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On the modal interpretation of the connective of realisation

The connective of realisation associates propositions with names of contexts, at which they are said to be realised. Realisation is usually understood as relativised truth-connective, thus under most accounts it distributes over all Boolean connectives. Nonetheless, there are good reasons to consider weaker kinds of realisation, which lack some distributive laws. Three kinds of semantics for such weak realisation were provided: many-valued, set-theoretic and relational. The paper addresses the problem of Jarmużek's interpretation of the connective of realisation in relational models. It is argued that Jarmużek truth condition for the connective of realisation is tantamount to imposing symmetry on the relation of accessibility in a model. Moreover, a simplified semantics as well as axiom system for Jarmużek system are presented. Finally, a unification with Tkaczyk's – most general -- results is provided by proving completeness result for the axiom system with respect to Tkaczyk-style set-theoretic semantics.