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Bimodal Logic with Contingency and Accident: Bisimulation and Axiomatizations

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Abstract

In this paper, a suitable notion of bisimulation is proposed for the bimodal logic with contingency and accident. We obtain several van Benthem Characterization Theorems, and axiomatize the bimodal logic over the class of Eulidean frames and over some more restricted classes, showing their strong completeness via a novel strategy, thereby answering two open questions raised in the literature. With the new bisimulation notion, we also correct an error in the expressivity results in the literature.

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Notes

  1. In fact, [2] gives a complete axiomatization for strong noncontingency logic \({\mathcal {L}}(\blacktriangle )\) over the class of Euclidean frames, thereby answering an open question posed in [11]. However, since as shown in [11], \({\mathcal {L}}(\blacktriangle )\) is equally expressive as accident logic over the class of arbitrary models, thus one can translate the axiomatization of Euclidean strong noncontingency logic into an axiomatization of Euclidean accident logic.

  2. It is worth remarking that the canonical relation is equivalent to the one given in the following way: \(sR^ct\) iff for all \(\varphi \), if \(\Delta (\varphi \vee \chi )\wedge \circ (\lnot \psi \rightarrow \varphi )\in s\) for all \(\psi \) and \(\chi \), then \(\varphi \in t\).

  3. Here what it means for a bisimulation notion to be tailored for a logic consists in two respects: firstly, bisimilarity implies logical equivalence; secondly, on a certain class of saturated models, logical equivalence implies bisimilarity.

  4. There was an error in the proof of [9, Prop. 7], where the same models were used to demonstrate that \(({\mathcal {M}},s)\) and \(({\mathcal {M}}',s')\) are \({\mathcal {L}}(\nabla ,\bullet )\)-logically equivalent, which are in fact not. We will give a correct answer in Proposition 3.4.

  5. Note that now we do not need a translation from \({\mathcal {L}}(\nabla )\) to \({\mathcal {L}}(\nabla ,\bullet )\), because all formulas in \({\mathcal {L}}(\nabla )\) are obviously also formulas in \({\mathcal {L}}(\nabla ,\bullet )\).

  6. The compactness of \({\mathcal {L}}(\nabla ,\bullet )\) follows from the strong completeness of the minimal logic of \({\mathcal {L}}(\nabla ,\bullet )\), for which we refer to [9].

  7. Here the standard translation \(ST_x\) of an \({\mathcal {L}}(\nabla ,\bullet )\)-formula, where x is a first-order variable, is defined as follows:

    $$\begin{aligned} \begin{array}{lll} ST_x(p)&{}=&{}Px\\ ST_x(\lnot \varphi )&{}=&{}\lnot ST_x(\varphi )\\ ST_x(\varphi \wedge \varphi )&{}=&{}ST_x(\varphi )\wedge ST_x(\psi )\\ ST_x(\nabla \varphi )&{}=&{}\exists y\exists z(xRy\wedge xRz\wedge ST_y(\varphi )\wedge \lnot ST_z(\varphi ))\\ ST_x(\bullet \varphi )&{}=&{}ST_x(\varphi )\wedge \exists y(xRy\wedge \lnot ST_y(\varphi ))\\ \end{array} \end{aligned}$$

    One may verify that every \({\mathcal {L}}(\nabla ,\bullet )\)-formula is equivalent to a first-order formula containing three variables, in contrast to only two variables in the case of standard modal logic \({\mathcal {L}}(\Diamond )\).

  8. An alternative proof method is using the notion of ‘mirror reduction’ proposed in [9, Footnote. 5]. In detail, we can define the notion of mirror reduction of a model as follows: let \({\mathcal {M}}=\langle S,R,V \rangle \) be a model. Its mirror reduction \({\mathcal {M}}^m=\langle S,R^m,V \rangle \), where \(R\backslash \{(x,x)\mid R(x)=\{x\}\}\subseteq R^m\subseteq R\). [9, Prop. 10] has shown that \({\mathcal {M}}^m,s\vDash \varphi \) iff \({\mathcal {M}},s\vDash \varphi \) for all \(\varphi \in {\mathcal {L}}(\nabla ,\bullet )\). Then this proposition follows from the fact that \({\mathcal {M}}^c\) is the mirror reduction of \({\mathcal {M}}^+\). Similar remarks apply to Proposition 6.14.

  9. \(\mathbf{K4^{\nabla \bullet }}\) is proposed in [9, Sec. 6.2], which is the smallest extension of \(\mathbf{K^{\nabla \bullet }}\) with the following axioms:

    $$\begin{aligned} \begin{array}{ll} \text {A4-1} &{} \circ \Delta \varphi \\ \text {A4-2} &{} \bullet \psi _1\wedge \Delta \varphi \wedge \circ (\lnot \psi _1\rightarrow \varphi )\rightarrow \circ \circ (\lnot \psi _2\rightarrow \varphi ).\\ \end{array}\end{aligned}$$

    It is unknown whether axiom A4-1 is dispensable without affecting the completeness.

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Acknowledgements

This research is supported by the Project 17CZX053 of National Social Science Fundation of China.

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  1. Institute of Philosophy, Chinese Academy of Sciences, Beijing, China

    Jie Fan

  2. School of Humanities, University of Chinese Academy of Sciences, Beijing, China

    Jie Fan

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Fan, J. Bimodal Logic with Contingency and Accident: Bisimulation and Axiomatizations. Log. Univers. 15, 123–147 (2021). https://doi.org/10.1007/s11787-021-00270-9

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