1 Introduction
Under Kripke semantics, contingency logic (CL for short) is nonnormal, less expressive than standard modal logic (ML for short), and the five basic frame properties (seriality, reflexivity, transitivity, symmetry, Eucludicity) cannot be defined in CL. This makes the axiomatizations of CL nontrivial: although there have been a mountain of work on the axiomatization problem since the 1960s [11, 9, 10, 15, 12], over , , , , and any combinations thereof, no method in the cited work can uniformly handle all the five basic frame properties. This job has not been addressed until in [5], which also contains an axiomatization of CL on and its multimodal version. This indicates that Kripke semantics may not be suitable for CL.
Partly inspired by the above motivation (in particular, the nonnormality of CL), and partly by a weaker logical omniscience in Kripke semantics (namely, all theorems are known to be true or known to be false), a neighborhood semantics for CL is proposed in [4], which interprets the noncontingency operator in a way such that its philosophical intuition, viz. necessarily true or necessarily false, holds. However, under this (old) semantics, as shown in [4], CL is still less expressive than ML on various classes of neighborhood models, and many usual neighborhood frame properties are undefinable in CL. Moreover, based on this semantics, [1] proposes a bisimulation (called ‘nbhbisimulation’ there) to characterize CL within ML and within firstorder logic (FOL for short), but the essence of the bisimulation seems not quite clear.
In retrospect, no matter whether the semantics for CL is Kripkestyle or neigborhoodstyle in the sense of [4], there is an asymmetry between the syntax and models of CL: on the one hand, the language is too weak, since it is less expressive than ML over various model classes; on the other hand, the models are too strong, since its models are the same as those of ML. This makes it hard to handle CL.^{1}^{1}1Analogous problem occurs in the setting of knowingvalue logic [14, 13].
Inspired by [6], we simplify the neighborhood semantics for CL in [4], and meanwhile keep the logic (valid formulas) the same by restricting models. This can weaken the too strong models so as to balance the syntax and models for CL. Under this new perspective, we can gain a lot of things, for example, bisimulation notions and their corresponding HennessyMilner Theorems, and frame definability. Moreover, we show that one of bisimulation notions is equivalent to the notion of nbhbisimulation, which helps us understand the essence of nbhbisimulation. We also obtain some simple axiomatizations.
2 Preliminaries
2.1 Language and old neighborhood semantics
First, we introduce the language and the old neighborhood semantics of CL. Fix a countable set Prop of propositional variables. The language of CL, denoted , is an extension of propositional logic with a sole primitive modality , where .
is read “it is noncontingent that ”. , read “it is contingent that ”, abbreviates .
A neighborhood model for is defined as that for the language of ML. That is, to say is a neighborhood model, if is a nonempty set of states, is a neighborhood function assigning each state in a set of neighborhoods, and is a valuation assigning each propositional variable in Prop a set of states in which it holds. A neighborhood frame is a neighborhood model without any valuation.
There are a variety of neighborhood properties. The following list is taken from [4, Def. 3].
Definition 1 (Neighborhood properties).
: contains the unit, if .
: contains its core, if .
: is closed under intersections, if implies .
: is supplemented, or closed under supersets, if and implies . We also call this property ‘monotonicity’.
: is closed under complements, if implies .
: implies .
: implies .
: implies .
: implies .
: implies .
Frame (and the corresponding model) possesses such a property P, if has the property P for each , and we call the frame (resp. the model) Pframe (resp. Pmodel).
Given a neighborhood model and , the old neighborhood semantics of [4] is defined as follows, where .
2.2 Existing results on old neighborhood semantics
Under the above old neighborhood semantics, it is shown in [4, Props.27] that on the class of models or the class of models, is equally expressive as ; however, on other class of models in Def. 1, is less expressive than ; moreover, none of frame properties in the above list is definable in CL.
Based on the above semantics for CL, a notion of bisimulation is proposed in [1], which is inspired by the definition of precocongruences in [8] and the old neighbourhood semantics of .
Definition 2 (nbhbisimulation).
