A family of neighborhood contingency logics
Abstract
This article proposes the axiomatizations of contingency logics of various natural classes of neighborhood frames. In particular, by defining a suitable canonical neighborhood function, we give sound and complete axiomatizations of monotone contingency logic and regular contingency logic, thereby answering two open questions raised by Bakhtiari, van Ditmarsch, and Hansen. The canonical function is inspired by a function proposed by Kuhn in 1995. We show that Kuhn’s function is actually equal to a related function originally given by Humberstone.
Keywords: contingency logic, neighborhood semantics, axiomatization, monotone logic, regular logic
1 Introduction
Compared to standard modal logic, nonstandard modal logics usually have many disadvantages, such as weak expressivity, weak frame definability, which brings about nontriviality of axiomatizations. Contingency logic is such a logic [15, 4, 9, 12, 19, 18, 17, 7, 8]. Contingency logic is concerned with the study of principles of reasoning involving contingency, noncontingency, and related notions. Since it was introduced, contingency logic has mainly been investigated within the framework of Kripke semantics. However, a known pain for axiomatizing this logic over various Kripke frames is the absence of axioms characterizing frame properties. Moreover, although there have been many results on the axiomatizations of contingency logic which are extensions of minimal logic , there have been no yet much axiomatizations weaker than , for which we need neighborhood semantics. Since it was independently proposed by Scott and Montague in 1970 [16, 14], neighborhood semantics has been a standard semantical tool for handling nonnormal modal logics [3].
A neighborhood semantics of contingency logic is proposed in [6]. According to the interpretation, a formula is noncontingent, if and only if the proposition expressed by is a neighborhood of the evaluated state, or the complement of the proposition expressed by is a neighborhood of the evaluated state. This interpretation is in line with the philosophical intuition of noncontingency, viz. necessarily true or necessarily false. It is shown that contingency logic is less expressive than standard modal logic over various neighborhood model classes, and many neighborhood frame properties are undefinable in contingency logic. This brings about the difficulties in axiomatizing this logic over various neighborhood frames.
To our knowledge, only the classical contingency logic, i.e. the minimal system of contingency logic under neighborhood semantics, is presented in the literature [6]. It is left as two open questions in [1] what the axiomatizations of monotone contingency logic and regular contingency logic are. In this paper, we will answer these two questions.
Besides, we also propose other proof systems up to the minimal Kripke contingency logic, and show their completeness with respect to the corresponding neighborhood frames. This will give a complete diagram which includes 8 systems, as [3, Fig. 8.1] did for standard modal logic. It is a subdiagram of a larger diagram of 16 logics, due to the introduction of a property of being closed under complements.
The remainder of the paper is structured as follows. Section 2 introduces some basics of contingency logic, such as its language, neighborhood semantics, axioms and rules. Sections 3 and 4 deal with the completeness of proof systems mentioned in Sec. 2, with or without a special axiom. The completeness proofs rely on the use of canonical neighborhood functions. In Sec. 3, a simple canonical function is needed, while in Sec. 4 we need a more complex canonical function, which is inspired by a crucial function used in a Kripke completeness proof in the literature. We further reflect on this in Section 5, and show it is in fact equal to a related but complicated function originally given by Humberstone. We conclude with some discussions in Section 6.
2 Preliminaries
Throughout this paper, we fix P to be a nonempty set of propositional variables. The language of contingency logic is defined recursively as follows:
is read “it is noncontingent that ”. The contingency operator abbreviates . It does not matter which one of and is taken as primitive. We use to mean that is an formula, and we always leave out the reference to and simply say that is a formula.
The neighborhood semantics of is interpreted on neighborhood models. We say that is a neighborhood model if is a nonempty set of states, is a neighborhood function, and is a valuation assigning a set to each propositional variable . A neighborhood frame is a neighborhood model without valuations.
Given a neighborhood model and a state , the semantics of is defined recursively as follows [6],
where is the truth set of (i.e. the proposition expressed by ) in . Formula is valid in a frame , notation: , if for all models based on and all in , we have that ; is valid on a class of frames, notation: , if for all in , we have that . Notions of validity of a set of formulas in a frame and on a class of frames are defined similarly. Moreover, given a class of frames, we say is definable in , if there is a such that iff .
Definition 1 (Neighborhood frame properties).
Let be a neighborhood frame. For every and every :
: is supplemented, or closed under supersets, if and implies .
: is closed under intersections, if implies .
: contains the unit, if .
: is closed under complements, if implies .^{1}^{1}1The property was introduced in [6, Def. 3], named ‘’ therein.
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). Especially, a frame is called quasifilter, if it possesses and ; a frame is called filter, if it has also . The property is needed for the following soundness and completeness results, and it provides us a new perspective (see [5]) for the neighborhood semantics of . All properties listed above are shown to be undefinable in [6, Prop. 7]. In contrast, they are definable in standard modal logic .^{2}^{2}2 extends the language of propositional logic with the necessity operator , formally defined as follows: where the neighborhood semantics of is The proofs of the first three can be found in [3, Thm. 7.5, Thm. 9.2], and the proof of the last one is similar to [5, Prop. 5].
Fact 2.
The frame properties on the left are respectively defined by the formulas on the right:
Recall the axioms and rules in 8 classical modal systems and the classes of frames determining them listed below, see e.g. [3, Chap. 8].
Our discussions will mainly be based on the following axioms and rules.
We will show that the following systems are sound and strongly complete with respect to their corresponding frame classes.
The notion of theorems in a system is defined as normal.
By comparison, one can easily see that almost all of systems and the corresponding systems mentioned above are determined by the same class of frames, but with two exceptions: even though and are respectively determined by the class of frames and the class of frames, we have only systems which are respectively determined by the class of frames and the class of frames, that is — and . We do not know whether there are axiomatizations of contingency logics over frames and over frames.
Given a logic (that is, logic in ), one can define a logic, denoted , as , where is defined inductively, with . In other words, proves exactly those formulas whose translations are provable in . It is easy to show that if is the logic of some class of frames (in symbol, ), that is, is the set of formulas that are valid in , then is the logic of (in symbol, ). Note that one cannot obtain the axiomatization of from the axiomatization of , since there is no translation function from to .
Recall that Def. 1 listed 4 frame properties, which constitutes different combinations of such properties. Since every frame class can define a logic, namely , we should have 16 different logics. In this paper, we axiomatize 10 logics as listed above, and leave the axiomatizations of the remaining 6 logics open. We will defer a summary of 16 logics with some remarks to the end of Sec. 4.
In what follows, we also use to denote , which is clearly ‘stronger’ than , . Note that is equivalent to in , since they are interderivable with the rule of monotony in the system in question.
Let denote the class of all frames, denote the class of frames, denote the class of frames, and similarly for other properties.
Proposition 3.
We have the following validities and invalidities:

