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Expressive power of “now” and “then” operators
Igor Yanovich
July 9, 2014
Abstract Natural language provides motivation for studying modal backwards- looking operators such as “now”, “then” and “actually” that evaluate their argument formula at some previously considered point instead of the current one. This paper investigates the expressive power over models of both propo- sitional and first-order basic modal language enriched with such operators.
Having defined an appropriate notion of bisimulation for first-order modal logic, I show that backwards-looking operators increase its expressive power quite mildly, contrary to beliefs widespread among philosophers of language and formal semanticists. That in turn presents a strong argument for the use of operator-based systems in the semantics of natural language, instead of systems with explicit quantification over worlds and times that have become a de-facto standard for such applications. The popularity of such explicit- quantification systems is shown to be based on the misinterpretation of a claim by [Cresswell, 1990], which led many philosophers and linguists to as- sume (wrongly) that introducing “now” and “then” is expressively equivalent to explicitly quantifying over worlds and times.
Keywords “now” operator, backwards-looking operators, bisimulation, first-order modal logic, hybrid logic
The purpose of this paper is to study the expressive power that is added to modal logic by the introduction of now and then operators.1 Logically speaking, it is actually not a particularly exciting subject. Once we apply relatively familiar techniques from the modern logical toolkit, it turns out
Igor Yanovich
Universit¨at T¨ubingen, Institute of Linguistics Wilhelmstraße 19, T¨ubingen, 72074 Germany E-mail: igor.yanovich@uni-tuebingen.de
1 From here on, I talk only about expressive power over models. For the application to natural language, that is arguably a more important kind of expressivity than expressivity over frames.
that now and then add extra expressive power only for first-order modal logic, as opposed to propositional modal logic.2 Moreover, in most cases that power is only added when models have an infinite number of individuals. Thus for a pure logician, the interest of this paper would mainly lie in the notion of bisimulation appropriate for first-order (=quantified) modal logic, cf. Def. 8 and Thm. 2.
But from the applied point of view, the systems with nowand thenare extremely important because of their role in the philosophy of language and in formal semantics. In those areas, it is often taken as a proven fact that modal logic with now and then is as expressive as a first-order multi-sortal logic with explicit quantification over times and worlds. But this near-consensus is very different from the actual mathematical state of affairs, as we will show below. Once the wrongful assumption is corrected, there are consequences for how linguists and philosophers of language might want to go about analyzing natural language phenomena. In particular, when the expressive power ofnow and thenis properly characterized, we become able to see more advantages of using operator-based systems for modality and temporality.
1 Introduction
Certain expressions of natural language prompted philosophers and linguists to introducenow andthenoperators which could shift the interpretation of an embedded subformula to a point (that is, world or time) introduced by a higher modal operator. Those expressions may be calledbackwards-looking operators.3 From the early days of formal semantics, it has become accepted that when such operators are added to quantified (that is, first-order) modal logic, its expressive power increases, as was shown by [Kamp, 1971]. But by how much? Since [Cresswell, 1990], it has also become accepted in the fields of formal semantics and philosophy of language that quantified modal logic enriched withnowandthenis as expressive as a full many-sortal first-order logic with unrestricted explicit quantification over worlds and times. And that formal understanding (or, in fact, misunderstanding, as we will show) in turn led philosophers and especially linguists to widely popularize the use of such explicit-quantification systems.
Explicit-quantification systems have become a de-facto standard to such an extent that it is hard to find contemporary semantic work that would use modal and temporal operators rather than explicit quantifiers over times and worlds. This sometimes leads to curious results: for instance, [Percus, 2000]
is an important and well-cited4work that is dedicated to formulating abind- ing theory for explicit world variables assumed to populate syntactic repre-
2 Quantified modal logic, also calledfirst-order modal logic, is related to (propositional) modal logic the same way as first-order logic is related to propositional logic.
3 The term “backwards-looking operators” is due to [Saarinen, 1978].
4 Currently with about 200 citations in the Google Scholar web service, which is a large number for a linguistic article.
sentations for natural language. The main content of that binding theory is as follows: verbs and adverbial modifiers combine with world variables that must be bound by the closest higher modal operator. Importantly, in a system where world variables are only manipulated implicitly through modal oper- ators and backwards-looking operators, no such constraint would be needed:
in the absence of an extra operator, such interpretation would be the de- fault. This underscores which kind of problems one runs into upon accepting without argument a more expressive system than one needs to: in a more ex- pressive system, more things can go wrong, and moreover, more additional constraints on the workings of the system are needed. But given the common misinterpretation of Cresswell’s result, this problem has flown under the radar of philosophers and linguists because it was assumed that there is no differ- ence in expressive power between operator-based and explicit-quantification systems.
What I aim to achieve in this paper is to bring the debate about explicit- quantification vs. operator modal systems for natural language semantics back onto a solid logical ground. The reading of[Cresswell, 1990] that led analysts of natural language to adopt explicit-quantification systems is based on a misun- derstanding. Cresswell never proved the result that the subsequent linguistic and philosophical literature took him to have proven. He addednowandthen operators not to the basic modal language, but to a language with universal modality — which few if any philosophers and linguists would posit for natural language (or, at least, for everyday natural language). In that language,then indeed increases expressivity up to full many-sortal FOL (first-order logic). But as is well-known to modal logicians, universal modality is a very powerful op- erator itself, cf. [Goranko and Passy, 1992], [Blackburn and Seligman, 1995], [Blackburn and Seligman, 1998], a.o. So Cresswell’s increase in expressive power happens to a system that is already far more powerful than what is currently assumed for natural language (which we will discuss further in Section 7). It is thus improper to apply Cresswell’s results to ordinary investigations of the properties of natural language.
But what happens if we analyze the expressive power of now and then properly, namely adding them to the basic modal language ML? (That lan- guage would be agreed upon as a proper basis for analysis of natural language.) The current paper answers this question. In the propositional case, no extra power is added bythen. In the quantified modal logic case, there is indeed an increase in power, but it is tiny.MLwith thenoperators added is the least expressive language of the hybrid family, and when identity is in the language of quantified modal logic, then’s extra power only manifests itself in models with an infinite number of individuals. Far from going all the way to many- sortal first-order logic, a system withnowandthenis perhaps themildest of the known systems expanding basic modal logic.
