1 Introduction
Compared to normal modal logics, nonnormal modal logics usually have many disadvantages, such as weak expressivity, weak frame definability, which leads to nontriviality of axiomatizations. Contingency logic is such a logic [13, 3, 9, 11, 15, 8]. Since it was independently proposed by Scott and Montague in 1970 [14, 12], neighborhood semantics has been a standard semantical tool for handling nonnormal modal logics [2].
A neighborhood semantics of contingency logic is proposed in [5]. 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 [5]. 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 contingency logic, and show their completeness with respect to the corresponding neighborhood frames. This will give a complete diagram which includes 8 systems, as [2, Fig. 8.1] did for standard modal logic.
The remainder of this note 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 note, we fix P to be a denumerable 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.
The neighborhood semantics of is interpreted on neighborhood models. To say a triple is a neighborhood model, if is a nonempty set of states, is a neighborhood function, and is a valuation.
The following list of neighborhood properties is taken from [5, Def. 3].
Definition 1 (Neighborhood properties).
: contains the unit, if .
: is closed under intersections, if implies .
: is supplemented, or closed under supersets, if and implies .
: is closed under complements, if 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). Especially, a frame is called quasifilter, if it possesses and ; a frame is called filter, if it has also .
Given a neighborhood model and a state , the semantics of is defined as follows [5], where is the truth set of (i.e. the proposition expressed by ) in .
Our discussions will be based on the following axioms and rules.
We will show that the following systems are sound and strongly complete with respect to the class of their corresponding frame classes.
Given a system and a maximal consistent set for , let be the proof set of in ; in symbol, . It is easy to show that . 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 2.
Let be a system excluding M. A tuple is a canonical neighborhood model for , if

,

,

.
Theorem 3.
[5, Thm. 1] is sound and strongly complete with respect to the class of all neighborhood frames.
In what follows, we will extend the canonical model construction to all systems excluding M listed above.
It is not hard to show that C is invalid on the class of all frames, and thus is not sound (and strongly complete) with respect to the class of all frames satisfying . Despite this, it is indeed sound and strongly complete with respect to a more restricted frame class.
Theorem 4.
is sound and strongly complete with respect to the class of all frames satisfying .
Proof.
By Thm. 3, it suffices to show that C is valid on frames, and that possesses and . The former follows from the fact that C is valid on the class of frames under a new semantics proposed in [4] and that on frames, the current semantics and the new semantics satisfies the same formulas.
As for the latter, Equ guarantees , and C provides . ∎
Theorem 5.
is sound and strongly complete with respect to the class of all frames satisfying .
Proof.
It suffices to show that possesses the property . This is immediate due to N and the definition of . ∎
Theorem 6.
is sound and strongly complete with respect to the class of all frames satisfying .
4 Systems including M
In this section, we show that the systems including M listed above are sound and strongly complete with respect to the corresponding frame classes.
We first consider the system . The following result tells us that is sound with respect to the class of frames satisfying .
Proposition 7.
M is valid on the class of frames satisfying .
Proof.
Let be a model and . Suppose that , then or . If , then by , , which implies ; if , then similarly, we can obtain . Either case gives us , as required. ∎
For the completeness, we construct the canonical neighborhood model for , where the crucial definition is the canonical neighborhood function. The definition of below is inspired by a function introduced in [11].^{1}^{1}1The difference between and lies in the codomains: ’s codomain is , whereas ’s is .
Definition 8.
Let be a system including M. A triple is a canonical model for , if

