First-order intensional logic

Annals of Pure and Applied Logic 127 (1-3):171-193 (2004)
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 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
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DOI 10.1016/j.apal.2003.11.014
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Melvin Fitting (2012). Prefixed Tableaus and Nested Sequents. Annals of Pure and Applied Logic 163 (3):291 - 313.
James W. Garson (2005). Unifying Quantified Modal Logic. Journal of Philosophical Logic 34 (5/6):621 - 649.
Melvin Fitting (2006). FOIL Axiomatized. Studia Logica 84 (1):1 - 22.

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Chris Fox & Shalom Lappin (2004). An Expressive First-Order Logic with Flexible Typing for Natural Language Semantics. Logic Journal of the Interest Group in Pure and Applied Logics 12 (2):135--168.
Melvin Fitting (2006). FOIL Axiomatized. Studia Logica 84 (1):1 - 22.
Melvin Fitting (2012). Barcan Both Ways. Journal of Applied Non-Classical Logics 9 (2-3):329-344.

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