What does this mean for applied logicians, such as philosophers and lin- guists, who use modal logic to analyze natural language? The bottom line is that an operator-based system is arguably superior to the explicit-quantification systems that have become the standard in the field. Operator-based systems
are less expressive, but the expressivity they carry is already enough to ac- count for the relevant natural language phenomena. Such systems are thus more restrictive and more predictive — the properties which linguists and philosophers value in formal systems. This does not necessarily mean that one should just throw explicit-quantification systems out of the window: it can be that for some purposes, they would be more intuitive to use. But what the results presented in this paper show is that such systems are far from innocent, and thus ought to be used with due caution.
The plan of the paper is as follows. In Section 2, I introduce important datapoints from natural language that motivate the introduction of backwards- looking operators. I also demonstrate how one would translate such natural language sentences into a formal language using both an operator-based and an explicit-quantification based systems. This section thus provides the applied motivation for the logical study to follow. In Section 3, I introduce languages Cr and CrF O resulting from the addition of generalized backwards-looking operators to the basic modal languageMLand its quantified versionMLF O. This section provides the definitions for the formal systems whose expressivity we will be studying. In Section 4, I provide a truth-preserving translation from the fragment of propositionalCrthat only features genuinely backwards- looking then, into ML. The existence of such translation shows that then operators do not actually increase expressive power in the propositional case.
Section 5 turns to the case of first-order modal logic with now and then.
We introduce a notion of bisimulation appropriate to quantified MLF O, and with its help prove that CrF O is strictly more expressive, but at the same time that extra expressivity only kicks in in a limited number of cases — in particular, when the domains of individuals are infinite. Section 6 closes the logical part of the paper: it it, I situate the Cr languages within the family of hybrid languages. It turns out that Cr is the mildest member of that clan. Finally, in Section 7 we return to the applied use of backwards- looking operators, discussing how the expressivity claim by [Cresswell, 1990]
was misinterpreted in the linguistic and philosohical literature, and what the practical consequences of learning the actual expressivity results may be.
2 Backwards-looking operators of natural language, and their formal analysis
In this section, I introduce natural language examples of the kind used to motivate the introduction ofnow andthenoperators. Then show how those examples can be analyzed with such operators, and also, alternatively, in a full many-sortal first-order logic with explicit quantification over worlds and times.
Mary [who is readingnow] came. (1)
In 1, “now” is embedded within a relative clause and signals that the em- bedded predicate “is reading” should be evaluated at the current time, outside
of the scope of the matrix past tense. Assuming operatornowwith appropriate semantics, we can represent the sentence withP(come(m)∧now(read(m))), whereP is the past operator. There is also a equivalent logical representation for 1 without now: read(m)∧P(come(m)), but while semantically correct, it does not respect the constituent structure of the original natural language sentence.
(One day in the future,) everyone [nowalive] will be dead (2) Unlike for 1, for 2 there is no translation into quantified modal logic without anowoperator. Or, rather, strictly speaking we will only be able toprovethat after we derive the results in Section 5, but even in the absence of a formal proof, it has been widely accepted as fact for decades that 2 is inexpressible unless we addnow (or allow explicit quantification over times).5
It was the case that everyone [thenalive] would all be dead one day (3) Everyone [actuallytall] might have been short (4) You might have considered yourself short while [actually being tall] (5) While “now” in 1 and 2 shifts evaluation back to the matrix time, “then”
in 3 shifts it to the moment introduced by the higher past tense. In 4, the word
“actually” forces the predicate “tall” to be evaluated at the actual world. In 5, the same word returns the evaluation to the counterfactual world introduced by the higher “might have been” operator, similarly to how “then” in 3 refers back to the past moment introduced by a higher operator.
In all of the cases above, natural language expressions “now”, “then” and
“actually” shift the evaluation index to some index that was used while evalu- ating the higher levels of the sentence. This can be the matrix index as in 1, 2 and 4, or an index introduced by a higher temporal or modal operator, as in 3 and 5. In both cases, those words may be said to belooking back into the series of indices introduced earlier, and shifting the interpretation of their argument formula to one of the previously used indices and away from the current one.
In an operator-based system, we would account for such natural language expressions as follows. We would introduce a family of logical backwards- looking operators theni (which is exactly what we do formally in the next section). The idea would be thattheni would shift the interpretation to the i-th evaluation index from the ones that were used earlier. The special case of now would be defined as then0 that would always go back to the initial evaluation time. Then we would analyze 2 as follows:
[[now]] =λPet.now(P) (6)
[[alive now]] =λxe.now(alive(x)) (7) [[everyone now alive will be dead]] =F(∀x:now(alive(x))→dead(x)) (8)
5 Contrary to that, [Verkuyl, 2008, pp. 130-132] argues thatnowis semantically superflu- ous in modal logic. But Verkuyl does not discuss any quantificational examples like 2 which pose a true expressivity problem, and only considers sentences such as 1 for whichnowis indeed semantically superfluous.
In an explicit-quantification system, there are certain design choices to be made when implementing “now” and “then”. In most current systems used by formal semanticists (cf. [Percus, 2000] for an example), predicates such as
“alive” have special syntactically represented slots for time and world vari- ables. Usually such a slot would be filled by a postulated covert constituent introducing an explicit time or world variable. There are several options as to what exactly an adverb such as “now” (for the temporal case) or “actually”
(in the world-variable case) would be doing in this setup. If “now” or “then”
would modify the predicate after it has combined with the covert variable, it would have to force abstraction over that variable. Such a system would look as follows:6
∧tφ:= expression that denotes the temporal intension ofφ (9) [[now]] = λPet.∧tP(t0),wheret0must refer to the current time (10) [[alive now]] = λxe.λt3.(alive(x)(t3))(t0) =λxe.alive(x)(t0) (11) [[2]] = ∃t1t0: (∀x: (alive(x)(t0)→dead(x)(t1)) (12) So what “now” does on such an analysis is essentially erasing any effect of the explicit syntactic time variable that combines with “alive”: the result is as if there was never such a variable in the first place.
Another version of the explicit-quantification story would go as follows:
we would say that “now” directly denotes a temporal variable, and that it occupies the same syntactic slot that is normally occupied by covert explicit temporal variables. This would result in the following analysis:
[[now]] =t0,wheret0 must refer to the current time (13)
[[alive now]] =λxe.alive(x)(t0) (14)
[[2]] =∃t1t0: (∀x: (alive(x)(t0)→dead(x)(t1)) (15) An immediate problem with this analysis is that empirically “now” seems to be a syntactic adjunct rather than an argument. For instance, it may occur both on the left and on the right of the modified expression: “everyone now alive” and “everyone alive now” are both OK. So some syntax-semantic inter- face story would have to be told about how come “now” would always fit into the proper slot for temporal variables — but let’s assume for the sake of the argument that such a story may somehow be told.