Let and be neighborhood models. A nonempty relation is a nbhbisimulation between and , if for all ,
(Atoms) iff for all ;
(Coherence) if the pair is coherent,^{2}^{2}2Let be a binary relation. We say is coherent, if for any , we have iff . We say is closed, if is coherent. It is obvious that is coherent for any . then
and is nbhbisimilar, notation , if there is a nbhbisimulation between and containing .^{3}^{3}3In fact, the notion of nbhbisimilarity is defined in a more complex way in [1]. It is easy to show that our definition is equivalent to, but simpler than, that one.
Although it is inspired by both the definition of precocongruences in [8] and the old neighbourhood semantics of , the essence of nbhbisimulation seems not so clear.
It is then proved that HennessyMilner Theorem holds for nbhbisimulation. For this, a notion of saturated model is required.
Definition 3 (saturated model).
[1, Def. 11] Let be a neighborhood model. A set is compact, if every set of formulas that is finitely satisfiable in is itself also satisfiable in . is said to be saturated, if for all and all closed neighborhoods , both and are compact.
Theorem 4 (HennessyMilner Theorem for nbhbisimulation).
[1, Thm.1] On saturated models and and states in and in , if , then
3 A new semantics for CL
As mentioned above, there is an asymmetry between the syntax and neighborhood models of CL, which makes it hard to tackle CL. In this section, we propose a new neighborhood semantics for this logic. This semantics is simpler than the old one, but the two semantics are equivalent in that their logics (valid formulas) are the same.
The new neighborhood semantics and the old one differ only in the case of noncontingency operator.
where . To say two models with the same domain are pointwise equivalent, if every world in both models satisfies the same formulas.
We hope that although we change the semantics, the validities are still kept the same as the old one. So how to make it out? Recall that noncontingency means necessarily true or necessarily false, which implies that should be valid. However, although the formula is indeed valid under the old neighborhood semantics, it is invalid under the new one. In order to make this come about, we need make some restriction to the models. Look at a proposition first.
Proposition 5.
Under the new semantics, defines the property .
Proof.
Let be a neighborhood frame.
First, suppose possesses , we need to show . For this, assume any model based on and such that , thus . By , , i.e., , which means exactly . Now assume , we have , that is . By , , i.e. , and thus . Hence , and therefore .
Conversely, suppose does not possess , we need to show . By supposition, there exists such that but . Define a valuation on as . We have now , thus . On the other side, , thus . Hence , and therefore . ∎
This means that in order to guarantee the validity under new semantics, we (only) need to ‘force’ the model to have the property . Thus from now on, we assume to be the minimal condition of a neighborhood model, and call this type of model ‘models’.
Definition 6 (structures).
A model is a model, if it has the property ; intuitively, if a proposition is noncontingent at a state in the domain, so is its negation. A frame is a frame, if the models based on it are models.
The following proposition states that on models, the new neighborhood semantics and the old one coincide with each other in terms of satisfiability.
Proposition 7.
Let be a model. Then for all , for all , we have , i.e., .
Proof.
By induction on . The only nontrivial case is .
First, suppose , then . By induction hypothesis, . Of course, or . This entails that .
Conversely, assume , then or . Since is a model, we can obtain . By induction hypothesis, . Therefore, . ∎
Definition 8 (variation).
Let be a neighborhood model. We say is a variation of , if , where for all , .
The definition of is very natural, in that just as “” corresponds to the old semantics of , corresponds to the new semantics of . It is easy to see that every neighborhood model has a sole variation, that every such variation is a cmodel, and moreover, for any neighborhood model , if is already a model, then .
Proposition 9.
Let be a neighborhood model. Then for all , for all , we have , i.e., .
Proof.
The proof is by induction on , where the only nontrivial case is . We have
∎
Let denote that entails over the class of all models, i.e., for each model and each , if for every , then . With Props. 7 and 9 in hand, we obtain immediately that
Corollary 10.
