊆ · · · ⊆ Z01
Zα2⊆ · · · ⊆ Z02
and when i,j ∈ {1,2},0β < αand0γ α,
(1) x Zγiy only if y∗jZγjx∗i
(2) If x Z(β+1)iy and Rjybc for some b,c∈Wj, there are b,c∈Wisuch that Rixbc,
b Zβjband cZβic.
(3) If x Zγiy and p∈PROP,
Mi,x p only if Mj,yp.
PROPOSITION4.4. Let(M1, w1)and(M2, w2)be two arbitrary Routley–Meyer models,
αan ordinal and i,j ∈ {1,2}. Then,(i)for each relevant formulaφof L→∞ω with degree
α, Mi, wi φonly if Mj, wj φiff(ii)there is a relevant directedα-bisimulation
(Zβi,Zβj)βαsuch thatwiZβjwj for eachβ α.
Proof. (ii) ⇒ (i): Assume that (ii). We argue for (i) for all α simultaneously, by
induction on the complexity ofφ.
The atomic cases as well as⊥are obvious from (3) in Definition 4.3 and the fact that ⊥ is never true. For negation, letφ = (∼ ψ) and suppose that Mi, wi (∼ ψ), so
Mi, wi∗i ψ. Butw∗jjZαjwi∗i by (1) in Definition 4.3 sincewiZαjwjby assumption, and,
by inductive hypothesis,Mj, w∗jj ψ, soMj, wj (∼ψ)as desired. Conjunction and
disjunction are routine exercises.
The only remaining case is whenφ = ψ → χ. By Definition 4.1, say thatdg(φ) =
β +1 α, whereβ = sup{dg(ψ),dg(χ)}. Suppose that Mi, wi ψ → χ, which
means that if Riwibc for some b,c, and Mi,b ψ, then Mi,c χ. Now, let
Rjwjbcfor arbitraryb,c. We need to show thatMj,bψonly ifMj,cχ. To get the
contrapositive, we will suppose that Mj,c χ. By the assumption (ii),wiZβ+1iwj, so
using property (2) in Definition 4.3, there areb,csuch thatRiwibc,b Zβjb, andcZβic.
Note that Z0i,Z0j, . . . ,Zβi,Zβj is a directed β-bisimulation between Mi and Mj.
This follows readily from our assumption that Z0i,Z0j, . . . ,Zαi,Zαj is a relevant
directed α-bisimulation between Mi and Mj by verifying (1)–(3) in Definition 4.3. By
inductive hypothesis, sinceMj,cχ anddg(χ) β,Mi,c χ. Given that Mi, wi
ψ →χ, it must be thatMi,b ψ. But by inductive hypothesis again using the fact that
b Zβjbanddg(ψ)β,Mj,bψ. Hence,Mj, wj ψ→χ.
(i)⇒(ii): For a modelS, and worldwfromS, we denote byr elγ-t pS(w)the relevant
type up to degreeγ ofw, i.e., the set of all infinitary relevant formulas such thatS, wφ anddg(φ)γ. We claim that, on the assumption that (i), the following system of relations defines a relevant directedα-bisimulation betweenMi andMj:
x Zβiyiffr elβ-t pMi(x)⊆r elβ-t pMj(y)(0βα)(i = j,i,j ∈ {1,2}).
Let us first note that Zαm ⊆ · · · ⊆ Z0m (m ∈ {1,2}). By the asumption (i), Zαi is
nonempty, sincewiZ0iwj, but the latter also implies thatw∗jjZαjw∗ii as we will see below,
Let 0 β α,i,j ∈ {1,2}. If x Zβiy, i.e., r elβ-t pMi(x) ⊆ r elβ-t pMj(y), we
see thatr elβ-t pMj(y∗j) ⊆r elβ-t pMi(x∗i), i.e., y∗jZβjx∗i. It suffices to show that if
Mj,y∗j φthenMi,x∗i φfor everyφwithdg(φ)β. We prove the contrapositive.
Suppose that Mi,x∗i φ, soMi,x (∼ φ)and sincedg(∼ φ) = dg(φ)andr elβ
-t pMi(x) ⊆ r elβ-t pMj(y), also Mj,y (∼ φ). Consequently, Mj,y∗ j
φ as we
wanted. This takes care of (1) in Definition 4.3.
