toward model-theoretic modal logics

Front. Philos. China 2010, 5(2): 294–311 DOI 10.1007/s11466-010-0017-2

Translated from Luojixue Yanjiu 逻辑学研究 (Researches of Logics), 2009, (1): 62–77 MA Minghui ( ) Department of Philosophy, Tsinghua University, Beijing 100084, China E-mail: [email protected]

RESEARCH ARTICLE

MA Minghui

Toward Model-Theoretic Modal Logics

© Higher Education Press and Springer-Verlag 2010

Abstract Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic logics can be transferred to modal logics. Keywords model theory, first-order logic, modal logic, graded modalities

1 Introduction

An approach to investigate the abstract model theory of modal logic will be explored in this paper. The book Model-Theoretic Logics, edited by Jon Barwise and S. Feferman (1985), proposed two approaches to study abstract model theory: One is to extend (first-order) logic and the other is to investigate relationships between logics in an abstract way. The first path will be taken in this paper and applied to studies in modal logic. Adding infinite cardinality quantifiers or arbitrary conjunctions (disjunctions) into first-order logic will give substantially more expressive languages. Basic modal logic can also be extended in the same way, and thus certain mathematical quantitative concepts such as countably many and finiteness can be explained from the modal perspective.

2 Model Theoretic Logics

We explain briefly the meaning of extending logic in model-theoretic logics in

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this section. Notions of logic, logics, and the basics of model theory will be discussed in section 2.1. In section 2.2, we consider the meaning of and problem with the thesis that logic is first-order logic. Finally, in section 2.3, we introduce basic ideas of Mostowskian cardinality quantifiers and infinite logic. 2.1 Logic, Logics and Model Theory The general notion of logic is that it investigates valid forms of inference. This sort of understanding has been in place since Aristotle, who used the notion of necessarily following from as the core of logic—that is, preserving validity in the terms of modern logic. However, the notion of validity is based on the notion of truth. A sentence φ of a language L is valid if it is true in every L-model. Thus, as Frege observed, “logic is the science of most general laws of truth” (Frege 1979, p. 128). We use the word logic mainly in such a sense. The word logics also occurs often. If we ask what the logic of a certain (mathematical) notion is, people will expect the answer to be the logic of using this notion. Different logics will be constructed for different notions. For instance, what are the logics implied in mathematical notions like finiteness, infiniteness, countable and uncountable? What are the mathematical structures with such properties? Which languages should be used to talk about them? The answers to such questions lie in the study of logics of these notions or relevant properties.

The basic idea of model theory is to acquire the logic of mathematical structures by investigating the relationship between these structures and the linguistic expressions used to describe them. The change of mathematical notion or language will produce different logics or logical systems. In a word, model theory investigates the interaction between syntactic and semantic notions. The most fundamental notion of the classical first-order model theory is that of a model M satisfying a formula φ (written: M φ). A structure consists of a domain, relations and operations (functions) on this domain, and constants. Domains in first-order structures are confined to individuals. Formulas in first-order language are built by using logical constants like ¬, ∧, ∨, ∀ and ∃ where ∧ and ∨ are binary boolean operators. The universal quantifier is interpreted as the whole given domain, and the existential quantifier is interpreted as the class of all non-empty subsets of the given domain. If richer classes of structures are allowed—for example, relations or properties which are allowed to enter the domain—then the language used to talk about such structures must contain relational or predicate variables, and even second-order quantifiers. If we want to extend the language, for example, to add conjunction (disjunction) which is an unary operation on sets of formulas (like the generalization of the binary union x ∪ y to the unary union operation ∪x on sets), then infinitary logics are obtained. More extensions of first-order logic can be obtained by adding

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quantifiers which are not definable in first-order logic. We discuss basic ideas of such extensions in section 2.3. 2.2 The Thesis Logic = First-Order Logic Attempts to define logic as the logic implied in logical constants or as being equivalent to first-order logic actually exclude that which cannot be defined in first-order logic from the logical area. This is partly due to the belief that most mathematical concepts can be expressed in first-order logic or set theory. On the other hand, some philosophers believe that Quine’s thesis is right. Quine claimed that what is usually expressed by classes, relations, and even numbers can be easily re-casted in quantification theory, probably with identity theory (Quine 1953, p. 116). First-order logic indeed has considerable expressive power. The following two examples, however, show that there are limitations to the expressive power of first-order logic.

Example 1. There is no first-order sentence which can be used to define the two complementary notions countable and uncountable. The Löwenheim- Skolem theorem says that for any countable set of first-order sentences T, if it has a model then it has a countable model. Suppose that the first-order sentence χ defines the notion countable. Then for any first-order model M, M is countable iff M χ. Consider the sentence ¬χ. Given any uncountable model M ′, M ′ ¬χ. By the Löwenheim-Skolem theorem, ¬χ has a countable model, but all models of ¬χ are countable, a contradiction.