For all , Therefore, for all ,
In this way, we strengthened the expressive power of CL, since it is now equally expressive as ML, and kept the logic (valid formulas) the same as the old neighborhood semantics. The noncontingency operator can now be seen as a package of and in the old neighborhood semantics; under the new neighborhood semantics, on the one hand, it is interpreted just as ; on the other hand, it retains the validity .
4 Bisimulation
Recall that the essence of the notion of nbhbisimulation proposed in [1] is not so clear. In this section, we introduce a notion of bisimulation, and show that this notion is equivalent to nbhbisimulation. The bisimulation is inspired by both Prop. 5 and the definition of precocongruences in [8, Prop. 3.16]. Intuitively, the notion is obtained by just adding the property into the notion of precocongruences.
Definition 11 (cbisimulation).
Let and be models. A nonempty relation is a cbisimulation between and , if for all ,

iff for all ;

if the pair is coherent, then
We say and are cbisimilar, written , if there is a cbisimulation between and such that .
Note that both bisimulation and bisimilarity are defined between models, rather than between any neighborhood models. formulas are invariant under bisimilarity.
Proposition 12.
Let and be models, and . If , then for all ,
Proof.
Let and be both models. By induction on . The nontrivial case is .
follows from the fact that is coherent plus the condition of bisimulation. To see why is coherent, the proof goes as follows: if for any , i.e., , then by induction hypothesis, iff , i.e., iff . ∎
Now we are ready to show the HennessyMilner Theorem for bisimulation. Since bisimulation is defined between models, we need also to add the property into the notion of saturated models in Def. 3.
Definition 13 (saturated model).
Let be a model. A set is compact, if every set of formulas that is finitely satisfiable in is itself also satisfiable in . is said to be saturated, if for all and all closed neighborhood , is compact.^{4}^{4}4Note that we do not distinguish here from that in Def. 3 despite different neighborhood semantics. This is because as we show in Prop. 7, on models the two neighborhood semantics are the same in terms of satisfiability. Thus it does not matter which semantics is involved in the current context.
In the above definition of saturated model, we write “ is compact”, rather than “both and are compact”, since under the condition that and the property , these two statements are equivalent. Thus each saturated model must be a saturated model.
We will demonstrate that on saturated models, equivalence implies bisimilarity, for which we prove that the notion of cbisimulation is equivalent to that of nbhbisimulation, in the sense that every nbhbisimulation (between neighborhood models) is a cbisimulation (between models), and vice versa. By doing so, we can see clearly the essence of nbhbisimulation, i.e. precocongruences with property .
Proposition 14.
Let and be neighborhood models. If is a nbhbisimulation between and , then is a cbisimulation between and .
Proof.
Suppose that is a nbhbisimulation between and , to show is a cbisimulation between and .
First, one can easily verify that and are both models.
Second, assume that . Since and have the same domain and valuation, item (i) can be obtained from the supposition and (Atoms). For item (ii), let be coherent. We need to show that iff . For this, we have the following line of argumentation: iff (by definition of ) ( or ) iff (by (Coherence)) iff ( or ) iff (by definition of ) . ∎
Proposition 15.
Let and be models. If is a cbisimulation between and , then is a nbhbisimulation between and .
Proof.
Suppose that is a cbisimulation between models and , to show is a nbhbisimulation between and . Assume that , we only need to show (Atoms) and (Coherence) holds. (Atoms) is clear from (i).
For (Coherence), let the pair is coherent. Then by (ii), . We also have that is coherent. Using (ii) again, we infer that iff . Therefore, ( or ) iff ( or ), as desired. ∎
Since every variation of a model is just the model itself, by Props. 14 and 15, we obtain immediately that
Corollary 16.
Let and be both models. Then is a bisimulation between and iff is an nbhbisimulation between and .
Theorem 17 (HennessyMilner Theorem for bisimulation).
Let and be saturated models, and , . If for all , , then .
5 Monotonic bisimulation
This section proposes a notion of bisimulation for CL over monotonic, models. This notion can be obtained via two ways: one is to add the property of monotonicity into bisimulation, the other is to add the property into monotonic bisimulation (for ML).^{5}^{5}5For the notion of monotonic bisimulation, refer to [7, Def.
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