We establish completeness and the finite model property for logics featuring the pooling modalities that were introduced in Van De Putte and Klein (Pooling modalities and pointwise intersection: semantics, expressivity, and applications). The definition of our canonical models combines standard techniques with a so-called “puzzle piece construction”, which we first illustrate informally. After that, we apply it to the weakest classical logics with pooling modalities and investigate the technique’s potential for the axiomatization of stronger logics, obtained by imposing well-known frame conditions on the models.
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Open Access funding provided by Projekt DEAL. We are indebted to Eric Pacuit and Fengkui Ju for insightful discussions on the completeness proofs. We are also indebted to two anonymous referees for their incisive comments on an earlier version. Frederik Van De Putte’s work on this paper was supported by the European Comission through a Marie Skłodowska-Curie Fellowship (Grant Agreement ID: 795329) and by the Flemish Research Foundation (FWO-Vlaanderen). The work of Dominik Klein was partially supported by the Deutsche Forschungsgemeinschaft (DFG) and Agence Nationale de la Recherche (ANR) as part of the joint project Collective Attitude Formation [RO 4548/8-1], by DFG and Grantová Agentura České Republiky (GAČR) through the joint project From Shared Evidence to Group Attitudes [RO 4548/6-1] and by the National Science Foundation of China as part of the project Logics of Information Flow in Social Networks [17ZDA026].
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Van De Putte, F., Klein, D. Pooling Modalities and Pointwise Intersection: Axiomatization and Decidability. Stud Logica 109, 47–93 (2021). https://doi.org/10.1007/s11225-020-09901-6
- Pointwise intersection
- Pooling modalities
- Classical modal logics
- Finite model property
- Puzzle piece construction