Example 2. There is no set of first-order sentences which can be used to define the two complementary notions finiteness and infiniteness. The compactness theorem says that for any set of first-order sentences T, if every finite subset of T has a model then T has a model. One of its corollaries is the following: If a set of first-order sentences T has arbitrarily large finite models then it has an infinite model. Suppose that T defines the notion finiteness. Then the class of models of T is just the set of all finite models. Then T has arbitrarily large finite models. Thus, by compactness, T has an infinite model. But all models of T are finite, a contradiction.

If we allow the first-order thesis, the above quantitative notions cannot be handled. First-order logic is only one of formal languages designed by us to investigate logic, just as the telescope is designed to observe stars in the heavens. It can be extended by introducing richer structures and languages. In this paper, only the extending of languages to gain expressive power will be discussed. That is, we will search for logics of structural properties which cannot be expressed in the original logic. We will not discuss richer mathematical structures, like structures with properties and relations discussed in second-order logic.


2.3 Mostowskian Cardinality Quantifiers and Infinitary Logic Since notions like finitely many and countably many cannot be defined in first-order logic, the language needs to be extended to cover such notions. Investigations about generalized quantifiers proposed by Mostowski (1957, pp. 12–36) properly open the door to model theory of extensions of first-order language. Given any infinite cardinal number ℵα, the new language L(Qα) is obtained from the first-order language L by adding the quantifier Qα. There are also new formulas of the form Qαx φ(x), which is interpreted as follows:

M Qαx φ(x) iff there exists ℵα elements b in M such that M φ(x)[b]. Let φ(x1 … xn) M = {(a1 … an) : M φ(x1 … xn)[a1 … an]}. Then M Qαx φ(x) iff Card( φ(x1 … xn) M ) ≥ ℵα where Card(X) denotes the cardinality of the set X. L(Q0) is the language obtained by adding the cardinality quantifier Q0 into first-order language, and M Q0x φ(x) iff Card( φ(x) M) ≥ ℵ0. A. Tarski proposed to investigate numerical quantifiers before Mostowski suggested to investigate extensions of first-order logic with cardinality quantifiers. An example of numerical quantifiers is the following: There are at least 2 numbers z such that z + 2 < 6. In his work Introduction to Logic and to the Methodology of Deductive Sciences, Tarski wrote: “Hitherto this theory [predicate logic] has primarily concerned itself with the universal and existential quantifiers, while the numerical quantifiers have been largely neglected” (Tarski 1941, p. 64). For any given natural number n, let ∃≥n x φ(x) express that there are at least n elements which satisfy the complex predicate φ(x). In the first-order language with identity, ∃≥n x φ(x) can be defined recursively as follows:

∃≥0 x φ(x) := ∀x (x = x); ∃≥1 x φ(x) := ∃x φ(x);

∃≥n+2 x φ(x) := ∃x (φ(x) ∧ ∃≥n+1 y (φ(y) ∧ y ≠ x)). Thus we have ∃≤n x φ(x) := ¬∃≥n+1 x φ(x), ∃=n x φ(x) := ∃≥n x φ(x) ∧ ∃≤n x φ(x). The formula ∃=n x (x = x) means that there are exactly n elements.

We can express counting notions by using these numerical quantifiers and add them into first-order language without identity. Such an extension does not go beyond the expressive power of first-order logic with identity, but it is a sort of generalized quantifier, and we will see later that it is useful in many cases. Another way of extending first-order language is to change the meaning of conjunction (disjunction), and define the generalized unary conjunction (disjunction) operation ∧Φ (∨Φ). If Φ is infinite, then the meaning of ∧Φ (∨Φ) goes beyond first-order logic, and we achieve what is called infinitary

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logic: M ∧Φ iff M φ for all φ ∈ Φ, and M ∨Φ iff M φ for some φ ∈ Φ. Note that ∧∅ ↔ and ∨∅ ↔ ⊥. In the literature, the notation L∞ω denotes the infinitary language where the index ∞ denotes arbitrary conjunction or disjunction, and ω denotes that the number of occurrences of quantifiers in a formula is finite. Similarly, Lωω corresponds to first-order language. The language Lλκ allows that the cardinality of the set Φ in the formula ∧Φ is less than λ, and the number of occurrences of quantifiers in formulas is less than κ.