Comparing the operator-based and the explicit-quantification lines of anal- ysis, what can we say? Both lines would have to make a stipulation about the
6 Perhaps more in the spirit of current explicit-quantification systems, in particular of the branch of formal semantics called LF semantics, would be the following alternative. There would be no intension operator ∧t; nowwould take as arguments functions from times;
there will be a rule of freely applyingλti operators; and finally, a constraint would force the explicit variable next to “alive” to be bound by the closestλ-restrictor specifically when
“alive” is modified by “now”, but not otherwise. The problem with such an account is that this last constraint is, so to speak, non-compositional: the variables on predicates like “alive”
should generally not be subject to the closest-binder requirementm and only when we know that that predicate is in the scope of “now”, would a different constraint be imposed.
fact thatnowmust specifically refer to the current time, so on this count the two are equal. Beyond that, the operator story works right away (assuming the operators are properly defined, of course — but we will define a formal system with them right in the next section.) The explicit-quantification story cannot stop yet, though: it has to introduce a number of syntactic assumptions.7
This difference in the amount of additional work is actually related to a genuine difference in the expressive power of the two underlying formal systems. A system with backwards-looking operators, as we will show below, is a very mild logical system. It only increases the expressivity of basic first- order modal logic by a tiny bit. As a consequence, there are plenty of things that we cannot express in such a system — but the good news is that for analysis of natural language, we do not evenwant toexpress them. In contrast to that, the explicit-quantification system is very powerful. And as the result of that, we need to constrain its behavior in order to tailor it to the observed facts. Hence all the additional constraints on the binding theory of world and time variables, and a fair number of further issues to be resolved. For instance, why doesn’t natural language have operators that have meaning∃ti, without any connection to the current evaluation time? After all, it is often assumed in that line of theorizing that we have freely applyingλtioperators...
If natural language is as expressive as a full many-sortal first-order logic, then the absence of this, and many other kinds of meanings, is mysterious, and needs to be explained with yet further constraints.
What this comparison tries to demonstrate is ultimately that expressive power matters. Normally, a linguist or a philosopher of language would not be interested in the issues of expressivity, and for a good reason: if natural language demands a very expressive system, then we as analysts cannot help it, and would have to adopt it. That would just be an empirical fact about human language. But with backwards-looking operators, we have a different kind of case: the now-standard tools for treating them are vastly more expressive than is actually required by natural language data. In such a case, moving back to a less expressive system may would give to us greater explanatory adequacy.
So from the applied point of view, the task of this paper is to develop the logical theory that shows why and how exactly the operator-based story is more restrictive than the explicit-quantification story. Sections 3-6 below will take care of that logical part, and then in Section 7, we will return to the application to natural language semantics. I tried my best to make the logical part accessible for a linguist or a philosopher with applied interests in mind (to the possible frustration for the logician readers, for which I apol- ogize; textbook-level explanations have only been included for those topics
7 In fairness, some of those assumptions would be “independently justified”, in the sense that for many intensional phenomena of natural language, we would need similar ones any- way. For example, adding an extra binding-theoretic constraint specifically for time variables in the scope of “now” and “then” is not such a wild idea if we already adopted a number of such constraints anyway. But if we do not have to introduce any of this apparatus for restricting the enormous expressive power of the full FOL in the first place, that’s a different story.
that are not widely known among linguists and philosophers.) But in case I failed nevertheless, the main technical results are informally reformulated at the beginning of Section 7 for the reader’s convenience.
3 Adding backwards-looking operators to ML
In this section, we define languages that formalize backwards-looking oper- ators, Cr and CrF O (for Cresswell, as our system is very close to his sys- tem with “now”, “then” and “actually”).CrandCrF O result from enriching the basic modal language ML and its quantified counterpart MLF O with backwards-looking operators. It should be clear how to add such operators to other underlying modal languages (e.g., languages having more than one♦).
Definition 1 (The syntax of Cr)
LetP ROP be a non-empty set of propositional variables, e.g.p,q, ...
Then wff-s of Crare:
φ := P ROP | > | ¬φ|φ∧ψ|♦φ| thenk(φ), wherek∈N.
⊥,∨,→, andare defined as usual, andnow:=then0.
Formulas of MLare evaluated in a Kripke model, —consisting of the do- main W of points, an accessibility relation R, and a valuation function V,—
at a point from W. Informally, backwards-looking operators thenk shift the evaluation of their argument formula back to some point considered earlier.
In order to return to such points, we need to store them, and we do that in denumerableevaluation sequences ρof points from the domainW of modelM. Formulas ofCrare evaluated atpointed sequenceshρ, ii, where thei-th mem- ber ofρ, also writtenρ(i), functions as the current evaluation point in standard Kripke semantics. We callρ1 and ρ2 n-variants (in symbols, ρ1 ∼n ρ2) if for anym6=nwe have ρ1(m) =ρ2(m).
Definition 2 (The semantics of Cr)
For Kripke modelM =hWM, RM, VMi, sequenceρfromWM, andi∈N, M,hρ, ii |=Cr q iffρ(i)∈VM(q)
M,hρ, ii |=Cr > always
M,hρ, ii |=Cr ¬φ iff it is not the case thatM,hρ, ii |=Cr φ M,hρ, ii |=Cr φ∧ψ iffM,hρ, ii |=CrφandM,hρ, ii |=Crψ M,hρ, ii |=Cr ♦φ iff there is ρ0∼i+1ρ s.t.
ρ(i)Rρ0(i+ 1) andM,hρ0, i+ 1i |=Crφ M,hρ, ii |=Cr thenk(φ) iffM,hρ, ki |=Cr φ
The non-modal clauses of our semantics do exactly the same job as the cor- responding standard clauses, with ρ(i) playing the part of the current point.
The clause for♦, in addition to the standard truth conditions, also produces
“side effects”: it writes down the R-accessible point to which ♦ shifts the evaluation as the next member of the sequence, and stores the previous eval- uation point for future use. As these side effects only affect then-operators, the following easily follows:
Proposition 1 For allφ∈Cr that are also in ML, M,hρ, ii |=Cr φ iff M, ρ(i)|=MLφ
thenoperators shift the pointer to a different member ofρ. When the shift is to a point stored earlier, thenk functions as a genuine backwards-looking operator. But if we never “overwrote” ρ(k) while evaluating our formula by the time we encounterthenk, the point that we shift to is determined by theρ we started with. Evaluation sequences thus work pretty much like assignment functions, and we can think of thenk operators as implicitly introducing a variable over points. When thenk retrieves a previously stored ρ(k), the im- plicit variable is bound by a higher♦. Whenthenk accessesρ(k) determined by the initial evaluation sequence, the implicit variable is free.
