A sequence of neighborhood contingency logics
Abstract
This note proposes various axiomatizations of contingency logic under neighborhood semantics. 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 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 [MR66, Cresswell88, Humberstone95, DBLP:journals/ndjfl/Kuhn95, DBLP:journals/ndjfl/Zolin99, Fanetal:2015]. Since it was independently proposed by Scott and Montague in 1970 [Scott:1970, Montague:1970], neighborhood semantics has been a standard semantical tool for handling nonnormal modal logics [Chellas:1980].
A neighborhood semantics of contingency logic is proposed in [FanvD:neighborhood]. According to the interpretation, a formula \varphi is noncontingent, if and only if the proposition expressed by \varphi is a neighborhood of the evaluated state, or the complement of the proposition expressed by \varphi 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 [FanvD:neighborhood]. It is left as two open questions in [Bakhtiarietal:2017] 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 [Chellas:1980, 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 \lambda used in a Kripke completeness proof in the literature. We further reflect on this \lambda in Section LABEL:sec.reflectionlambda, and show it is in fact equal to a related but complicated function originally given by Humberstone. We conclude with some discussions in Section LABEL:sec.concl.
2 Preliminaries
Throughout this note, we fix P to be a denumerable set of propositional variables. The language \mathcal{L}(\Delta) of contingency logic is defined recursively as follows:
\varphi::=p\in\textbf{P}\mid\neg\varphi\mid\varphi\land\varphi\mid\Delta\varphi. 
\Delta\varphi is read “it is noncontingent that \varphi”. The contingency operator \nabla abbreviates \neg\Delta. It does not matter which one of \Delta and \nabla is taken as primitive.
The neighborhood semantics of \mathcal{L}(\Delta) is interpreted on neighborhood models. To say a triple \mathcal{M}=\langle S,N,V\rangle is a neighborhood model, if S is a nonempty set of states, N:S\to\mathcal{P}(\mathcal{P}(S)) is a neighborhood function, and V is a valuation.
The following list of neighborhood properties is taken from [FanvD:neighborhood, Def. 3].
Definition 1 (Neighborhood properties).
(n): N(s) contains the unit, if S\in N(s).
(i): N(s) is closed under intersections, if X,Y\in N(s) implies X\cap Y\in N(s).
(s): N(s) is supplemented, or closed under supersets, if X\in N(s) and X\subseteq Y\subseteq S implies Y\in N(s).
(c): N(s) is closed under complements, if X\in N(s) implies S\backslash X\in N(s).
Frame \mathcal{F}=\langle S,N\rangle (and the corresponding model) possesses such a property P, if N(s) has the property P for each s\in S, and we call the frame (resp. the model) Pframe (resp. Pmodel). Especially, a frame is called quasifilter, if it possesses (i) and (s); a frame is called filter, if it has also (n).
Given a neighborhood model \mathcal{M}=\langle S,N,V\rangle and a state s\in S, the semantics of \varphi\in\mathcal{L}(\Delta) is defined as follows [FanvD:neighborhood], where \varphi^{\mathcal{M}}=\{s\in S\mid\mathcal{M},s\vDash\varphi\} is the truth set of \varphi (i.e. the proposition expressed by \varphi) in \mathcal{M}.
\begin{array}[]{lll}\hline\mathcal{M},s\vDash p&\iff&s\in V(p)\\ \mathcal{M},s\vDash\neg\varphi&\iff&\mathcal{M},s\nvDash\varphi\\ \mathcal{M},s\vDash\varphi\land\psi&\iff&\mathcal{M},s\vDash\varphi\text{ and % }\mathcal{M},s\vDash\psi\\ \mathcal{M},s\vDash\Delta\varphi&\iff&\varphi^{\mathcal{M}}\in N(s)\text{ or }% S\backslash\varphi^{\mathcal{M}}\in N(s)\\ \hline\end{array} 
Our discussions will be based on the following axioms and rules.
\begin{array}[]{ll}\text{TAUT}&\text{all instances of tautologies}\\ \Delta\text{Equ}&\Delta\varphi\leftrightarrow\Delta\neg\varphi\\ \Delta\text{M}&\Delta\varphi\to\Delta(\varphi\vee\psi)\vee\Delta(\neg\varphi% \vee\chi)\\ \Delta\text{C}&\Delta\varphi\land\Delta\psi\to\Delta(\varphi\land\psi)\\ \Delta\text{N}&\Delta\top\\ \text{RE}\Delta&\dfrac{\varphi\leftrightarrow\psi}{\Delta\varphi% \leftrightarrow\Delta\psi}\\ \end{array} 
We will show that the following systems are sound and strongly complete with respect to the class of their corresponding frame classes.
\begin{array}[]{lc}\hline\text{systems}&\text{frame classes}\\ \hline{\bf E^{\Delta}}=\text{TAUT}+\Delta\text{Equ}+\text{RE}\Delta&\text{all}% \\ {\bf M^{\Delta}}={\bf E^{\Delta}}+\Delta\text{M}&(s)\\ {\bf(EC)^{\Delta}}={\bf E^{\Delta}}+\Delta\text{C}&(i)\&(c)\\ {\bf(EN)^{\Delta}}={\bf E^{\Delta}}+\Delta\text{N}&(n)\\ {\bf R^{\Delta}}={\bf M^{\Delta}}+\Delta\text{C}&\text{quasifilters}\\ {\bf(EMN)^{\Delta}}={\bf M^{\Delta}}+\Delta\text{N}&(s)\&(n)\\ {\bf(ECN)^{\Delta}}={\bf(EC)^{\Delta}}+\Delta\text{N}&(i)\&(c)\&(n)\\ {\bf K^{\Delta}}={\bf R^{\Delta}}+\Delta\text{N}&\text{filters}\\ \hline\end{array} 
Given a system \Lambda and a maximal consistent set S^{c} for \Lambda, let \varphi_{\Lambda} be the proof set of \varphi in \Lambda; in symbol, \varphi_{\Lambda}=\{s\in S^{c}\mid\varphi\in s\}. It is easy to show that \neg\varphi_{\Lambda}=S^{c}\backslash\varphi_{\Lambda}. We always omit the subscript \Lambda when it is clear from the context.
3 Systems excluding \DeltaM
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 \Sigma be a system excluding \DeltaM. A tuple \mathcal{M}^{c}=\langle S^{c},N^{c},V^{c}\rangle is a canonical neighborhood model for \Sigma, if