For clause (2) in Definition 4.3, suppose that x Z(β+1)iy (β+1 α), i.e.,r elβ+1
-t pMi(x) ⊆ r elβ+1-t pMj(y), and Rjybc for some worlds b,c from Mj. Where
Fmla(L→∞ω) stands for the class of propositional relevant formulas ofL→∞ω, consider
nr elβ-t pMj(c)= {φ: Mj,cφ, φ∈Fmla(L→∞ω),dg(φ)β}.
By Proposition 4.2 we see thatr elβ-t pMj(b)as well asnr elβ-t pMj(c)can be taken
as sets. It is clear that
Mj,yr elβ-t pMj(b)→
nr elβ-t pMj(c)
sinceRjybc,Mj,br elβ-t pMj(b)butMj,c
nr elβ-t pMj(c). Observe that dg(r elβ-t pMj(b)→ nr elβ-t pMj(c)) =sup{dg(r elβ-t pMj(b)),dg( nr elβ-t pMj(c))} +1, but dg(r elβ-t pMj(b))=sup{dg(δ):δ∈r elβ-t pMj(b)}β and dg(nr elβ-t pMj(c))=sup{dg(σ ):σ ∈nr elβ-t pMj(c)}β, so dg(r elβ-t pMj(b)→ nr elβ-t pMj(c))β+1.
Thus, sincer elβ+1-t pMi(x)⊆r elβ+1-t pMj(y), contraposing,
Mi,x
r elβ-t pMj(b)→
nr elβ-t pMj(c),
which means that there are b and c such that Rixbc, Mi,b
r elβ-t pMj(b),
andMi,c
nr elβ-t pMj(c). Hence,r elβ-t pMj(b)⊆r elβ-t pMi(b), i.e.,b Zβjb.
On the other hand, we have that if Mj,c φ then Mi,c φ wheneverdg(φ) β.
Contraposing,r elβ-t pMi(c)⊆r elβ-t pMj(c), i.e.,cZβic.
Condition (3) in Definition 4.3 follows given that atomic formulas have degree 0. DEFINITION4.5. Let M1= W1,R1,∗1,V1and M2= W2,R2,∗2,V2be two models.
A relevant directed bisimulation for PROP between M1 and M2 is a pair of nonempty
relationsZ1,Z2where
Z1⊆W1×W2and Z2⊆W2×W1
such that when i,j ∈ {1,2},
(1) x Ziy only if y∗jZjx∗i
(2) If x Ziy and Rjybc for some b,c ∈ Wj, there are b,c ∈ Wi such that Rixbc,
b Zjband cZic.
(3) If x Ziy and p∈PROP,
Next we show an analogue of Karp’s celebrated theorem characterizing L∞ω -equivalence in terms of partial isomorphisms. The corresponding result for modal logic is regarded as a “folklore” theorem.
THEOREM4.6 (Relevant Karp’s Theorem). Let(M1, w1)and(M2, w2)be two models and
i,j ∈ {1,2}. Then the following are equivalent:
(i) for every formulaφof L→∞ω, Mi, wi φonly if Mj, wj φ.
(ii) there is a relevant directed bisimulationZi,Zj between M1 and M2such that
wiZiwj.
Proof. (ii) ⇒ (i): This direction follows from Proposition 4.4, and the facts that
Zi,Zjcan be taken to be a relevant directedα-bisimulation for any αand that every
formula ofL→∞ωhas some degreeα.
(i)⇒(ii): We claim that
x Ziyiffrel-tpMi(x)⊆rel-tpMj(y)(i = j,i,j ∈ {1,2})
defines a relevant directed bisimulation whererel-tpMi(x)(i =1,2) is the collection of all
formulas ofL→∞ωholding atxinMi.
For clause (1) in Definition 4.5, suppose x Zβiy, i.e.,rel-tpMi(x) ⊆ rel-tpMj(y). We
have that that rel-tpMj(y∗j) ⊆ rel-tp Mi(x∗
i), i.e., y∗jZβ
jx∗i. It suffices to show that if
Mj,y∗j φthen Mi,x∗i φ for everyφ. We prove the contrapositive. Suppose that
Mi,x∗i φ, soMi,x (∼φ)and sincerel-tpMi(x)⊆rel-tpMj(y), alsoMj,y(∼φ).
Consequently,Mj,y∗ j
φas we wanted.