A formula ofCris asentenceiff, evaluated athρ,0i, it only depends on the pointρ(0). In other words, the truth of a sentence ofCris semantically relative to a single point, while the truth of a non-sentence is relative to multiple points.
In yet other words, a sentence of Cr would have no implicit free variables over points.8 Note that it is crucial that we restrict our attention to formulas evaluated at hρ,0i: whether a Cr formula features an implicit free variable depends on the initial index. Thus ♦♦then1(p) would not use points not introduced by♦s when evaluated athρ,0i, but it would do so when evaluated at hρ,2i: in that case the point ρ(1) would not have been overwritten by the clause for♦.
Summing up, ♦then3(p) is not a sentence, and ♦♦then1(p) is. We will write Crsent for the sentence fragment of Cr. For a sentence φ, we sayφ is true atρinM ifφis true athρ,0iinM. Whenρ(0) =w, we can also say that sentence φ is true at w, as we do forML. When the context makes it clear which model is to be used, we may suppress it.
A standard technique in modal logic is to relate the modal language we are working with to the first-order logic whose domain is points/worlds/times.
That technique uses the so-calledstandard translation which maps modal op- erators to FOL operators in a specially defined language. E.g., for propositional variablespiin modal logic, the corresponding language will have corresponding
8 It is usual to give a syntactic definition of a sentence, where a sentence is a formula without free variables. Semantically, such a formula does not depend on the assignment of values to variables. It is easy to give a purely syntactic definition of aCrsentence, but I find that the semantic definition in the main text makes the intuition behind the notion more prominent.
1-place predicatesPi over worlds. The points of a Kripke model may be refer- enced inL0using individual variablesxi, and the accessibility relation is repre- sented by a 2-place predicateR. (See, e.g., [Blackburn and van Benthem, 2007, Sec. 2.2] or [Blackburn et al., 2001, Ch. 2.4] for an introduction.) The standard translation of standard modal logicMLessentially shows that in a sense,ML is just a special notation for a particular fragment of FOL: the formulas that may be output by the translation are a very small subset of the full FOL. In particular, such FOL formulas will all feature exactly one free variable (corre- sponding to the initial evaluation world), and all new variablesxj will always be introduced using a link to an already introduced pointxi, by a construction
∃xj:xiRxj, corresponding to ♦.
To study the properties of our systemCr, we can easily extend the standard translation:
STi(p) =P(xi) STi(>) =>
STi(¬φ) =¬STi(φ)
STi(φ∧ψ) =STi(φ)∧STi(ψ)
STi(♦φ) =∃xi+1(xiRxi+1∧STi+1(φ)) STi(thenk(φ)) =STk(φ)
The standard translation forMLonly requires two variables over points.9 However, the translation for Cr may require any finite number of variables.
The corresponding fragment of FOL is thus much greater forCrthan forML.
However, we will see in the next section that despite appearances, for any sentenceofCrthere is an equivalentMLformula, so standard translations of Crsentences are equivalent to formulas in the two-variable fragment of FOL.
We now turn to the quantified languageCrF O: it is the quantified version that is needed to adequately model meanings of NL sentences such as 1-5.
Syntactically, CrF O is obtained by using a supply of individual variables xk
and n-place relation symbolsq instead of just propositional variables (which are retained as 0-place relation symbols), and quantifier∀over individuals. In addition to that, we may or may not want to add an existence predicate E and identity of individuals.
Definition 3 (The syntax of CrF O)
Let{P REDn}, for finiten, be a collection of setsP REDn, each containing n-ary predicate symbols, with at least one P REDN non-empty; let V ARbe an infinite supply of individual variablesx0,x1, ...., also written asx,y,z, ...;
and letE be the optional existence predicate.
Then the wffs of CrF O are defined as follows:
φ := q(x0, ..., xn−1) | ∀xφ | > | ¬φ | φ∧φ | ♦φ | thenk(φ), where q∈P REDn, and k∈N.
9 That only two variables are needed for the standard translation ofMLwas noted by [Gabbay, 1981]. The case of two-variable logics is special. See, e.g., [Gr¨adel and Otto, 1999]
on semantically two-variable logics and corresponding two-pebble games.
Optionally, Exandxi=xj may be well-formed wffs.
There are many design options when it comes to the semantics of quanti- fied modal logic (see [Fitting and Mendelsohn, 1998], [Blackburn et al., 2007, Ch. 9]). For domain semantics, I choose varying domain semantics wherein there are no restrictions on the relations of individual domains at different points — the most general setting possible. For quantifiers, I use untensed quantifiers ∀ (also called possibilist), which range over all individuals in the model regardless of which points they exist at. In the special case when the language has existence predicate E, we can also definetensed, or actualist, quantifiers ∀tensed using untensed ∀ and E: ∀tensedxφ := ∀x(Ex → φ) (see [Cresswell, 1991]). Of course, tensed quantifiers can also be defined as primi- tive.10
A first-order Kripke model with varying domains for CrF O is a structure hW, R,{δw∈W},{Vw∈W}i, where eachδwis the individual domain of the point w∈W, andVwis a valuation relative tow, that is, a function from predicate symbols in P REDn to n-ary relations over point w’s individual domain δw. For convenience, we also define the domainD of all individuals asS
w∈Wδw. Formulas ofCrF O are evaluated in a first-order Kripke modelM at a pointed sequence hρ, ii relative to an assignment h of individuals to individual vari- ables. For the interpretation of the modal component the presence of the as- signment functionhdoes not make a difference: we just pass it down. We call two assignment functions hand h0 x-variants,h ∼x h0, iff they agree on all variables butx. Instead ofCr’s propositional variable clauseM,hρ, ii |=q, we have two clauses for predicates and for the universal quantifier. The defined semantics is bivalent: any q is false of a tuple if it contains individuals not existing at the current point. All omitted clauses are as forCr.
Definition 4 (Varying domain semantics for CrF O) M, h,hρ, ii |=CrF O q(¯x) iff hh(x1), ..., h(xn)i ∈Vρ(i)(q)
M, h,hρ, ii |=CrF O ∀xφ iff ∀h0 s.t.h0∼xh, we haveM, h0,hρ, ii |=φ M, h,hρ, ii |=CrF O E(x) iff h(x)∈δρ(i)
M, h,hρ, ii |=CrF O xi=xj iff h(xi) =h(xj)
As CrF O has both explicit variables over individuals and implicit then- variables over points, we have two notions of sentencehood: a wff φofCrF O is athen-sentence iff, evaluated athρ,0i, it only depends on the pointρ(0).