3 Modal Logic and First-Order Logic

In this section, we explain how model theory of first-order logic can be transferred to modal logic. We discuss the tool for transferring in section 3.1. In sections 3.2 and 3.3, we discuss the most fundamental definability problem in modal model theory. In these discussions, the relationship between modal logic and first-order logic plays a crucial role. 3.1 Basic Modal Logic Basic modal language ML(◇, Φ) consists of the given set of propositional letters Φ and an unary modality ◇. Its formulas are given by the following rule:

φ ::= p | ⊥ | φ → ψ | ◇φ, where p ∈ Φ. Given a (Kripke) model M = (W, R, V) and w ∈ W, a modal formula ◇φ is true at w, i.e., M, w ◇φ, if there exists v ∈ W such that Rwv and M, v φ. Boolean operators are interpreted as classical propositional logic. Thus M, w ◇φ iff ∃v (Rwv ∧ M, v φ). Such a first-order expression of the semantics of modality shows directly the relation between modal and first-order logic: A model M = (W, R, V) can be taken as a first-order structure where

(a) W is the domain of individuals, and points in W can taken as individuals; (b) R ⊆ W × W is the interpretation of a binary relation symbol R; (c) V(p) is the interpretation of the corresponding unary predicate P.

Then we have w ∈ V(p) iff M Px [w], where x is a individual variable. There is a sort of correspondence between modal and first-order satisfaction relations. Modalities can be seen as restricted quantifiers over the domain of possible worlds. Such a restriction is the accessibility relation between possible worlds.

This shows that there is no essential difference between modal models and first-order structures. The only difference lies in the expressive power of modal


and first-order languages. That is, although relational structures are the same, the languages used to talk about them are not the same. Modal languages are highly expressive languages which talk about relational structures. The first-order language based on a relational vocabulary (without constants and functional symbols) can also talk about relational structures. We express such an idea formally as follows. Given a first-order language L1 with certain restrictions, its non-logical symbols only include unary predicates P corresponding to propositional letters p and a binary relational symbol R. For basic modal formulas φ, we define recursively its translation ST(φ, x) with respect to the individual variable x into first-order language as follows:

(1) ST(p, x) = Px; (2) ST(⊥, x) = ⊥; (3) ST(φ → ψ, x) = ST(φ, x) → ST(ψ, x); (4) ST(◇φ, x) = ∃y (Rxy ∧ ST(φ, y)), where y is a new variable.

By induction on the construction of formula φ, it is easy to prove that M, w φ iff M ST(φ, x)[w]. Thus, some model-theoretic properties of first-order logic can be transferred into modal logic in terms of this bi-condition. For any set of modal formulas Σ, let ST(Σ) = {ST(σ, x) : σ ∈ Σ}. If Σ has a model, it is easy to prove that Σ has a countable model according to the above bi-condition and the Löwenheim-Skolem theorem of first-order logic. Hence, modal logic also has the Löwenheim-Skolem property. A similar argument shows that the compactness of first-order logic can also be transferred into modal logic.

Note that the definition of ST(φ, x) involves the use of new variables. However, if we exploit the properties of bounded variables, the bound of the number of variables in the translation can be found: All modal formulas can be translated into two-variable fragments FO

2 of first-order logic. This leads us to investigate fragments of first-order logic with a bounded number of variables. This is a new field of inquiry.

Let us come back to the theme of the present inquiry, i.e., the relationship between modal and first-order logics. It is not the case that every first-order formula is the standard translation of some modal formula. For instance, the formula Rxx is not equivalent to any translation of modal formulas (The proof of this claim involves the problem of modal definability and model constructions which will be explained in section 3.2). The natural question then is how to determine the necessary and sufficient condition for a first-order formula α(x) to be equivalent to the standard translation of some modal formula. The van Benthem characterization theorem says that such a condition is invariance under bisimulation. Given any two models M1 = (W1, R1, V1) and M2 = (W2, R2, V2), w1 ∈ W1 and w2 ∈ W2, a non-empty binary relation Z ⊆ W1×W2 is called a

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bisimulation between M1 and M2 (written: Z : M1 M2), if it satisfies the following conditions:

(i) if w1Zw2 then w1 ∈ V1(p) iff w2 ∈ V2 (p) for all propositional letters p; (ii) if w1Zw2 and R1w1v1 then there exists v2 ∈ W2 with R2w2w2 and v1Zv2; (iii) if w1Zw2 and R2w2v2 then there exists v1 ∈ W1 with R1w1w1 and v1Zv2.