Now we have to take care of clause (2) in Definition 4.5. Assume that x Ziy, i.e.,
rel-tpMi(x)⊆rel-tpMj(y), andRjybcfor some worldsb,cfromMj. Suppose for reductio
that there are nob,c∈Wi such thatRixbc,b Zjb(i.e.,rel-tpMj(b)⊆rel-tpMi(b)) and
cZic(i.e.,rel-tpMi(c)⊆rel-tpMj(c)). We first notice that{b,c∈Wi :Rixbc} = ∅, for
otherwiseMi,x → ⊥, soMj,y → ⊥, which implies thatMj,c⊥, which is
impossible. Now, for anyb,c∈Wi such thatRixbcthere are formulasφb andφc such
that either (i)Mj,bφb andMi,b φb or (ii)Mi,c φc andMj,cφc. For any
b,c∈Wi such thatRixbcdefine the transformationτ as follows:
τ(φb)=
if (i) does not hold,
φb otherwise.
τ(φc)=
⊥ if (ii) does not hold,
φc otherwise.
Next, it suffices to consider the formula ∃vRxbv b∈Wi τ(φb )→∃vRi xvc c∈Wi τ(φc).
A moments reflection shows that
Mj,y ∃vRi xbv b∈Wi τ(φb)→ ∃vRi xvc c∈Wi τ(φc) but Mi,x∃vRi xbv b∈Wi τ(φb)→ ∃vRi xvc c∈Wi τ(φc),
contradicting the assumption thatrel-tpMi(x)⊆rel-tpMj(y).
Theorem 4.6 is nothing but the infinitary version of Theorem 13.5 from [22]. Quite frequently in infinitary logic we are able to obtain counterparts to results provable for finitary languages with the restriction that the models under consideration be finite.
§5. Scott’s theorem. Next we establish a result implying a corollary analogous to Scott’s isomorphism theorem in classical infinitary logic. The corresponding theorem for modal logic was proven in [8].
Since finitary relevant logic is considerably weaker than first order logic and modal logic in terms of expressive power, it only seems natural that to get a version of Scott’s isomorphism theorem one has to go beyond the expressive power gained by merely adding countable conjunctions. In fact, Corollary 5.2 requires us to add conjunctions of cardinality at most|2ω|.
There is another difference between the following result and Scott’s isomorphism the-orem or van Benthem’s modal version of it. Scott’s thethe-orem gives a formulaφM
charac-terizing up to isomorphism a given countable model M among the class of all countable models, so Scott’s formula only depends on the model M. In contrast, we give a formula that implies that there is a relevant directed bisimulation between two arbitrary countable models but which depends on both. This difference is due to the nature of relevant directed bisimulations. Contrary to isomorphism or bisimulation, a relevant directed bisimulation betweenM1andM2demands things from both models. Recall that it is not a relation from
W1×W2but a pair of relations fromW1×W2andW2×W1respectively.
THEOREM5.1. Let(M1, w1)and(M2, w2)be two models in some K such that K ⊆VB,κ
the least infinite cardinalsup{|W1|,|W2|}, andλ=sup{|PROP|,2κ}. Then, when i,j ∈
{1,2}, there is a formulaθwi of L→λ+ωsuch that(1)Mi, wi θwi, and(2)Mj, wj θwi iff
there is a relevant directed bisimulation(Zi,Zj)between Miand Mj such thatwiZiwj.
Proof. We start by defining for each worldaof Mi the formulaφηMaj—simultaneously
withφηMb
i forb∈Wj—by induction on the ordinalη < λ
+as follows: φ0a
Mj = the set of all literals satisfied by(Mi,a), φηa Mj = ξ < ηφξMaj ifηis a limit ordinal, φη+1a Mj = φ ηa Mj ∧ b∈W j, X⊆Wi, Mi,aφηMib→d∈XφηM jd φηb Mi → d∈Xφη d Mj ∧ b∈W j, X⊆Wi, Mi,a(∼(φηMib→d∈XφηM jd)) (∼(φηb Mi → d∈Xφη d Mj)).
Observe that whenγ < β < λ+,
Mj,aφβMaj implies that Mj,a
φγa Mj.