.

.

.

.^{3}^{3}3It is worth remarking that in the case of frames, we need the property to provide the validity of C (see the proof of item (iii) in this proposition); by comparison, in the case of quasifilters, we do not need , since the validity of C is now guaranteed by and together.

.

.

.

.

.

.

.
Proof.

Let be an model and . By , , that is, , and thus or , and hence . By the arbitrariness of and , we conclude that .

Let be an model and . Suppose that , then or . If , then by , , which implies ; if , then similarly, we can obtain . Either case gives us , as required.

Let be a model and . Suppose and , to show . From it follows that or . Using , we can infer . Similarly, from we can obtain . Now an application of gives us , that is, , and thus .

Let be a quasifilter model and . Suppose that , then or , and or . Consider the following three cases:

and . By , we obtain , i.e. , which gives .

. By , we infer , i.e. , which implies .

. Similar to the second case, we can derive that .


Consider an instance of C: and a model where , , , and . It should be obvious that is a model. On the other hand, since , we have , also, as , we infer ; however, and , thus . Therefore, . We have thus found a model which falsifies an instance of C, and it can be concluded that .

Consider an instance of C: and a model where , , , and , . It should be clear that is a model. Since , we have ; since , thus . However, and , thus . Therefore, . We have thus found a model which falsifies an instance of C, and it can be concluded that .

Consider the following instance of sM: , and a model where , , , and , . One may easily verify that is an model. However, : on one hand, as , we have ; on the other hand, since and , thus .

Follows directly from item (vii), since models are also models, models and models.

Let be an model and . Suppose that , to show that . By supposition, we have or . By , it follows that . Then by , it follows that , viz. , and therefore .
∎
Corollary 4 (Soundness).
The aforementioned 10 logics are sound with respect to their corresponding class of frames.
From the next section, we will start to show the completeness results of these systems, with the aid of canonical neighborhood model constructions. As one will see, all the above systems may not be handled by a uniform canonical neighborhood function; instead, we need to distinguish systems excluding axiom M from those including it.
Given a system and the set of all maximal consistent sets for , let be the proof set of relative to , in symbol, .^{4}^{4}4The terminology ‘the proof set’ can be found on [3, p. 57]. It is easy to show that and . We always omit the subscript when it is clear from the context.
3 Systems excluding M
Given a proof system, a standard method of showing its completeness under neighborhood semantics is constructing the canonical neighborhood model, where one essential part is the definition of canonical neighborhood function.
Definition 5.
Let be a system excluding M. A tuple is the canonical neighborhood model for , if

