Abstract
In 1951 in his book An Essay in Modal Logic, Georg Henrik von Wright strongly called attention to the analogies between quantifiers and modal operators. In 1984 I published a paper in Synthese examining the analogy formally. Confession: the presentation in that paper was badly done, and there is a significant (though correctable) error. Its time to repair the damage, present the ideas in a better way, and continue the investigation further.
There are natural sublogics of classical first-order logic that are direct analogs of standard, basic modal logics. The behavior of quantifiers can be given a possible world semantics, some analogous to normal models, some to regular models, and some to neighborhood models. The first-order logics have axiom systems and generally also tableau systems, paralleling those of modal logics. Many have the interpolation property. This gives concrete substance to von Wright’s observations. But then, what is the crucial difference between modal operators and quantifiers? This turns out to be surprising in its simplicity, and leads to an interesting way of looking at the familiar Henkin style completeness proof for first-order logic.