This can be seen by induction onβ. The case whenβ =0 is true by antecedent failure. Ifβ =η+1, eitherη=γorγ < η. If the first, since
Mj,aφγMaj ∧ b∈W j, X⊆Wi, Mi,aφγb Mi→ d∈Xφγ d Mj φγb Mi → d∈Xφγ d Mj implies that Mj,a φγa Mj, we have that Mj,aφβMa j implies that Mj,a φγa Mj.
If the second, since
Mj,aφβMaj implies that Mj,aφηMaj,
and, by inductive hypothesis,
Mj,aφηMaj implies that Mj,aφγMaj,
we get what we needed.
Now let us define a map f :W1×W2−→λ+in the following way:
f(a,a)=
the least ordinalξ < λ+such thatM2,aφξMa2 if there is some,
0 otherwise.
Given that|W1×W2| =κ < cf(λ+)=λ+, we see that there must beξ0 < λ+such that
the range of f is a subset ofξ0. Consequently, for everyβ such thatξ0 < β < λ+,
M2,aφξM0a
2 implies that M2,a φβa
M2,
for otherwise we have that there is an ordinalγ withξ0 < γ β which is the smallest
ordinal such thatM2,aφMγa2, contradicting the fact that the range of f is a subset ofξ0.
Similarly, we defineg:W2×W1−→λ+as
f(a,a)=
the least ordinalξ < λ+such thatM1,aφξMa1 if there is some,
0 otherwise,
and obtainξ1 < λ+such that the range ofgis a subset ofξ1. As before, for everyβsuch
thatξ1 < β < λ+,
M1,aφξM1a
1 implies that M1,a φβa
M1.
Chooseξto be sup{ξ0, ξ1}. By the above, whenξ < β < λ+,
M2,aφξMa2 implies that M2,a φβa M2, and M1,aφξMa1implies thatM1,a φβa M1.
We claim that the relationsu Z1viff M2, v φMξu2 andu Z2viff M1, v φ ξu
M1 satisfy all
clauses in Definition 4.5.
For (1) in Definition 4.5, we will show by induction that wheni,j ∈ {1,2}, for allβ, ifuis a world ofMi andMj, v φMβuj thenMi,u∗i φβv
∗j Mi . In particular, ifu Ziv, i.e., Mj, vφξMuj thenMi,u ∗i φξv∗j Mi , i.e.,v ∗jZ ju∗i.
Letβ =0, and assume that Mj, v φ0Muj. We need to show that every literal satisfied
byv∗j atM
j is also satisfied byu∗i atMi, that is: (a) Mj, v∗j ponly ifMi,u∗i p,
and (b) Mj, v∗j (∼ p)only if Mi,u∗i (∼ p). To prove the contrapositive of (a)
assume that Mi,u∗i p, soMi,u (∼ p), but Mj, v φ0Muj, hence Mj, v (∼ p),
i.e, Mj, v∗j p. Now, for the contrapositive of (b) assume that Mi,u∗i (∼ p), so
Mi,u∗i∗i pbutu∗i∗i =u, soMi,u p. However,Mj, v φ0Muj, which implies that
Mj, v p, i.e.,Mj, v∗j∗j p, henceMj, v∗j (∼p)as desired.
Ifβ is a limit ordinal andMj, v φβMuj, then Mj, v φγMuj for allγ < β, and by
inductive hypothesis,Mi,u∗i φγ v ∗j
Mi for allγ < β, which implies thatMi,u
∗i φβv∗j Mi .
Ifβ=γ+1 andMj, v φγM+j1u,Mj, v φγMuj, and by inductive hypothesis,Mi,u∗i φγ vMi∗j. Recall that φγ+1v∗j Mi = φ γ v∗j Mi ∧ b∈Wi, X⊆W j, M j,v∗jφγM jb→d∈XφγMid φγb Mj → d∈Xφγ d Mi ∧ b∈Wi, X⊆W j, M j,v∗j(∼(φγM jb→d∈XφγMid)) (∼(φγb Mj → d∈Xφγ d Mi)).
Hence, it remains to show that (a)Mj, v∗j φγMb j →
d∈Xφγ d
Mi for some worldbof Mi
and X ⊆ Wj only if Mi,u∗i φγMbj → d∈Xφγ d Mi, and (b) Mj, v ∗j (∼ (φγb Mj → d∈Xφγ d
Mi)) for some world b of Mi and X ⊆ Wj only if Mi,u
∗i (∼ (φγb Mj →
d∈Xφγ d
Mi)). These two follow similarly to (a) and (b) in the case whenβ=0.
