A central motivation for modern logic is that it provides a tool to represent natural language meaning and inference. One of the motivations of first-order logic is precisely that one can translate significant aspects of language into it, and in addition one can use it in connection with the foundations of mathematics. This talk returns to the topic of language and logic, and proposes new logical systems for the area. One leading idea is to propose logics with a decidable validity problem, ruling out full first-order logic. Indeed, we are interested in finding decidable fragments of language, just as others have asked for decidable fragments of first-order logic. We also axiomatize the logics, just to see what they look like. The overall topic of this research could be interesting to those pursuing natural language semantics and also to people in computational linguistics who work on inference. Indeed, I hope to report on how the ideas in the talk are (or are not) useful to those in the field of 'textual inference'. The talk mentions a number of technical results, and they are closest to algebraic logic, model theory and descriptive complexity theory.