,

For each , ,

For each , .
We need to show that is welldefined.
Lemma 9.
If , then iff .
Proof.
Suppose that , then , then for every , . By RE, we have , thus for every , iff for every , , and therefore iff . ∎
Lemma 10.
Let be a canonical model for . Then for all , for all , we have i.e. .
Proof.
By induction on . 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, . ∎
Note that is not necessarily supplemented. Thus we need to define a notion of supplementation, which comes from [2].
Definition 11.
Let be a neighborhood model. The supplementation of , denoted , is a triple , where for each , is the superset closure of , i.e. for every and ,
It is easy to see that is supplemented. Moreover, . The proof below is a routine work.
Proposition 12.
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 . To demonstrate the completeness of with respect to the class of frames, we need only show that is a canonical model for . That is,
Lemma 13.
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 14.
For all , for all , we have i.e. .
With a routine work, we obtain
Theorem 15.
is sound and strongly complete with respect to the class of frames satisfying .
We are now in a position to deal with the sound and strong completeness of . First, the soundness follows from the following result.
Proposition 16.
C is valid on the class of quasifilters.
Proof.
Let be a 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 .
∎
Proposition 17.
Let be a canonical model for . Then possesses the property . As a corollary, also possesses the property .
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. . Then it follows that also possesses the property from Prop. 12. ∎
Theorem 18.
is sound and strongly complete with respect to the class of quasifilters.
Proposition 19.
Let be a canonical model for . Then possesses the property . As a corollary, also possesses .
Theorem 20.
is sound and strongly complete with respect to the class of frames satisfying .
It is straightforward to obtain the following result.
Theorem 21.
is sound and strongly complete with respect to the class of filters.
By constructing countermodels, we can obtain the following cube, which summarizes the deductive powers of the systems in this paper. An arrow from a system to another means that is deductively stronger than .
5 Reflection: how does the function arise?
As noted, in order to show the completeness of proof systems including M, a crucial part is to define a suitable canonical function, i.e. , which is inspired by the function in [11]. The is very important for the definition of canonical relation and thus for the completeness proof in the cited paper. It is this function that helps find simple axiomatizations for the minimal contingency logic and transitive contingency logic under Kripke semantics, so to speak. Despite its importance, the author did not say any intuitive idea about . And this function was thought of as ‘ingenious’ creation by some other researchers, say Humberstone [10, p. 118] and Fan, Wang and van Ditmarch [8, p. 101]. But how does the function arise? In this section, we unfold the mystery of , and show that it is actually equal to a related function proposed in Humberstone [9].
To show completeness of minimal noncontingency logic under Kripke semantics, Humberstone [9, p. 219] defined the canonical relation as iff , where, denoted by H’s ,
The reason for defining the function in such a way, is that the author would like to ‘simulate’ the canonical relation of the minimal modal logic, which is defined via iff , where . This can be seen from several passages:
The intuitive idea is that for , is the set of formulas which are necessary at .
The idea of the entry condition on , that only such (with ) should be labeled as Necessary if all their consequences are noncontingent, is that , those noncontingencies which qualify as such because they, rather than their negations, are necessary and have only noncontingent consequences, since those consequences are themselves necessary. [9, p. 219]
Then the function was simplified, and accordingly, the completeness proof was simplified in [11]. There, , denoted by K’s , is defined as:
In the sequel, we will demonstrate that, in fact, K’s is equal to H’s .
To begin with, notice that , thus the part following ‘and’ in the H’s definition entails . Therefore, the H’s is equal to a simplified version:
Then it is sufficient to show that the simplified is further equal to K’s , even in the setting of arbitrary neighborhood contingency logics (as opposed to Kripke contingency logics).
Proposition 22.
Let be a maximal consistent set. Given the rule RE, the following statements are equivalent.^{3}^{3}3RE is just (Cong) in [9].
(1) For every , .
(2) For every , .
Proof.
: suppose (1) holds. Since , then it is immediate by (1) that , namely (2).
: suppose (2) holds, to show (1). For this, assume that , then , by RE, . By (2), we obtain that , as desired. ∎
6 Concluding Discussions
In this note, by defining suitable neighborhood canonical functions, we presented a sequence of contingency logics under neighborhood semantics. In particular, inspired by Kuhn’s function in [11], we defined a desired canonical neighborhood function, and then axiomatized monotone contingency logic and regular contingency logic and other logics including the axiom M, thereby answering two open questions raised in [1]. We then reflected on the function , and showed that it is actually equal to Humberstone’s function in [9], even in the setting of arbitrary neighborhood contingency logics.
Moreover, as we observe, in , M can be replaced by , and in , C can be replaced by .^{4}^{4}4The hard part is the direction from to C. The proof details for this, we refer to [6, Prop. 50], where knowing whether operator Kw is the epistemic reading of . Thus we can also adopt these two alternative formulas to axiomatize monotone contingency logic and regular contingency logic. Therefore, it was wrong to claim that “one cannot fill these gaps with the axioms and ” on [1, p. 62].
Recall that an ‘almost definability’ schema, , is proposed in [7], and shown in [8] to be applied to axiomatize contingency logic over much more Kripke frame classes than Kuhn’s function and other variations. Therefore, it may be natural to ask if the schema can also work in the neighborhood setting. The canonical neighborhood function inspired by the schema seems to be
Unfortunately the answer seems to be negative. The reason can be explained as follows. Although in Def. 8 is almost monotonic in the sense that if and , then , as can be easily seen from the proof of Lemma 13, in contrast, as one may easily verify, is not almost monotonic in the above sense, i.e., it fails that if and , then . This can also explain why works well for monotone and regular contingency logics and other logics including the axiom M. Despite this fact, this does not apply to systems excluding the axiom M, since we need this axiom to ensure the truth lemma (Lemma 10). It is also worth noting that this is smaller than that in the case of classical contingency logic (Def. 2), thus we cannot address all neighborhood contingency logics in a unified way.^{5}^{5}5In contrast, the canonical neighborhood function used in the completeness proof of classical modal logic is the smallest neighborhood function among canonical neighborhood functions used in the completeness proofs of all neighborhood modal logics. Cf. e.g. [2]. This indicates that the completeness proofs of these logics are nontrivial. Besides, seems not workable for proper extensions of , which we leave for future work.
7 Acknowledgements
This research is supported by the youth project 17CZX053 of National Social Science Fundation of China. The author would like to thank Lloyd Humberstone for careful reading of the manuscript and insightful comments.
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