The proof of (2) in Definition 4.5 requires us to notice first that fori ∈ {1,2},Mi,u
φβu
Mj for allβ. We argue by induction onβ. The caseβ =0 is trivial. Ifβis a limit ordinal
and, by inductive hypothesis, Mi,u φγMuj for allγ < β, then clearly Mi,u φβMuj.
Finally let β = γ +1. By inductive hypothesis, Mi,u φγMuj. But trivially both (a)
Mi,u φγMb
i →
d∈Xφγ d
Mj for some world b of Mj and X ⊆ Wi only if Mi,u
φγb Mi → d∈Xφγ d Mj, and (b) Mi,u (∼(φ γb Mi → d∈Xφγ d
Mj))for some worldb of Mj
andX ⊆Wi only ifMi,u(∼(φγMbi →d∈Xφγ d
Mj)). Hence,Mi,uφ βu Mj.
Now, suppose that u Ziv, i.e., Mj, v φMξuj, which implies that Mj, v φ ξ+1u Mj by
choice of ξ. Assume further that Rjvbc and consider the disjunction
d∈Xφξ d Mj where
d ∈ Wi is such that Mj,c φMξdj. By a previous observation, Mj,b φξMbi and clearly
Mj,c d∈XφξMd j, so Mj, v φ ξb Mi → d∈Xφξ d
Mj. Hence, given thatMj, v φ ξ+1u Mj , Mi,u φξMbi → d∈Xφξ d
Mj. Thus, there areb
,c ∈W i such that Riubc,Mi,bφξMbi, i.e.,b ZibandMi,c d∈Xφξ d
Mj. The latter means that ifd∈WiandMj,cφ ξd Mj then
Mi,cφξMd
j. Again by a previous observationMi,c
φξc
Mj, so we see thatMj,cφ ξc Mj
contraposing the previous sentence, i.e.,cZic.
Clause (3) in Definition 4.5 follows as ifi,j ∈ {1,2}andu Ziv, i.e.,Mj, v φξMuj then
Mj, vφ0Muj, so every propositional variable satisfied atuinMiis also satisfied atvinMj.
The right to left direction of the theorem follows since if Mj, wj φξwiMj then Z1and
Z2are both nonempty, so we have the required relevant directed bisimulation betweenMi
andMj.
For the other direction if there is one such relevant directed bisimulationMj, wj φβwiMj
for allβ, so in particular,Mj, wj φξwiMj . This can be be seen by recalling that for anyβ,
Mi, wi φβwiMj and sincewiZiwj by assumption, Mj, wj φβwiMj since all formulas of
L→λ+ωare preserved under relevant directed bisimulations.
COROLLARY5.2 (Relevant Scott’s Theorem). Let(M1, w1)and(M2, w2)be two
variables. Then, when i,j ∈ {1,2}, there is a formulaθwi of L→|2ω|+ωsuch that Mj, wj
θwi iff there is a relevant directed bisimulation(Z
i,Zj)between Mi and Mj such that
wiZiwj.
§6. Interpolation, preservation, and Beth definability. In this section, following the analogous case for modal logic [5, 7], we obtain a preservation theorem for relevant infini-tary formulas as a corollary to a generalized interpolation result. Interpolation theorems have a history of implying preservation results (some examples in infinitary logic can be found in [19]).
LetMbe a structure for a languageL∞ω, ifX ⊆dom(M)andXis closed under all the functions in the signature ofM, then [X]M is the submodel obtained by restricting all the relations in the signature of M to X. Note that if X fails to be closed under the required functions, then [X]M is not defined.
LEMMA6.1 (Relativization Lemma). Let L∞ωbe a language with a unary predicate P.
Then for any formulaφ(x)L∞ω not containing P there is a first order formulaφP such
that if M is a structure where[PM]Mis defined then for every sequence a of elements from
[PM]M,
M φP[a]iff [PM]M φ[a].
Proof. This is just Theorem 5.1.1 from [17].
Given a languageL, by11(L)and11(L)we will mean the languages resulting from admitting, respectively, second order existential quantifications in front of a formula of L and second order universal quantifications in front of a formula ofL.