If Z : M1 M2 and w1Zw2 then let Z : M1, w1 M2, w2 denote that there exists a bisimulation relation Z between the two pointed models. A first-order formula α(x) is invariant under bisimulation, if for any Z : M1 M2 and w1Zw2, M1 α(x)[w1] iff M2 α(x)[w2]. The notion of bisimulation is the tool used to determine the corresponding part of modal logic in first-order logic. Moreover, a frame F = (W, R) can be seen as a second-order structure like a model seen as a first-order structure: F φ iff for any valuation V and point w in F we have F, V, v φ. Each valuation V(p) can be taken as the assignment to the predicate variable P corresponding to the propositional letter p. We then get a second order translation ST

2(◇φ, x) = ∀P1…Pn ∃y(Rxy ∧ ST 2(φ, y)) where P1…Pn are

unary predicates corresponding to propositional letters occurring in the formula φ. It can also prove that for any modal formula φ, F, w φ iff F ST

2(◇φ, x)[w]. Here we will not discuss the relation completely. 3.2 Model Definability On the level of models, bisimulation is the fundamental tool for investigating modal model theory. Classical model constructions include disjoint union, generated submodels, p-morphic images and tree unraveling. Apart from model constructions, another main problem in model theory is definability, i.e., which structural properties or classes of models can be defined in the language under consideration? Clearly, in a language used to talk about relation structures, any formula (class of formulas) can be used to determine a structural class, i.e., the class of structures on which the formula (class of formula) is true. But, conversely, a structural class or property is not necessarily definable by a formula (class of formulas). The expressive power of a language may have limitations, as we have seen that the expressive power of first-order language has limits. The definability of structural properties involves the problem of equivalence. Two structures A and B are elementarily equivalent if A β iff B β for any first-order sentence β. Two pointed models (M, w) and (N, v) are modally equivalent if M, w φ iff N, v φ for all modal formulas φ. The definition of model constructions or operations between models generally requires the result that these constructions imply the equivalence between models, and hence the definable class of structures is closed under these operations. The notion of


bisimulation given above implies modal equivalence: If Z : M, w N, v then (M, w) and (N, v) are modally equivalent. By this result, we can explain the problem mentioned in section 3.1: Rxx is not equivalent to the standard translation of any modal formula. Suppose that ST(φ, x) = Rxx for some basic modal formula φ. Consider the following two models M and N (in which the interpretation of each propositional letter is the whole domain): The frame M consists of the reflexive relation on the singleton {w}; and the frame N consists of the successor relation on the set of all natural numbers. Then the relation Z = {(n, w) : n ∈ ω} is a bisimulation. It is clear that M Rxx[w], and by semantic equivalence, M, w φ. Then by bisimulation invariance, we have N, n φ and N Rxx [n]. This is impossible.

Generally, a class K of pointed models is definable by a set of modal formulas if K is closed under bisimulations and ultraproducts, and the complementary class of K is closed under ultrapowers. K is definable by some modal formula if both K and the complementary class of K are closed under bisimulations and ultraproducts. These two definability results transfer definability results in model theory of first-order logic into modal logic. 3.3 Frame Definability Disjoint unions, generated subframes, and subjective bounded morphisms preserve validity of formulas on frames. For a family of frames {Fi}i ∈ I, let iFi be their disjoint union. Then Fi φ implies iFi φ. Let G be a generated subframe of F, then F φ implies G φ. If G is a bounded morphic image of F, then F φ implies G φ. By using such results, it can be shown for instance that the class of all finite frames is not modally definable since the disjoint union of infinitely many finite frames is infinite.

A special construction is the ultrafilter extension. For any frame F = (W, R), its ultrafilter extension ueF = (Uf (W), Rue) is defined as follows: (a) Uf (W) is the set of all ultrafilters on W; (b) Rueuv iff mR(X) ∈ u for all X ∈ v where the operation mR is defined on the power set of W:

mR(X) = {w ∈ W : there exists x ∈ X such that Rwx}.

The ultrafilter extension ueM = (ueF, Vue) of the model M = (F, V ) has the valuation Vue(p) = {u ∈ Uf (W) : V(p) ∈ u}. By using the ultrafilter existence theorem, i.e., any proper filter can be extended to an ultrafilter, it can be shown that ueM, u φ iff V(φ) ∈ u. Thus we can obtain that ueF φ implies F φ.

Generally, a first-order definable class of frames K is modally definable (by a set of basic modal formulas) iff K is closed under disjoint unions, generated subframes, and bounded morphic images, and reflecting ultrafilter extensions

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(i.e., ueF ∈ Κ implies F ∈ Κ). Frame and model definability are basics of investigations in modal model theory. On the level of frames, the relationship between first-order and modal definability is determined by using closure conditions. On the level of models, the standard translation and bisimulation are used to determine the modal fragment of first-order logic and some good model-theoretic properties possessed by first-order logic. The conditions of model definability are also determined.