S^{c}=\{s\mid s\text{ is a maximal consistent set for }\Sigma\},

N^{c}(s)=\{\varphi\mid\Delta\varphi\in s\},

V^{c}(p)=p.
Theorem 3.
[FanvD:neighborhood, Thm. 1] {\bf E^{\Delta}} 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 \DeltaM listed above.
It is not hard to show that \DeltaC is invalid on the class of all (i)frames, and thus {\bf(EC)^{\Delta}} is not sound (and strongly complete) with respect to the class of all frames satisfying (i). Despite this, it is indeed sound and strongly complete with respect to a more restricted frame class.
Theorem 4.
{\bf(EC)^{\Delta}} is sound and strongly complete with respect to the class of all frames satisfying (i)\&(c).
Proof.
By Thm. 3, it suffices to show that \DeltaC is valid on (i)\&(c)frames, and that N^{c} possesses (i) and (c). The former follows from the fact that \DeltaC is valid on the class of (i)\&(c)frames under a new semantics proposed in [Fan:2017] and that on (c)frames, the current semantics and the new semantics satisfies the same formulas.
As for the latter, \DeltaEqu guarantees (c), and \DeltaC provides (i). ∎
Theorem 5.
{\bf(EN)^{\Delta}} is sound and strongly complete with respect to the class of all frames satisfying (n).
Proof.
It suffices to show that N^{c} possesses the property (n). This is immediate due to \DeltaN and the definition of N^{c}. ∎
Theorem 6.
{\bf(ECN)^{\Delta}} is sound and strongly complete with respect to the class of all frames satisfying (i)\&(c)\&(n).
4 Systems including \DeltaM
In this section, we show that the systems including \DeltaM listed above are sound and strongly complete with respect to the corresponding frame classes.
We first consider the system {\bf M^{\Delta}}. The following result tells us that {\bf M^{\Delta}} is sound with respect to the class of frames satisfying (s).
Proposition 7.
\DeltaM is valid on the class of frames satisfying (s).
Proof.
Let \mathcal{M}=\langle S,N,V\rangle be a (s)model and s\in S. Suppose that \mathcal{M},s\vDash\Delta\varphi, then \varphi^{\mathcal{M}}\in N(s) or (\neg\varphi)^{\mathcal{M}}\in N(s). If \varphi^{\mathcal{M}}\in N(s), then by (s), \varphi^{\mathcal{M}}\cup\psi^{\mathcal{M}}\in N(s), which implies \mathcal{M},s\vDash\Delta(\varphi\vee\psi); if (\neg\varphi)^{\mathcal{M}}\in N(s), then similarly, we can obtain \mathcal{M},s\vDash\Delta(\neg\varphi\vee\chi). Either case gives us \mathcal{M},s\vDash\Delta(\varphi\vee\psi)\vee\Delta(\neg\varphi\vee\chi), as required. ∎
For the completeness, we construct the canonical neighborhood model for {\bf M^{\Delta}}, where the crucial definition is the canonical neighborhood function. The definition of N^{c} below is inspired by a function \lambda introduced in [DBLP:journals/ndjfl/Kuhn95].^{1}^{1}1The difference between N^{c} and \lambda lies in the codomains: N^{c}’s codomain is \mathcal{P}(\mathcal{P}(S^{c})), whereas \lambda’s is \mathcal{P}(\mathcal{L}(\Delta)).
Definition 8.
Let \Gamma be a system including \DeltaM. A triple \mathcal{M}^{c}=\langle S^{c},N^{c},V^{c}\rangle is a canonical model for \Gamma, if