LEMMA6.2. If L∞ω has a signature containing a binary symbol<,φ(x)andψ are
formulas of L∞ωand11(L∞ω)respectively such that for each ordinalαthere is a model
M such that<M is a linear ordering onφ(M)in order typeα, thenψhas a model N
such that<Nis a linear ordering onφ(N)which is not well-ordered.
Proof. This is essentially Theorem 11.5.4 in [17] or Theorem 1. 8 in [6].
Lemma 6.2 is known as the property of the model-theoretic language L∞ω of being
bounded, a substitute for compactness when establishing that a property is not expressible
inL∞ω([17], p. 581). It is a useful property that can be seen to characterizeL∞ωin terms of expressive power via a Lindström theorem (cf. [4]).
LetR,S be a pair of binary relations between two structures M1 and M2, while φ
andψ are formulas of Lcorr∞ω. Following [5, 7] we say thatφ implies ψ alongR,S if whenever M1R M2,M1 φonly ifM2 ψ and ifM2S M1, M2 φ only ifM1 ψ.
This can be seen as a generalization of the usual notion of consequence (note that standard consequence is the case when R andS are the identity). When the relation in question is relevant directed bisimulations,φimpliesψalong relevant directed bisimulations if when Z1,Z2is a relevant directed bisimulation between two models M1 and M2, anda Zib
(i,j ∈ {1,2}) for elementsa,bof the domains ofMi andMj respectively, thenMi φ[a]
only ifMj φ[b].
Ifφis a formula ofLcorr∞ω, we will writePROPφfor the collection of predicates appearing inφcorresponding to propositional variables inPROP.
LEMMA 6.3. Let φ, ψ be formulas of11(L∞corrω), 11(Lcorr∞ω)respectively. Supposeφ
impliesψalong relevant directed bisimulations forPROPφ∩PROPψ over some class of
αsuch that for every M,N ∈K if M φ[w]and u satisfies in N all the infinitary relevant
formulas of degreeαsatisfied bywin M, then N ψ[u].
Proof. Suppose for reductio that for eachαthere are(M1, w1)and(M2, w2)such that
M1φ[w1] andM2ψ[w2] whilew2satisfies inM2all the infinitary relevant formulas
of degreeαsatisfied byw1inM1. Hence, by Proposition 4.4, there is a relevant directed α-bisimulation(Zβ1,Zβ2)βαsuch thatw1Zβ1w2for eachβ α.
Suppose for simplicity that PROPφ ∩PROPψ has a single nonlogical symbol p. So the correspondence language Lcorr
∞ω has signature K = {∗,R,P,Q0,Q1, . . .}where Qi
(i =0,1, . . .) are the predicates corresponding the propositional variables not inPROPφ∩ PROPψ. Expand this signature by adding the set of symbols{U1,U2, <,O,B1,B2,I,G},
whereU1,U2, ,B1,B2andOare unary predicates,<andI are binary predicates, while
Gis a ternary predicate.
Consider the infinitary formula, where is the theory containing the following formulas:
σU1,σU2
“There arex,ysuch thatU1x,U2y, φU1x,¬ψU2yand for allz,usuch thatO z,B1u
andI zu, we have thatGuxy”
“<is a discrete total ordering with first and last elements” “Ois the field of<”
“IfUix, thenUix∗” (i ∈ {1,2})
“IfUixandRx yz, thenUiyandUiz” (i ∈ {1,2})
“IfBiz,Ou,I uzandGzx y, thenUixandUjy” (i ∈ {1,2})
“For allzsuch thatO z, there isuwithBiuandI zu” (i ∈ {1,2})
“If Biz,Ou,I uz and Gzx y, then there is v such that Bjv,I uv, andGvy∗x∗”
(i ∈ {1,2})
“IfBiz,Ou,I uzandGzx y, then P xonly if P y” (i ∈ {1,2})
“IfUix,Ujy,Ujb,Ujc,O z,I uz,Biz,Gzx y,R ybc,Ovandv <u, then there are
w, w such that Ivw,Ivw,Bjw,Biw and there are bc such that Uib,Uic,
Rxbc,GwbbandGwcc” (i ∈ {1,2}).
The last three classes of sentences described in our presentation ofare simply restate-ments in first order logic of conditions appearing in the definition of a directed
α-bisimulation.