,

,

.
Notice that thanks to axiom Equ, the function in the above definition has the property , that is, for all and , if , then .
Theorem 6.
[6, Thm. 1, Thm. 2] is strongly complete with respect to the class of all neighborhood frames and also w.r.t. the class of all frames. Therefore, .
In what follows, we will extend the canonical model construction to all systems excluding M listed above.
Theorem 7.
is strongly complete with respect to the class of all frames.
Proof.
By Thm. 6, it suffices to show that possesses . This is guaranteed by axiom C: suppose and , then by definition of , and for some and , thus and , which implies because of axiom C, and therefore , that is, . ∎
Theorem 8.
is strongly complete with respect to the class of all frames and also w.r.t. the class of all frames. Therefore, .
Proof.
By Thm. 6, it suffices to show that possesses the property . This is immediate due to N and the definition of : since , we have that for all , , and then , that is, . ∎
Theorem 9.
is strongly complete with respect to the class of all frames.
4 Systems including M
In this section, we show that the systems including M listed above are strongly complete with respect to the corresponding frame classes. For this, we construct the canonical neighborhood model for any system extending , where the crucial definition is the canonical neighborhood function. The definition of below is inspired by a function introduced in [12].^{5}^{5}5The difference between and lies in the codomains: ’s codomain is , whereas ’s is .
Definition 10.
Let be a system extending . A triple is a canonical neighborhood model for , if

,

For each , iff ,

For each , .
We need to show that is welldefined.
Lemma 11.
Let as defined in Def. 10. If , then iff .
Proof.
Suppose that , then , then for every , . By RE, we have , thus (for every , ) iff (for every , ). ∎
Def. 10 does not specify the function completely; besides the sets of the form that satisfy this definition, may contain other sets that are not of the form for any formula . Therefore, each logic under consideration has many canonical models.
Lemma 12.
Let be an arbitrary canonical model for any system extending . Then for all , for all , we have i.e. .
Proof.
By induction on . The base case and Boolean cases are straightforward by Def. 10 and induction hypothesis. The only nontrivial case is .
Suppose, for a contradiction, that but . Then by induction hypothesis, we obtain , and , i.e. . Thus for some , and for some . Using axiom M, we obtain : a contradiction.
Conversely, assume that , to show that . By assumption and induction hypothesis, we have , or , i.e. . If , then for every , . In particular, ; if , then by a similar argument, we obtain , thus . Therefore, . ∎
Given a system extending , the minimal canonical model for , denoted , is defined where . Note that is not necessarily supplemented. Thus we need to define a notion of supplementation, which comes from [3].
Definition 13.
Let be a neighborhood model. The supplementation of , denoted , is a triple , where for each , is the superset closure of , i.e. for every ,
Intuitively, differs from only in that contains every proposition in that includes any proposition in .
It is easy to see that is supplemented. Moreover, . The proof below is a routine work.
Proposition 14.
Let be a neighborhood model. If possesses the property , then so does ; if possesses the property , then so does .
We will denote the supplementation of by . By definition of , is an model. To demonstrate the completeness of any system extending with respect to the class of frames, we need only to show that is a canonical neighborhood model for any system extending . That is,
Lemma 15.
For each ,
Proof.
‘’: immediate by for each and the definition of .
‘’: Suppose that , to show that . By supposition, for some . Then there is a such that , and thus for every , in particular . From follows that , thus , and hence by RE. Therefore for every . ∎
Lemma 16.
For all , for all , we have i.e. .
With a routine work, we obtain
Theorem 17.
is strongly complete with respect to the class of frames.
We are now in a position to deal with the strong completeness of .
Proposition 18.
For any system extending , the minimal canonical model has the property . Hence, its supplementation is an model.
Proof.
Suppose and , to show that . By supposition, there exist and such that and , and then for every , and for every . Using axiom C, we infer for every . Therefore, , i.e. . Thus has the property . Then it follows that also possesses the property from Prop. 14. ∎
Theorem 19.
is strongly complete with respect to the class of quasifilters.
Now for .
Proposition 20.
For any system extending , the minimal canonical model has the property . Hence, its supplementation is an model.
Proof.
Let . By axiom N, we have . This implies that for every . Then , and thus . We have now shown that has the property . Then it follows that is an model from Prop. 14. ∎
Theorem 21.
is strongly complete with respect to the class of frames.
Finally, for . For any system , we have and . By Prop. 18, has the property ; by Prop. 20, has the property ; by the definition of , has the property . Therefore, is a filter. Then combining Lemma 16, we conclude that
Theorem 22.
is strongly complete with respect to the class of filters.
Remark 23.
We conclude this section with a diagram and some remarks. By constructing countermodels, we can obtain the following cubes, which summarize the deductive powers of the 16 logics mentioned in this paper. Among these systems, is the weakest system, and is the strongest system. An arrow from a system to another means that is deductively stronger than . This is in line with the case of classical modal logics, cf. e.g. [3, Fig. 8.1].
Remark 24.
Let Z denote .

In this paper, among 16 logics in the above diagram, 10 logics were axiomatized (labeled with blue), and axiomatizations of the remaining 6 logics are open. Our conjecture is that if we replace M in the axiomatizations of logics on the second level with sM, namely , or equivalently , then we will obtain the axiomatizations of logics on the third (topmost) level.

holds for any classical modal logics . This is because for any classes of frames and , if and , then . For instance, although we do not know yet what the axiomatizations of logics and are, we do know and , since and .

However, the equation does not hold for any classical modal logics , but only holds for the cases when or (in comparison, and as is easily verified). This may be explained with help of a notion of ‘