4 Model Theoretic Modal Logics

Now, let us apply the first part’s approach to extending first-order logic to modal logic, which can also be extended. M. de Rijke (1993) investigated the model theory of extended modal logics. Most extensions are formed by adding new modalities into basic modal or propositional logic. For instance, we can add the universal modality A into basic modal language which is interpreted as follows: M, w Aφ iff M, v φ for all states v in M. This modality cannot be defined in basic modal logic. This extended logic can be translated into first-order logic, and hence its expressive power does not go beyond first-order logic.

Let us discuss extensions of modal logic by adding cardinality quantifiers. Consider the basic semantic clause in basic modal logic. Let M = (W, R, V ) be a model and w ∈ W, M, w ◇φ iff ∃v (Rwv ∧ M, v φ). Let R[w ] = {v ∈ W : Rwv}. We have

M, w ◇φ iff Card(R[w ] ∩ V(φ)) ≥ 1,

where V(φ) = {u ∈ W : M, u φ} is the truth set of φ in the model M. Make a minor change on the right side of the equivalence, and new modalities can be defined. First, consider the modalities ◇n defined as follows for each and every natural number n:

(*) M, w ◇n φ iff Card(R[w] ∩ V(φ)) ≥ n. ◇n φ means that there are at least n successor states where φ is true. When n = 0, it is clear that ◇0 φ is logically equivalent to the logical constant . When n = 1, ◇1 φ is logically equivalent to ◇φ. For n > 1, ◇n is not definable in basic modal language. The argument is the following: for models M1 = ({w1, u1, …, un}, {(w1, ui) : i ≥ 0}, V1) and M2 = ({w2, u}, {(w2, u)}, V2) such that V1( p) = V2(p), it is clear that there is a bisimulation Z such that w1Zw2. But the modal formula ◇n distinguished the states w1 and w2, and this contradicts to the fact that bisimulation implies modal equivalence in basic modal logic.


Thus such an extension makes a substantially more expressive modal language. Although it can be shown that these modalities can be defined in first-order language with identity, we can add numerical quantifiers into first-order logic without identity and then investigate the modal logic of finite cardinality quantifiers. This is completely different from basic modal logic since it needs an essentially new notion of bisimulation.

We may generalize the above modalities and define the modalities ◇κ for any cardinal number κ as follows:

(**) M, w ◇κ φ iff Card(R[w] ∩ V(φ)) ≥ κ.

In the case of infinite cardinal numbers, the extension goes beyond the expressive power of first-order language. If we introduce infinite conjunction and disjunction, four sorts of logic are obtained:

(a) modal logic of finite cardinal numbers (graded modal logic); (b) infinitary modal logic of finite cardinal numbers; (c) modal logic of infinite cardinal numbers; (d) infinitary modal logic of infinite cardinal numbers.

The simplest case is graded modal logic, which is obtained by adding countably many operators ◇n into basic modal logic. It can be used to express the notion of counting. ◇n means that we must take at least n successor states. In next section, we will explain some investigations of this. There are few results on the other three types of logic. Moreover, we can also introduce modal logics of certain cardinal numbers, for instance, the modal logic of the notion countable. By adding the modality ◇ω into propositional logic or basic modal logic, ◇ω φ is true at the state w if there are at least countably many successor states of w where φ is true. Such a type of logic needs more investigations.

5 Graded Modal Logic

Goble introduced a logic of modalities with fixed numbers. Each modality Ni is related with a natural number i, and used to express different degrees of necessity. For instance, the formula Nmα ∧ Nnβ (m > n) says that both α and β are necessary but that α is more necessary than β (Goble 1970, pp. 323–334). Kit Fine introduced the so-called numerical modalities ◇n (n ∈ ω), which are inspired from numerical quantifiers. Such modalities express numerical possibilities. Kit Fine introduced some modal logics and proved their soundness and completeness results (Fine 1972, pp. 516–520).

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Fattorosi-Barnaba and de Caro (1985, pp. 197–221), de Caro (1988, pp.1–10), and Cerrato (1990, pp. 241–252) provided various Hilbert-style axiomatizations of graded modal logics and proved completeness by using canonical models. Van der Hoek and Meyer discussed the application of graded modal logic to epistemic logic. They gave the epistemic interpretation of the formula Knφ so as to express uncertain knowledge (van der Hoek and Meyer 1992, pp. 503–514). Under the new computational view, de Rijke (2000, pp. 271–283) defined the notion of bisimulation for graded modal logic and so provided a tool for investigating the model theory of graded modal logic. M. de Rijke re-proved the finite model property by using finite bisimulation and selection method, and also proved invariance and definability results on models (de Rijke 2000, pp. 271–283). Under the inspiration of M. de Rijke’s definitions, we can define some elementary model constructions and prove some invariant results of all graded modal formulas. 5.1 Graded Modalities There are many applications of graded modalities in epistemic logic, knowledge representation, and generalized quantifier theory. We take two examples. Example 3. (van der Hoek and Meyer 1993, pp. 503–514) In a S5-model, the formula Kn φ is interpreted as follows:

M, w Kn φ iff Card(R[w ] ∩ V(¬φ)) < n, that is, the agent holds that there are fewer than n exceptions of φ. The formula K1 φ corresponds to the standard epistemic logical formula Kφ. In the formula Kn φ, n is larger, the degree of the agent’s acceptance of φ is smaller. That is, the formula expresses the degrees of uncertain knowledge. The formula Knφ → Kn+1φ is also intuitively plausible since if the agent knows that φ has less than n exceptions, then he knows of course that φ has less than n + 1 exceptions. We use the following figure to show the change of degrees of uncertainty of knowledge: K1 φ → … → Kn φ → Kn+1 φ → … ⇒ … → Mn+1 φ → Mn φ → … → M1 φ,

where Mn is the dual of Kn . The meaning of Mn φ is that the agent holds that φ is not impossible. Example 4. (van der Hoek and de Rijke 1993, pp. 19–58) In the generalized quantifiers theory, numerical quantifiers can be used to express syllogisms and inferences involving numerical quantification. For instance, define Q !nA as ◇nA ∧ ¬◇n+1A. Then


(1) All A are B. Some C are not B. Therefore, some C are not A. This syllogism can be expressed as follows: □1(A → B), ◇1(C ∧ ¬B) ◇1(C ∧ ¬A);

(2) There are 10 A. At least 7 B are A. At least 4 C are A. Therefore, at least 1 B is C. Such a inference can be expressed as follows: Q !10 A, ◇7(B ∧ A), ◇4(C ∧ A) ◇1(B ∧ C).

5.2 Language and Semantics Graded modal language GML(τ, Φ) consists a set of unary modalities τ = {◇k : k ∈ ω} and a (countable) set of propositional letters Φ. The set of formulas Form(τ, Φ) of GML is given by the following rule:

φ ::= p | ⊥ | φ → ψ | ◇k φ, where p ∈ Φ and k ∈ ω. Other connectives are defined as usual, and the dual □k of ◇k is defined as ¬◇k ¬.

Graded modal language is a highly expressive variant of basic modal language. In basic modal logic, the fundamental semantic concept is the truth of a formula at a state on a model. In graded modal logic, however, the basic semantic notion is the truth at a set of states. Given a model M = (W, R, V ), x ∈ W and X ⊆ W, let RxX denote Rxu for all u ∈ X. Note that we only require that the set of states X is finite. Thus, we generally consider the set Fin(W) of all finite subsets of W. The following are some semantic facts about graded modal formulas.

(1) Let M be a transitive model. Then there exists a natural number m such

that M ◇k◇j φ → ◇k + j − m φ; (2) Let M be a transitive tree model with root w and k, j > 0. Then we have M,

w ◇k◇j φ → ◇k + j φ; (3) Let F = (W, R) be a frame. F ◇φ → □φ iff R is not branching, namely,

any state w is a dead point or has unique R-successor. Thus F ◇φ → □φ iff Card(R[w ]) ≤ 1 for any state w. Similarly, F !◇ k φ → □φ iff Card(R[w]) = k for any state w.

5.3 Standard Translation of GML Graded modal logic can be translated naturally into first-order logic with identity. Define the translation GST(φ, x) recursively:

(1) GST(P, x) = Px for every propositional letter p ∈ Φ; (2) GST(⊥, x) = ⊥; (3) GST(φ → ψ, x) = GST(φ, x) → GST(ψ, x);

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(4) GST(◇k φ, x) = ∃y1…yn(∧i < j ≤ k(yi ≠ yk)∧∧i ≤ k(Rxyi∧GST(φ, yi))).

It is easy to check that M, w φ iff M GST(φ, x)[w ] for any pointed model (M, w) and graded modal formula φ. Thus GML has compactness and the Löwenheim-Skolem property. Note that there are problems here. Each ◇n– formula needs at least n(n − 1)/2 non-equations and n + 1 variables.1 Thus we translate GML naturally into first-order logic with finite cardinality quantifiers. By adding finite cardinality quantifiers which are definable in first-order language with identity into first-order logic without identity, we obtain the language FOL(∃≥n). In the standard translation of GML, the clause of graded modalities is replaced by GST(◇k φ, x) = ∃≥k y (Rxy ∧ GST(φ, y)). This translation strategy matches exactly with the semantics of graded modalities.