For each ordinalα,has a modelMαsuch that the ordering<Mα onOMαhas order
typeα. To see this consider(M1, w1)and(M2, w2)as given by our reductio assumption,
that is, M1 φ[w1] andM2 ψ[w2] while there is a relevant directedα-bisimulation (Zβ1,Zβ2)βα such thatw1Zβ1w2for eachβα.
We can suppose without loss of generality that W1∩W2 = ∅(if this is not the case
already simply take isomorphic copies of M1andM2satisfying the proviso). Let Mα be
any modelM3such that:
W3=W1∪W2∪α+1∪ {Zβi :β α,i∈ {1,2}}, R3=R1∪R2, ∗3= ∗1∪ ∗2, UM3 i =Wi (i∈ {1,2}), PM3 =PM1∪PM2,
QM3 i =Q M1 i ∪Q M2 i (i =0,1, . . .), BM3 i = {Zβi :β α} (i ∈ {1,2}), OM3 =α+1,
<M3 is the natural ordering onα+1,
IM3βyiffβαandy=Zβ i for somei ∈ {1,2}, GM3xabiffx=Z ∈ {Zβ i :β α,i∈ {1,2}}anda Z b. It follows thatM3
. The sentencesσU1,σU2hold inM
3by Lemma 6.1, the fact that
bothM1andM2makeψtrue, and that [U1M3]M3 =M1and [U2M3]M3 =M2.
Since for each ordinalα,has a modelMαsuch that the ordering<Mα onOMαhas
order type α, by Lemma 6.2,has a model M4such that<M4 is a linear ordering
which is not well ordered. This means thatOM4 being the field of<M4 contains an infinite
descending sequence:
(∗) . . .e3 <M4 e2 <M4 e1 <M4 e0.
LetM4|K be the restriction ofM4to the signatureK. Now, sinceM4makes
hold,
there are a ∈ UM4
1 andb ∈ U M4
2 such that M4 φU1[a] (i.e., [U M4
1 ]M4|K φ[a]),
M4ψU2[b] (i.e., [U2M4]M4|K ψ[b]) and for allz,u such thatz ∈ OM4,u ∈ B M4 1 and
M4 I[zu], we have thatM4G[uab].
The pairZ1,Z2defines a relevant directed bisimulation forPROPφ∩PROPψbetween
[UM4
1 ]M4|K and [U M4
2 ]M4|K where
x Z1yiff there isen (n ∈ω) in the sequence(∗)such that there isu ∈ B1M4,M4
I[enu] andM4G[ux y],
x Z2yiff there isen (n ∈ω) in the sequence(∗)such that there isu ∈ B2M4,M4
I[enu] andM4G[ux y].
First note thatZ1 = ∅ = Z2. For alluand arbitraryen such thatu ∈ B1M4 andM4
I[enu], we have that M4 G[uab], and given that there is such au, we have thata Z1b.
But one of the formulas inimplies that there is alsov ∈ BM4
2 such M4 I[env] and
M4G[vb∗4a∗4]. Hence,a Z1bandb∗4Z2a∗4, i.e.,b
∗ [UM4 2 ]M4|KZ2a ∗ [UM4 1 ]M4|K.
To show (1) in Definition 4.5 suppose that i ∈ {1,2}and x Ziy. By essentially the
argument in the above paragraph it follows thaty
∗ [UjM4]M4|K Zjx ∗ [UM4 i ] M4|K .
For clause (2) in Definition 4.5, suppose thati ∈ {1,2}andx Ziy, so there isen(n ∈ω)
in the sequence(∗)such that there isu ∈ BM4
i ,M4 I[enu] andM4 G[ux y]. Now
letR
[UM4 j ]M4|K
ybcfor someb,c∈UM4
j , i.e.,R4ybcby Lemma 6.1. But sinceen+1<en,
there is formula inwhich implies that there arew, wsuch thatM4 I[en+1w],M4
I[en+1w], w∈ BMj 4, w ∈ BiM4 and there arebcsuch thatb,c ∈UiM4,R4xbc(so, by
Lemma 6.1, R
[UM4 i ]M4|K
xbc),M4 G[wbb] and M4 G[wcc] (hence b Zjb and
cZic).
Condition (3) in Definition 4.5 follows as ifi ∈ {1,2}anda Zib, there is formula in
implying that M4 P[a] only if M4 P[b], and, by the Lemma 6.1, [UiM4]M4|KP[a]
only if [UM4
j ]
M4|KP[b].
