Abstract
First-order modal logic is not just propositional modal logic plus classical quantifier machinery. The situation is much subtler than that. The addition of quantifiers to propositional modal logic may lead to many difficulties. In this paper we aim to solve one of them — the problem of rigidity versus non-rigidity for variables, that is, how to determine the denotations for each variable in different possible worlds or the connections among the denotations of each variable in different possible worlds. Since all the currently proposed semantics for first-order modal logic are not suitable to solve this problem, we proposed an ontology-based first-order modal semantics, in which ontologies are introduced to restrain the modal logic frames and models. An ontology-based counterpart relation S is introduced into each model. By requiring that the assignments of each variable in different possible worlds must accord with the relation S, we can correctly characterize the connections among the denotations of each variable in different worlds.
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