Moreover, we can translate GML into the 2-variable fragment of FOL(∃≥n), as in the case of basic modal language. For instance, GST(◇m◇n p, x) = ∃≥m y (Rxy ∧ GST(◇n p, y)) = ∃≥m y (Rxy ∧ ∃≥n x (Ryx ∧ Px)). Such a translation provided a way of investigating the fragment of FOL(∃≥n) in terms of guarded quantification. Thus we get the guarded fragment of graded modal logic as if the standard translation of basic modal logic induces the idea of investigating the guarded fragment of first-order logic.2 In such a fragment, quantifiers are only allowed to occur in the formula of the form ∃≥n x (G(x, z1, …, zm) ∧ φ(x, z1, …, zm)) where G(x, z1, …, zm) is atomic. Certain logical aspects of the fragment which can be investigated, especially the decidability, are still unexplored. 5.4 Model Constructions In the model-theoretic study of graded modal logic, we can prove some preservation results under model constructions as in basic modal logic. Such model constructions include disjoint unions, generated substructures, morphisms, tree unraveling, and so on.

Disjoint unions of models will not change the accessibility relations of the original models. Thus, they automatically fit with graded modal logic. Generated substructures also do not change the accessibility relations, since the domain of a substructure is closed under the original R-accessibility relation. Tree unraveling is also the same, since all original accessibility relations are inherited. Thus,

1 This feature leads to considerations about other sorts of economic translations. Actually, proof-makers cannot at present cope with so much non-equations. Ohlbach and Schmidt introduced the set-theoretic translation which supports automatic reasoning (Ohlbach and Schmidt 1996, pp. 253–291). 2 See Blackburn et al. 2001, section 7.4 for a brief introduction to guarded fragments. Also see Goranko and Otto 2007, pp. 249–330, for discussions of the model theory of this fragment, especially the notion of bisimulation.


discussions about such model constructions are simple. Conversely, the notion of bounded morphism is different from the classical

one since the classical notion will not preserve semantics for GML. We redefine the new notion of graded strong homomorphism and graded bounded morphism. Definition 1. Given any two models M = (W, R, V ) and M ′ = (W ′, R ′, V′), define a function f : Fin(W) → Fin(W ′) as follows:

(1) if f ({w}) = {w ′}, then w ∈ V(p) iff w ′ ∈ V ′(p) for all propositional letter p;

(2) if f (X) = Y, then Card(X) = Card(Y) = i ≥ 1; (3) RwX iff R ′w ′f (X), where w ′ ∈ f ({w}); (4) if f (X) = Y, then for all x ∈ X there exists y ∈ Y with f ({x}) = {y}, and

vice versa. Then f is called a graded strong homomorphism from M to M ′. Proposition 1. All graded modal formulas are invariant under surjective graded strong homomorphism, i.e., for any surjective strong homomorphism f : M → M ′ with f ({w}) = {w ′}, we have M, w ≡g M ′, w ′. Definition 2. A function f : M → M ′ is called a g-bounded morphism if it satisfies the following conditions:

(1) if f ({w}) = {w ′}, then w ∈ V(p) iff w ′ ∈ V ′(p) for all propositional letter p;

(2) if f (X) = Y, then Card(X) = Card(Y) = i ≥ 1; (3) if RwX, then R ′w ′f (X), where w ′ ∈ f ({w}); (4) if R ′w ′Y and f ({w}) = {w ′}, then there exists X ∈ Fin(W) with f (X)

= Y and RwX; (5) if f (X) = Y, then for all x ∈ X there exists y ∈ Y with f ({x}) = {y},

and vice versa. Proposition 2. All graded modal formulas are invariant under g-bounded morphic images, i.e., for any g-bounded morphism f : M → M ′ with f ({w}) = {w ′}, M, w ≡g M ′, w ′. Finally, let us consider the tree model property of graded modal logic. A path in a frame F = (W, R) is a sequence s = (w0, …, wk) such that wi−1Rwi for all 1 ≤ i ≤ k. The point w0 is called the start point of s, and w the end point of s. The length of s is denoted by lh(s) which is k for the above sequence s. Let S[u] be the set of all paths starting from the point u. Define a function t : S [u] → W such

