Definability of Functional Properties in the Basic Modal-Temporal Language over Ordered Frames

Abstract

We study the expressive power of the simplest modal-temporal language, obtained by adding Prior's temporal operators \(G\) and \(H\) to the basic modal language with \(\Box\). This language is the standard bimodal combination of modal and tense logic; under its functional interpretation it is denoted \(L_{T\times W}\). To analyse its definability across five order types, we consider two semantic readings of the temporal operators: the standard reading (\(G,H\)), which includes the current instant, and the strict reading (\(G^{\ast},H^{\ast}\)), which always excludes it. We examine nine functional properties -- totality, non-totality, injectivity, surjectivity, monotonicity, strict monotonicity, antitonicity, strict antitonicity, and constancy -- over preorders, strict preorders, partial orders, linear orders, and strict linear orders. Our analysis reveals two different levels of expressive power. In the original multiflow setting, the language is quite weak and the two readings coincide. When we restrict the semantics to minimal functional frames (the \(O^{2}\) family), many properties become definable, and the choice of reading becomes crucial: the strict reading can define properties such as injectivity even in reflexive orders. The same definability patterns appear with indexed languages and with the Uniform Domain condition on the semantics of \(L_{T\times W}\). That three such different ways of controlling functional multiplicity lead to identical definability patterns indicates that the expressive limitations of the original framework come from the uncontrolled multiplicity of functions, not from any weakness of the operators. Even after controlling functional multiplicity, a set of properties remains undefinable in all non-linear orders, showing that the lack of connectivity is a fundamental obstacle.