(The notion of then-sentence is thus parallel to the notion of sentence for Cr.) Furthermore, aCrF Othen-sentence is aCrF Osentenceiff it has no free individual variables. We say that a then-sentence is true at ρifhρ,0imakes
10 Another design choice is whether to add any sort of counterpart theory, cf. [Lewis, 1968].
Counterpart theory is often used to identify individuals at different points when point do- mains are disjoint, but can be added to any other kind of domain semantics as well. I will refrain from discussing counterpart theories altogether. See [Fara, 2008] and references therein for combining a counterpart theory with ‘now’ and ‘actually’.
it true. We may suppressM andhfor brevity when that is safe to do. We will writeCrF Osent for thethen-sentence fragment ofCrF O.
Extending the standard translation of Cr to a translation CrF O into a two-sortal corresponding first-order language is straightforward.
Now the stage is set. In the next section, we will show that the sentence fragment Crsent is expressively equivalent to ML by building an effective truth-preserving translation. Then in Section 5, we will define bisimulations for MLF O, and on the basis of that show that CrF O is genuinely more ex- pressive than MLF O. Finally, in Section 6, we will show which placeCrF O occupies in the expressive hierarchy of hybrid languages. Put together, we will have a theory of just how much expressive power adding “now” and “then” op- erators adds to a logic, and why that additional expressive power only arises in quantified modal logic, and crucially depends on models with infinite domains of individuals.
4 Eliminating then-operators in the propositional case
Special cases of eliminating backwards-looking operators in propositional modal systems have been discussed in the literature, cf. [Kamp, 1971], [Meyer, 2009].
In this section, we provide a truth-preserving translation from the sentence fragment Crsent of Cr into its underlying language ML (or, actually, two such translations). The existence of such translations shows that when we add
“now” and “then” to modell natural language operators, in the propositional case it does not actually increase the expressive power of modal logic. As long as we do not have quantification over individuals, no increase in expressive power occurs — unlike in the case when instead of “now” and “then”, we introduce explicit quantification over worlds and times. (Of course, if we con- sidernon-sentences ofCr, they are relative to more than a single point, and standard ML cannot express such meanings. But such cases are not what is usually taken to justify the linguistic and philosophical practice of using explicit quantification over worlds and times.)
What “bound”thenk-operators in aCrsentence do, is shift the evaluation back to ρ(k) introduced by some higher♦. We will provide two translations allowing us to eliminatethenkby bringing its argument to be in the immediate scope of the relevant♦. One translation is local, and works by “floating”thenk up one operator at a time until it reaches the level where we can eliminate thenk. The other translation is global, introducing at the level of the “binder”
♦ two disjoined cases, one for when φis true at ρ(k), another for when it is false. Both translations are complex: the first one in the worst case involves length increase exponential in the length of the translated sentence; the second involves length blow-up exponential in the number ofthenkφsubformulas. We start with the local translation as it allows us to better illustrate the working of theCr system. The reader not interested in such illustrations may safely skip to Thm. 1 and its second proof on p. 15.
For the first, local translation, we want to “float” each thenkψ into the immediate scope of the ♦ that introduces ρ(k) to which thenkψ is to be evaluated. Our first task then is to determine how we can transform Cr formulas while preserving their truth. Unlike in standard modal logic, with then-operators safety for substitution is determined relative to the evaluation sequence and the syntactic context in which substitution occurs. Thusφmay be safe to substitute forψin wffξ1, but not in wffξ2. For a simple example, consider♦then1pandthen1p. If both are evaluated athρ,0i, we can substi- tutethen1pwith justpin the first formula, but not in the second. So we will need to get a handle on such cases where substitution is OK.
Many substitutions, however, are always safe. It is easy to check that all MLvalidities define valid substitutions inCr: no matter the context, (p∨q)∧r is always equivalent to (p∧r)∨(q∧r) inCr. Similarly, the following equivalences hold regardless of the context, as can be easily checked from the truth clauses forCr:
¬thenk(φ) ⇔ thenk(¬φ) (16)
thenk(φ∧ψ) ⇔ (thenkφ)∧(thenkψ) (17)
thenl(thenk(φ)) ⇔ thenk(φ) (18)
But none of those allows us to “float” a thenk operator past a♦. What we need is to determine when the followingsemi-equivalence (∼) is valid:
♦(thenk(φ)∧ψ) ∼ thenk(φ)∧♦ψ (19) In some cases, e.g. 20, a substitution that instantiates the schema in 19 results in an equivalent formula. But in other cases, e.g. 21, it does not. In fact, the left formula in 21 is a sentence, while the right one is not: it depends onρ(2), not only on ρ(0).
♦♦(then0(p)∧q) =♦(then0(p)∧♦q) (20)
♦(then1(p)∧q)6=then1(p)∧♦q (21) For our purposes, the following simple case where the schema in 19 works will suffice:
Lemma 1 LetCrsentenceξcontain an occurrence of♦(thenk(φ)∧ψ), where (1)φcontains nothenoperators, and (2) for indexiat which♦(thenk(φ)∧ψ) would be interpreted inξ, k6= (i+ 1). Let ξ0 be the result of substituting that occurrence withthenk(φ)∧♦ψ. Then ifξ0 is a sentence, it is equivalent toξ.
Proof Suppose that ♦(thenk(φ)∧ψ) is true at hρ, ii. Then first, there is a point v s.t. ρ(i)Rv and ψ is true at ρ0 ∼i+1 ρ where ρ0(i+ 1) =v. Second, φ is true at ρ0(k), and as it contains no then operators, its truth does not depend on the rest ofρ0. Ask6= (i+ 1),φis also true atρ(k), making the first conjunct ofthenk(φ)∧♦ψtrue athρ, ii. The existence ofv makes the second conjunct true as well.
In the other direction, the equivalence is as easy.a
For the translation, we’ll need one more simple fact: whenthenkφis in a sentence, and it would be evaluated at indexkwhen the sentence is evaluated at somehρ,0i, thenthenk can be safely eliminated: it simply shifts the inter- pretation index from k to itself. Now we are ready to prove the following by building a local translation that floats eachthenk up one step at a time until it is eliminated:
Theorem 1 (Translation from Crsent into ML)
For each sentence ξ∈ Cr, there is a ξ0 ∈ML such thatM,hρ,0i |=Cr ξ iff M, ρ(0)|=MLξ0.