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that t(s) = wk for every s in S [u] and lh(s) = k. The function t is called the end point function in the unraveling model. Definition 3. The model unraveled from the point u of a model M = (W, R, V ) is M[u] = (S[u], RS, VS) where RS = {(s, sv) : s ∈ S[u] & Rs(t)v}, and VS(p) = f -1[V(p)]. Define a function g : Fin(S [u]) → Fin(W) such that g(X) = {v ∈ W : t(s) = v for some s ∈ X}. It is clear that Card(g(X)) = Card(X). Let M = (W, R, V ) is a model generated from the point u ∈ W. It can then be checked that the function g is a p-morphism from M [u] to M. We then have the following tree model property: each satisfiable graded modal formula is satisfied at the root of some tree model. 5.5 Bisimulation The notion of bisimulation is the fundamental notion for studying modal logic. The notion of bisimulation for basic modal language does not preserve semantics for GML. M. de Rijke redefined the notion of bisimulation so that it fits graded modal logic (de Rijke 2000, pp. 271–283). Definition 4. (Ibid.) Let M = (W, R, V ), and M ′ = (W ′, R ′, V ′ ) are models. A sequence of relations Z = (Z1, Z2, …) is called a g-bisimulation between M and M ′ (written: Z : M g M ′) if it satisfies the following conditions: (1) Z1 ≠ ∅ and Zi ⊆ Fin(W) × Fin(W ′) for all i ≥ 1; (2) if XZiY, then a) Card(X) = Card(Y) = i ; b) for all x ∈ X there exists y ∈

Y with {x}Zi{y}, and vice versa; (3) if {x}Z1{y} then x ∈ V(p) iff y ∈ V ′(p) for all propositional letter p;

(4) if RxX & Card(X) = i ≥ 1, then there exists Y ∈ Fin(W ′) with R ′yY and XZiY;

(5) if R ′yY & Card(Y) = i ≥ 1, then there exists X ∈ Fin(W) with RxX and XZiY.

The g-bisimulation between two pointed models (M, x) and (M, y) is written as Z : M, x g M, y where {x}Z1{y}. It is easy to check that M, x g M ′, y implies M, x ≡g M ′, y. De Rijke also proved that all ω-saturated models have the Hennessy-Milner property. The van Benthem characterization theorem for GML is then obtained.


Proposition 3. (Ibid.) For any two ω-saturated models (M, x) and (M ′, y),we have M, x ≡g M ′, y iff M, x g M ′, y. Theorem 1. (Ibid.) A formula α(x) in countable first-order language with identity is logically equivalent to the standard translation of some graded modal formula iff it is invariant under g-bisimulation. Moreover, M. de Rijke (Ibid.) also established the model-theoretic closure conditions for the modal definability of classes of pointed models:

(1) A class K of pointed models is definable by a set of graded modal formulas iff K is closed under g-bisimulations and ultraproducts, and the complementary class of K is closed under ultrapowers;

(2) K is definable by a single graded modal formula iff K is closed under g-bisimulations, and both K and its complementary class are closed under ultraproducts.

6 Further Questions

We have surveyed the basic ideas of model-theoretic modal logics and some new directions of inquiry. In the third part, we also pointed out some different sorts of graded modal logics. In part 4, we concentrated on studies of graded modal logic (i.e., the modal of finite cardinal numbers). In this final section we propose some further questions for study, which I hope will be interesting and noteworthy to readers.

(1) Some model-theoretic aspects of graded modal logic are worthy of study. For instance, we have not yet obtained conditions for the definability of elementary frame classes in graded modal logic, i.e., we need a Goldblatt-Thomason type theorem for GML (See Goldblatt and Thomason 1974, pp. 163–173 and section 3.8 of Blackburn et al. 2001).

(2) Ultrafilter extension is also a very important model construction. It can be used to characterize bisimulation. We need a definition of ultrafilter extension for GML and the corresponding invariance results on models (See chapter 2 in Blackburn et al. 2001). Note that we have the canonical model construction for GML. Thus, we should be able to prove the fundamental theorem for ultrafilter extensions. (3) Algebraic semantics is a powerful semantics for modal logic. Modal algebras as well as Boolean algebras with operators are used, respectively, to algebraize modal semantics and modal axiomatizations. These two sorts of algebra are connected through a representation theorem. Since we have some

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GML completeness results, we should be able to obtain Boolean algebras with operators for normal graded modal logic and prove the corresponding representation theorem. (4) Only one aspect of abstract model theory of modal logic has been displayed in this paper: extending modal logic. Abstract properties of logics are also worthy of research. Ten Cate et al. (2007) proved the Lindström theorem for graded modal logic on tree structures. What about the full characterization of graded modal logic?

(5) We also pointed out the potential for studying the fragment of first-order logic with finite cardinality quantifiers, which is inspired by graded modal logic. We may try to determine the graded modal fragment of FOL(∃≥n), and to study the decidability and bisimulation of it. (6) There are many applications in other fields. For instance, graded modal logic is used to cope with the degrees of uncertainty of knowledge in epistemic logics (See van der Hoek and Meyer 1992, pp. 503–514). In the investigations of generalized quantifiers, we can also study similar modal logics (See van der Hoek and de Rijke 1993, pp. 19–58). In knowledge representation theory or description logic, we can treat the expression of knowledge with number restrictions. W. van der Hoek and M. de Rijke (1995, pp. 325–345) introduced description logics with graded modalities.

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