Concerning nonstandard quantifiers such as most, we have already sketched the generalized quantifier approach of Montague Grammar, and pointed out the alternative of using restricted quantifiers; an example might be (Most x: dog(x))friendly(x). Instead of viewing most as a second-order predicate, we can specify its semantics by analogy with classical quantifiers: The sample formula is true (under a given interpretation) just in case a majority of individuals satisfying dog(x) (when used as value of x) also satisfy friendly(x). Quantifying determiners such as few, many, much, almost all, etc., can be treated similarly, though ultimately the problem of vagueness needs to be addressed as well (which of course extends beyond quantifiers to predicates and indeed all aspects of a formal semantic representation).  Vague quantifiers, rather than setting rigid quantitative bounds, seem instead to convey probabilistic information, as if a somewhat unreliable measuring instrument had been applied in formulating the quantified claim, and the recipient of the information needs to take this unreliability into account in updating beliefs. Apart from their vagueness, the quantifiers under discussion are not first-order definable (e.g., Landman 1991), so that they cannot be completely axiomatized in FOL.  But this does not prevent practical  reasoning, either by direct use of such quantifiers in the logical representations of sentences (an approach in the spirit of natural logic), or by reducing them to set-theoretic or mereological relations within an FOL framework.