Finally, since the pair Z1,Z2 defines a relevant directed bisimulation for PROPφ∩
PROPψ between [UM4 1 ] M4|K and [UM4 2 ] M4|K with a Z 1b, [U2M4]M4|K ψ[b] and [UM4
relevant directed bisimulations. Also, [UM4 1 ]
M4|Kand [UM4 2 ]
M4|Kare in the class of models
K sinceψholds in both by Lemma 6.1.
THEOREM6.4 (Interpolation). Letφ, ψbe formulas of11(Lcorr∞ω),11(Lcorr∞ω)respectively
and K a class of Routley–Meyer structures axiomatizable by some formulaσ of Lcorr∞ω.
Then,φimpliesψalong relevant directed bisimulations forPROPφ∩PROPψ over K iff
there is a relevant interpolantθforφandψover K according to the standard consequence
relation, with propositional variables inPROPφ∩PROPψ.
Proof. For the right to left direction of the theorem suppose that there is a relevant
infinitary interpolantθforφandψoverKwith propositional variables inPROPφ∩PROPψ. Thatφimpliesψalong relevant directed bisimulations forPROPφ∩PROPψoverKfollows from Theorem 4.6 and the fact thatθ is an interpolant forφandψaccording to the usual consequence relation.
For the converse, by Lemma 6.3, we know that there is an ordinalαsuch that for every M,N ∈ K ifM φ[w] andu satisfies inN all the infinitary relevant formulas of degree
α satisfied by w in M, then N ψ[u]. Consider the disjunction Mφ[w]
(r elα(M, w)), wherer elα is the set of all translations of formulas ofL→∞ω of degree
α with propositional variables in PROPφ ∩PROPψ. The class of all nonequivalent formulas of L→∞ω of degree α is a set according to Proposition 4.2. Thus,Mφ[w]
(r elα(M, w))is a perfectly good formula ofLcorr∞ω. This formula is the desired inter-polant ofφandψ. It is easy to see thatφimpliesMφ[w](r elα(M, w)), while the
latter impliesψby choice ofα.
COROLLARY6.5 (Preservation). Letφbe a formula of Lcorr∞ω and K a class of Routley–
Meyer structures defined by some formulaψof Lcorr∞ω. Then,φis preserved under directed
bisimulations in K iffφis equivalent to an infinitary relevant formula over K .
Proof. Right to left follows from Theorem 4.6. For the converse, just setφ = ψ in
Theorem 6.4.
COROLLARY6.6 (Beth definability). Let P be a unary predicate not in Lcorr∞ω,φ(P)a
formula of Lcorr∞ω∪{P}and K a class of Routley–Meyer structures defined by some formula
ψof Lcorr∞ω ∪ {P}. Then the following are equivalent:
(i) There is a relevant formulaθ(x)of Lcorr∞ω such thatθ(x)≡ P x is a logical
conse-quence ofφ(P)in the standard classical sense.
(ii) If(M1, w1,PM1)and(M2, w2,PM2)are models ofφ(P)such thatZ1,Z2is a
relevant directed bisimulation between the restrictions(M1, w1)and (M2, w2)of
(M1, w1,PM1)and(M2, w2,PM2)to Lcorr∞ω, thenZ1,Z2 is a relevant directed
bisimulation between(M1, w1,PM1)and(M2, w2,PM2).
Proof. (i) ⇒ (ii): It suffices to show that when (M1, w1,PM1)and (M2, w2,PM2)
are models of φ(P)such thatZ1,Z2 is a relevant directed bisimulation between the
restrictions (M1, w1) and (M2, w2) of (M1, w1,PM1) and (M2, w2,PM2) to Lcorr∞ω, if
x ∈ PMi and x Z
iy then y ∈ PMj. The result follows by the assumption (i) and the
easy direction of Proposition 4.4.
(ii)⇒(i): It is enough to establish that∃P(φ(P)∧P x)implies∀P(φ(P)⊃P x)along relevant directed bisimulations forPROP∃P(φ(P)∧P x)∩PROP∀P(φ(P)⊃P x) over K, since
then, by Theorem 6.4, it follows that there is a relevant formulaθ(x)ofLcorr∞ω which is an interpolant for∃P(φ(P)∧P x)and∀P(φ(P) ⊃ P x)overK according to the standard