S^{c}=\{s\mid s\text{ is a maximal consistent set for }\Gamma\},

For each s\in S^{c}, N^{c}(s)=\{\varphi\mid\Delta(\varphi\vee\psi)\in s\text{ for every }\psi\},

For each p\in\textbf{P}, V^{c}(p)=p.
We need to show that N^{c} is welldefined.
Lemma 9.
If \varphi=\psi, then \varphi\in N^{c}(s) iff \psi\in N^{c}(s).
Proof.
Suppose that \varphi=\psi, then \vdash\varphi\leftrightarrow\psi, then for every \chi, \vdash\varphi\vee\chi\leftrightarrow\psi\vee\chi. By RE\Delta, we have \vdash\Delta(\varphi\vee\chi)\leftrightarrow\Delta(\psi\vee\chi), thus for every \chi, \Delta(\varphi\vee\chi)\in s iff for every \chi, \Delta(\psi\vee\chi)\in s, and therefore \varphi\in N^{c}(s) iff \psi\in N^{c}(s). ∎
Lemma 10.
Let \mathcal{M}^{c} be a canonical model for {\bf M^{\Delta}}. Then for all \varphi\in\mathcal{L}(\Delta), for all s\in S^{c}, we have \mathcal{M}^{c},s\vDash\varphi\iff\varphi\in s, i.e. \varphi^{\mathcal{M}^{c}}=\varphi.
Proof.
By induction on \varphi. The only nontrivial case is \Delta\varphi.
Suppose, for a contradiction, that \Delta\varphi\in s but \mathcal{M}^{c},s\nvDash\Delta\varphi. Then by induction hypothesis, we obtain \varphi\notin N^{c}(s), and S^{c}\backslash\varphi\notin N^{c}(s), i.e. \neg\varphi\notin N^{c}(s). Thus \Delta(\varphi\vee\psi)\notin s for some \psi, and \Delta(\neg\varphi\vee\chi)\notin s for some \chi. Using axiom \DeltaM, we obtain \Delta\varphi\notin s: a contradiction.
Conversely, assume that \mathcal{M}^{c},s\vDash\Delta\varphi, to show that \Delta\varphi\in s. By assumption and induction hypothesis, we have \varphi\in N^{c}(s), or S^{c}\backslash\varphi\in N^{c}(s), i.e. \neg\varphi\in N^{c}(s). If \varphi\in N^{c}(s), then for every \psi, \Delta(\varphi\vee\psi)\in s. In particular, \Delta\varphi\in s; if \neg\varphi\in N^{c}(s), then by a similar argument, we obtain \Delta\neg\varphi\in s, thus \Delta\varphi\in s. Therefore, \Delta\varphi\in s. ∎
Note that \mathcal{M}^{c} is not necessarily supplemented. Thus we need to define a notion of supplementation, which comes from [Chellas:1980].
Definition 11.
Let \mathcal{M}=\langle S,N,V\rangle be a neighborhood model. The supplementation of \mathcal{M}, denoted \mathcal{M}^{+}, is a triple \langle S,N^{+},V\rangle, where for each s\in S, N^{+}(s) is the superset closure of N(s), i.e. for every s\in S and X\subseteq S,
N^{+}(s)=\{X\mid Y\subseteq X\text{ for some }Y\in N(s)\}. 
It is easy to see that \mathcal{M}^{+} is supplemented. Moreover, N(s)\subseteq N^{+}(s). The proof below is a routine work.
Proposition 12.
Let \mathcal{M} be a neighborhood model. If \mathcal{M} possesses the property (i), then so does \mathcal{M}^{+}; if \mathcal{M} possesses the property (n), then so does \mathcal{M}^{+}.
We will denote the supplementation of \mathcal{M}^{c} by (\mathcal{M}^{c})^{+}=\langle S^{c},(N^{c})^{+},V^{c}\rangle. To demonstrate the completeness of {\bf M^{\Delta}} with respect to the class of (s)frames, we need only show that (\mathcal{M}^{c})^{+} is a canonical model for {\bf M^{\Delta}}. That is,
Lemma 13.
For each s\in S^{c},
\varphi\in(N^{c})^{+}(s)\iff\Delta(\varphi\vee\psi)\in s\text{ for every }\psi. 
Proof.
‘\Longleftarrow’: immediate by N^{c}(s)\subseteq(N^{c})^{+}(s) for each s\in S^{c} and the definition of N^{c}.
‘\Longrightarrow’: Suppose that \varphi\in(N^{c})^{+}(s), to show that \Delta(\varphi\vee\psi)\in s\text{ for every }\psi. By supposition, X\subseteq\varphi for some X\in N^{c}(s). Then there is a \chi such that \chi=X, and thus \Delta(\chi\vee\psi)\in s for every \psi, in particular \Delta(\chi\vee\varphi\vee\psi)\in s. From \chi\subseteq\varphi follows that \vdash\chi\to\varphi, thus \vdash\chi\vee\varphi\vee\psi\leftrightarrow\varphi\vee\psi, and hence \vdash\Delta(\chi\vee\varphi\vee\psi)\leftrightarrow\Delta(\varphi\vee\psi) by RE\Delta. Therefore \Delta(\varphi\vee\psi)\in s for every \psi. ∎
Lemma 14.
For all \varphi\in\mathcal{L}(\Delta), for all s\in S^{c}, we have (\mathcal{M}^{c})^{+},s\vDash\varphi\iff\varphi\in s, i.e. \varphi^{(\mathcal{M}^{c})^{+}}=\varphi.
With a routine work, we obtain
Theorem 15.
{\bf M^{\Delta}} is sound and strongly complete with respect to the class of frames satisfying (s).
We are now in a position to deal with the sound and strong completeness of {\bf R^{\Delta}}. First, the soundness follows from the following result.
Proposition 16.
\DeltaC is valid on the class of quasifilters.
Proof.
Let \mathcal{M}=\langle S,N,V\rangle be a (s)model and s\in S. Suppose that \mathcal{M},s\vDash\Delta\varphi\land\Delta\psi, then \varphi^{\mathcal{M}}\in N(s) or (\neg\varphi)^{\mathcal{M}}\in N(s), and \psi^{\mathcal{M}}\in N(s) or (\neg\psi)^{\mathcal{M}}\in N(s). Consider the following three cases:

