First-order intensional logic

First-order modal logic is very much under current development, with many different semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently several semantics based on counterparts have been examined, in a development that goes back to David Lewis. There is yet another line of research, using intensional objects, that traces back to Richard Montague. I have been involved with this line of development for some time. In the present paper I briefly sketch several of the approaches to first-order modal logic. Then I present one that I call FOIL (for first-order intensional logic) in the Montague tradition that, I believe, is both expressive and natural. I briefly discuss in what sense it can be made to encompass the other approaches. Finally I provide tableau rules to go with the FOIL semantics.