Moreover, ξ0 can be effectively computed from an arbitrary ξ∈Crsent. Proof of Thm. 1 (local translation) We define a (local, bottom-up) transla- tion from an arbitraryξtoξ0. Whenξis a sentence, allthenkoperators return evaluation to a point introduced by a higher♦. If we float eachthenk(φ) up into the immediate scope of that♦, we can then safely eliminatethenk.
We start with an arbitrarily chosen thenk that has no other thenl in its scope, and float it one♦up. After each such step, we check if we can eliminate the movedthenkbecause we reached the immediate scope of the♦thatthenk
referred back to. If the check is positive, we eliminate that occurrence ofthenk. (Before the first step, of course, we need to check if we can eliminate anythenk
right away.)
Fix such athenkφwhereφdoes not containthen-operators. Thatthenkφ would be within subformula ♦(...thenkφ...) where the scope of♦ is a non- modal formula. (If there is no such ♦, then ourk= 0, and we can eliminate thenk right away.) If in (...thenkφ...), our thenkφ is embedded under an- otherthenl, we apply the equivalences for negation andthen-distribution, 16 and 17, to get to a configuration thenlthenkφ. At this point, we apply 18 to eliminate thenl. If there are more higher then operators, we repeat the procedure until we transform♦(...thenkφ...) so that there are no otherthenl betweenthenk and♦.
Then we normalize the resulting non-modal formula into the disjunctive normal form, treating allthen-formulas as propositional variables. After that, we apply modal equivalence ♦(φ∨ψ) ⇔♦φ∨♦ψ, and obtain a subformula where thenkφ may only be embedded under ¬ or ∧. For ¬, we apply 16.
Finally, embedding under∧ does not prevent us from applying Lemma 1; in fact, if thenkφ is not embedded under ∧, we need to apply the equivalence ψ ⇔(ψ∧ >) to create a subformula that meets the conditions of Lemma 1.
Finally, we move thenkφ over ♦ using that lemma. If thenk may now be eliminated, we do that, and otherwise we repeat.
Reapplying the same procedure, we can always select a then-formula to be moved one ♦ up, so eventually we will be able to eliminate all of them, obtaining anMLformulaξ0 as desired.a
Here is how the translation works in one particular case:
♦♦p∧then0(q∨ ¬then1r)
♦♦p∧(then0q∨then0¬then1r) (17)&(16)
♦♦p∧(then0q∨then0then1¬r) (16)
♦♦(p∧(then0q∨then1¬r)) (18)
♦♦((p∧then0q)∨(p∧then1¬r)) →DNF
♦(♦(p∧then0q)∨♦(p∧then1¬r)) ♦(φ∨ψ)⇔♦φ∨♦ψ
♦(♦(p∧then0q)∨(then1¬r∧♦p)) Lemma 1
♦((♦p∧then0q)∨(¬r∧♦p)) then1 elimination
♦(then0q∧♦p)∨(¬r∧♦p)) Lemma 1
♦(then0q∧♦p)∨♦(¬r∧♦p)) ♦(φ∨ψ)⇔♦φ∨♦ψ ((then0q)∧♦♦p)∨♦(¬r∧♦p)) Lemma 1
(q∧♦♦p)∨♦(¬r∧♦p)) then0 elimination
The complexity of the translation is high because of the normalization step which in the worst case leads to exponential blow-up of the formula length. If, on the other hand, we apply a global translation to be defined below instead of the local translation above, there would be guaranteed exponential formula growth, but only in the number ofthenkφsubformulas:
Proof of Thm. 1 (global translation) ConsiderCrsentence...♦(ψ(thenkφ))..., where (1)φis a formula ofML; (2)ψ(thenkφ) is a formula ofCr(thus possi- bly containing morethenoperators); and (3) the shown♦is the one to which thenkφrefers back to.
When that sentence is evaluated, the ♦ introduces the point ρ(k). As φ contains nothen-operators, it isρ(k) alone that determines whetherthenkφ will amount to > or ⊥; the rest of the sequence is irrelevant. We thus have two cases, one wherethenkφ=>and another wherethenkφ=⊥relative to ρ(k). In the first case,ψ(thenkφ) amounts to ψ(>), and in the second case, it amounts toψ(⊥). The original sentence is thus equivalent to the following:
...♦((φ∧ψ(>))∨(¬φ∧ψ(⊥))).
As in the local translation case, for an arbitrary Cr sentence ξ, we can repeat this procedure, each time selectingthenkφwithφnot containing other thenoperators, and eventually we obtainξ0∈ML.a
Here is an example of the global translation applied to the same Cr sen- tence as above (with some simplification steps added for readability):
♦♦p∧then0(q∨ ¬then1r)
then1elimination
♦[(r∧♦p∧then0(q∨ ¬>))∨(¬r∧♦p∧then0(q∨ ¬⊥)]
simplifying using propositional validities
♦[(r∧♦p∧then0q)∨(¬r∧♦p)]
then0elimination
(q∧♦[(r∧♦p∧ >)∨(¬r∧♦p)])∨(¬q∧♦[(r∧♦p∧ ⊥)∨(¬r∧♦p)])
simplifying using propositional validities
(q∧♦♦p)∨(¬q∧♦[¬r∧♦p])
Each elimination ofthenkφleads to a roughly 2-fold increase of the substi- tutedψ(thenkφ), and we need as many such operations as there are distinct thenkφ subformulas inξ. The translation is thus exponential in the number of then-operators. It is interesting if a less-than-exponential translation can be given, but very simple translations are unlikely to exist.11
To sum up, as non-sentences of Cr contain implicit free variables over points, it is trivial to find aCrformula that cannot be expressed in ML. For instance, p∧then1(¬p) can distinguish a model that contains a p-point and a non-ppoint not connected by the accessibility relation, while ML cannot do that, as a simple bisimulation argument can show. But if we only consider the fragment Crsent, where thenk only work as genuine backwards-looking operators, then Thm. 1 shows that the extra operators do not increase the range ofmeanings that the language can express.
In the next section we will see that once we move fromCrto CrF O, that will change: adding backwards-looking operators to quantified (i.e. first-order) modal logic leads to a genuine increase in expressivity even within the sentence fragment.
5 Bisimulation for quantified modal logic
Since [Kamp, 1971], it is known that backwards-looking operators are not elim- inable in quantified modal logic: Kamp presents a sentence ofMLF O+now that has no equivalentnow-less sentence. Our task in this section is not to just prove that CrF O is more expressive thanMLF O (Kamp’s proof is sufficient for that), but rather to pin down the exact amount of new expressivity which thenk operators bring in when added to quantified modal logic.