\varphi^{\mathcal{M}}\in N(s) and \psi^{\mathcal{M}}\in N(s). By (i), we obtain \varphi^{\mathcal{M}}\cap\psi^{\mathcal{M}}\in N(s), i.e. (\varphi\land\psi)^{\mathcal{M}}\in N(s), which gives \mathcal{M},s\vDash\Delta(\varphi\land\psi).

(\neg\varphi)^{\mathcal{M}}\in N(s). By (s), we infer (\neg\varphi)^{\mathcal{M}}\cup(\neg\psi)^{\mathcal{M}}\in N(s), i.e. (\neg(\varphi\land\psi))^{\mathcal{M}}\in N(s), which implies \mathcal{M},s\vDash\Delta(\varphi\land\psi).

(\neg\psi)^{\mathcal{M}}\in N(s). Similar to the second case, we can derive that \mathcal{M},s\vDash\Delta(\varphi\land\psi).
∎
Proposition 17.
Let \mathcal{M}^{c} be a canonical model for {\bf R^{\Delta}}. Then \mathcal{M}^{c} possesses the property (i). As a corollary, (\mathcal{M}^{c})^{+} also possesses the property (i).
Proof.
Suppose X\in N^{c}(s) and Y\in N^{c}(s), to show that X\cap Y\in N^{c}(s). By supposition, there exist \varphi and \chi such that X=\varphi and Y=\chi, and then \Delta(\varphi\vee\psi)\in s for every \psi, and \Delta(\chi\vee\psi)\in s for every \psi. Using axiom \DeltaC, we infer \Delta((\varphi\land\chi)\vee\psi)\in s for every \psi. Therefore, \varphi\land\chi\in N^{c}(s), i.e. X\cap Y\in N^{c}(s). Then it follows that (\mathcal{M}^{c})^{+} also possesses the property (i) from Prop. 12. ∎
Theorem 18.
{\bf R^{\Delta}} is sound and strongly complete with respect to the class of quasifilters.
Proposition 19.
Let \mathcal{M}^{c} be a canonical model for {\bf(EMN)^{\Delta}}. Then \mathcal{M}^{c} possesses the property (n). As a corollary, (\mathcal{M}^{c})^{+} also possesses (n).
Theorem 20.
{\bf(EMN)^{\Delta}} is sound and strongly complete with respect to the class of frames satisfying (s)\&(n).
It is straightforward to obtain the following result.
Theorem 21.
{\bf K^{\Delta}} 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 S_{1} to another S_{2} means that S_{2} is deductively stronger than S_{1}.
\xymatrix@!0{&{\bf R^{\Delta}}{}{}{}{}{}{}{}{}{}{}\xy@@ix@{{\hbox{}}}} 