  In Section 3.2 a first order modal logic was sketched, in which quantification was over objects. Now a second kind of quantification is added, over intensions. As has been noted several times, an intensional object, or individual concept, will be modeled by a function from states to objects, but now we get into the question of what functions should be allowed. Intensions are supposed to be related to meanings. If we consider meaning to be a human construct, what constitutes an intension should probably be restricted. There should not, for instance, be more intensional objects than there are sentences that can specify meanings, and this limits intensions to a countable set. Or we might consider intensions as ‘being there,’ and we pick out the ones that we want to think about, in which case cardinality considerations don’t apply. This is an issue that probably cannot be settled once and for all. Instead, the semantics about to be presented allows for different choices in different models—it is not required that all functions from states to objects be present. It should be noted that, while this semantical gambit does have philosophical justification, it also makes an axiomatization possible. The fundamental point is the same as in the move from first to second order logic. If we insist that second order quantifiers range over all sets and relations, an axiomatization is not possible. If we use Henkin models, in which the range of second order quantifiers has more freedom, an axiomatization becomes available.