We will use a standard tool that the modern modal logic uses for studying expressivity of modal languages: bisimulations. A bisimulation corresponding to a particular modal language L is a relation between the domains of two L-models such that if two points are bisimilar, then they are indistinguish- able by any formula ofL. Bisimulation may be informally thought of as a re- laxed version of isomorphism. Two isomorphic models cannot be distinguished no matter what. Two bisimilar models, under a fixed notion of bisimulation, cannot be distinguished by a particular logical language, though in a more expressive language we may be able to tell them apart.
Thus with a suitable notion of bisimulation in hand, it becomes easy to prove expressivity results. For instance, after we show that all bisimilar points,
11 See [ten Cate, 2005, Prop. 3.3.3], who shows that there is no polynomial normalization for hybrid @-operators, close cousins of our then-operators (cf. Sect. 6 on the relation between the two kinds).
under a fixed notion of bisimulation, are indistinguishable by language A, it suffices to show that languageB can distinguish some of such points to prove thatB is more expressive.
For a textbook-level review of bisimulations for propositional modal logic, see [Blackburn et al., 2001, Ch. 2]. Defining appropriate notions of bisimu- lation for richer propositional modal languages has become a routine step in modal-logical model-theoretic investigations (cf., e.g., [Areces et al., 2001], [ten Cate, 2005], [Areces et al., 2011]). But what we need to pin down the difference between MLF O and CrF O is a notion of bisimulation for a first- order modal language, and to my knowledge, such a notion so far has not been introduced in the literature. It will thus be worth spending some time on how exactly we can arrive at the right notion. Consider standard proposi- tional bisimulation first (and the reader interested in the new results may skip directly to Def. 7 and 8, and Thm. 2):
Definition 5 (Bisimulation for ML)
A bisimulation E between two Kripke models M and N is a non-empty relation inWM ×WN with the following properties:
Propositional Harmony: IfwEw0, then for any propositional symbolp, M, w|=piffN, w0|=p
Zig: IfwEw0 and∃v(wRMv), then∃v0(w0RNv0∧vEv0) Zag: IfwEw0 and∃v0(w0RNv0), then∃v(wRMv∧vEv0)
Points w ∈ M and v ∈ N are called bisimilar if there is a bisimulation E such that wEv. Models M and N are called bisimilar if there exists a bisimulation between them.
It is not hard to see why any two bisimilar points must be indistinguishable in ML. Suppose that we need to find out whether we are at w ∈ M or at v∈N, withwbisimilar tov, and that our only way of getting information is by testing for truthMLformulas at our current point. If we check the truth of propositional formulas, by Propositional Harmony and easy induction the results will be the same atwandv, so that doesn’t help. Now suppose we are actually atw, and we check if ♦φ is true. If it is, there is some accessiblew0 whereφis true. But then by Zig, in N there is also an accessiblev0 bisimilar to w0 whereφ is true. By induction onφ, we will never find out whether we are atw or atv. ThusMLis invariant under bisimulation. (Again, consider [Blackburn et al., 2001, Ch. 2] for formal proofs.)
Bisimulation is much more relaxed than isomorphism. E.g., the following models are bisimilar, though clearly not isomorphic:
Example 1 Bisimilar, but not isomorphic models
?>=<
89:;w
?>=<
89:;v1
''?>=<89:;v2
gg
M N
ML cannot distinguish M and N of Ex. 1, but first-order logic can: the formula∃u2(u1Ru2∧u16=u2) is false atwand is true atv1andv2. So while MLis invariant over bisimulations, its corresponding FO language is not. The corresponding language is thus more expressive.
What should the notion of bisimulation appropriate for MLF O look like?
It is clear that Zig and Zag from the propositional case should be preserved.
It is also clear that instead of requiring Propositional Harmony, we need to at the very least require “FOL harmony”: any bisimilar points should have the same non-modal theories (that is, they should make true exactly the same sets of formulas without modal operators). This leads us to the notion of FOL bisimulation. As we will see shortly, this notion is not yet quite adequate, but nevertheless it is useful as a first approximation:
Definition 6 (FOL bisimulation)
A FOL bisimulationEbetween two first-order Kripke modelsM andN is a non-empty relation inWM×WN with the following properties:
FOL Harmony: IfwEw0, then for anyφ∈FOL, M, w|=φiffN, w0|=φ
Zig: IfwEw0 and∃v(wRMv), then∃v0(w0RNv0∧vEv0) Zag: IfwEw0 and∃v0(w0RNv0), then∃v(wRMv∧vEv0)
Note thatφmay contain free variables. Thus for any tuple ¯aof individuals at pointw, at any bisimilarw0 there should be a corresponding tuple ¯bmaking precisely the same non-modal formulas true. However, FOL bisimulation does not ensure that such corresponding tuples would make the same sets ofmodal formulas true.
Example 2 Mismatch of individuals
?>=<
89:;w //?>=<89:;v GFED@ABCw0 //?>=<89:;v0
q:a
¬q:b
q:a
¬q:b
q:c
¬q:d
q:d
¬q:c
M N
Consider relation E={hw, w0i,hv, v0i} betweenM andN from Ex. 2. At all four points in the two models, the non-modal formulas∃xq(x) and∃x¬q(x) are true, and it is easy to see that FOL harmony is satisfied. Furthermore, Zig and Zag are also satisfied. Relation E is thus a FOL bisimulation. But the MLF O formula∃x(q(x)∧♦q(x)) is true atw inM, but false atw0 inN.
FOL bisimulation ensures that the internal FOL-theories of bisimilar points are the same, but it does not require that “harmony between individuals” holds across points. That is why we could easily distinguish betweenM andN from Ex. 2: we used the fact that for a at w, there is no corresponding a0 at w0 which would satisfy exactly the same modal formulas in one free individual variable.
To define a proper notion of bisimulation for MLF O, we need to make sure that for each tuple of individuals at a point, at a bisimilar point there is a corresponding tuple which makes exactly the sameMLF O formulas true.
It suffices that the correspondent make exactly the same non-modal formulas true at eachmodal path:
Definition 7 (Modal paths)
A modal path is a finite string of diamonds from the language. For w1, w2 points in modelM, a non-empty pathπ=♦i1...♦in leads from w1 to w2
(in symbols, w1πw2) iff there exist points vi1, ..., vvn−1 s.t. w1Ri1vi1∧...∧ vin−1Rinw2. For the empty modal pathΛ, by definition,∀w:wΛw.
Definition 8 (FOL path bisimulation)
A FOL path bisimulationEbetween two first-order Kripke modelsM and N is a non-empty relation inWM ×WN with the following properties:
FOL path harmony: (i) If wEw0, then for any finite tuple ¯a in DM, there is ¯binDN such that for any modal pathπ, if∃v∈WM to which pathπ leads fromw, then∃v0∈WN such thatw0πv0 and for any formulaφ∈FOL, M, v|=φ[¯a] iffN, v0 |=φ[¯b]. Similarly for any ¯batw0inN. We write ¯a!¯bfor such correspondent tuples. (ii) When ¯a!¯b atwandw0, it must be possible to extend those tuples to corresponding (¯a, a1)!(¯b, b1).
Zig: IfwEw0 and∃v(wRMv), then∃v0(w0RNv0∧vEv0).
Zag: IfwEw0 and∃v0(w0RNv0) , then∃v(wRMv∧vEv0).
Returning to Ex. 2, we can see thatw andw0 are FOL-bisimilar, but not FOL-path-bisimilar. There is no individual at w0 that could be a FOL-path- harmony correspondent of a from w: c is no good because there is no point accessible by the path♦ where c satisfies q(x), whiled does not satisfy q(x) at the empty pathΛ.
Theorem 2 If E is a FOL path bisimulationbetweenM andN, andwEw0 for w ∈M, w0 ∈ N, then for any φ of MLF O, there is a tuple ¯a such that M, w|=φ[¯a]iff there exists a tupleb such thatN, w0|=φ[¯b].
Proof Suppose towards contradiction that there exists φsuch that there is ¯a for which M, w |=φ[¯a], but for all ¯b, N, w0 6|=φ[¯b]. We fix some FOL-path- correspondent ¯bof ¯a, and thus have a pair of correspondents only one of which makes φ true. The proof goes by gradually disassembling φ so that we can finally derive a contradiction at the level of non-modal formulas. There are the following cases:φ=¬φ0, φ= (φ0∧φ00),φ=∀xφ0, andφ=♦φ0.
If φ =¬φ0, we have ¯afor which M, w6|=φ0[¯a], but for its correspondent
¯b that we fixed,N, w0 |=φ0[¯b]. Exchanging the roles for ¯aand ¯b, we can now considerφ0.
Forφ=φ0∧φ00, we haveM, w|=φ0∧φ00[¯a], butN, w06|=φ0∧φ00[¯b] where
¯
a ! ¯b. That means that either N, w0 6|= φ0 for a restriction ¯b0 of ¯b to the individuals substituted intoφ0, or similarlyN, w06|=φ00[ ¯b00].
— We can show that the restrictions ¯a0 and ¯b0 must be correspondents just as ¯aand ¯b. Suppose that is not so, and ¯a0 6!b¯0. Then by definition of FOL path harmony, there are someπ and (non-modal)ψthat M, w|=πψ[ ¯a0], but
N, w0 6|=πψ[ ¯b0]. Without loss of generality, let ¯a0 be the initial segment of ¯a, and let there benelements in the non- ¯a0part of ¯a. Then we can build formula ξ:=ψ∧(p(x1)∨ ¬p(x1))∧...∧(p(xn)∨ ¬p(xn)). As we only added tautologies to ψ, we have that M, w|=πξ[¯a], but N, w0 6|=πξ[¯b]. But that is contrary to assumption that ¯a!¯b. Thus all restrictions of corresponding tuples are also FOL-path-correspondents.
— Returning to φ0∧φ00, we note that either there are correspondent re- strictions ¯a0 and ¯b0 of ¯a and ¯b which disagree on φ0, or similarly for φ00. We then considerφ0 andφ00.
If φ=∀xφ0, we haveM, w|=∀xφ0[¯a], butN, w0 6|=∀xφ0[¯b]. We pick some extension (¯b, b1) such that N, w0 6|= φ0[(¯b, b1)]. By clause (ii) of FOL path harmony, we should be able to extend ¯ato some (¯a, a1) that is correspondent to (¯b, b1). As we haveM, w|=φ0[(¯a, a1)] for anya1, we now considerφ0, (¯a, a1), and (¯b, b1).
Whenφ=♦φ0, we move toφ0 andR-accessible v∈M andv0 ∈N thanks to the Zig and Zag conditions.
And finally, when φis non-modal, and we haveM, w|=φ[¯a], butN, w0 6|= φ[¯b], that directly contradicts FOL path harmony given that ¯a!¯b.a
It follows from Thm. 2 that when two points are FOL-path-bisimilar, then they are indistinguishable inMLF O.
Thm. 2’s converse does not hold in the general form: as is well-known, the converse fails for propositional ML, and that result carries over to MLF O. Thus there can beMLF O-models that are indistinguishable in the language, but nevertheless not FOL-path-bisimilar.12
Note that whether two models are bisimilar depends on the particular language used. E.g., M and N from Ex. 3 are FOL-path-bisimilar if identity is not in the language, and are not FOL-path-bisimilar if identity is included.
12 However, we can provide an analogue of the Hennessy-Milner theorem that states that the converse holds for a particular class of models. ForML, that is the class ofimage-finite models: those where every point has only a finite number ofR-successors for eachR. The propositional proof shows that the relation of modal equivalence is itself a bisimulation in this case. The condition of image-finiteness allows the following argument to come through:
suppose thatwandw0are bisimilar, but for somev:wRv, there is no bisimilarv0:w0Rv0. Then for eachu0i:w0Ru0i, there isφis.t.v0|=φi, butu0i6|=φi. As the set of allu0iis finite, we can build formula♦V
iφi, which is true atwthanks to the existence ofv, but is false at w0. This is contrary to assumption.
In the case ofMLF O, we need not only the assumption of finiteness for successor points, but also for domains of individuals. The argument for individuals would be along the following lines. Suppose that there are indistinguishablewandw0where ¯aatwhas no correspondent
¯b at w0. Then we collect all pairs ofπ and φ that witness that a particular ¯b does not correspond to ¯a, and as there is only a finite number of distinct ¯bs, we can collect them into one large formula∃¯xV
iπiφi(¯x). Atw, tuple ¯aensures that this formula is true, but atw0 by construction there is no ¯b that would witness that. But then wand w0 have different MLF Otheories, contrary to assumption. When the number of distinct tuples is not finite, we cannot gather allπandφinto asingle formula, hence the converse to Thm. 2 would not hold in such a case.
