The connection between quantification and entities outside language, be they universals or particulars, consists in the fact that the truth or falsity of a quantified statement ordinarily depends in part on what we reckon into the range of entities appealed to by the phrases 'some entity x' and 'each entity x'(flpov103)

A valid term schema, then, is one that will come out true of all objects of any chosen universe under all interpretations, within that universe, of its term letters. Such schemata are counted valid when they come out true under all interpretations in all nonempty universes. It is convenient now to make this exception of the empty universe because there are among the Boolean statement schemata, unlike the term schemata, certain ones such as 'backwardsEF V backwardsEFbar' which fail for the empty universe but are valid and useful otherwise(mol115)

The idea of changeable universes goes back to DeMorgan (1846), as does the phrase 'universe of discourse'. Exclusion of the empty universe was implicit already in Gottlob Frege's logic of 1879(mol120)

If we think of the universe as limited to a finite set of objects we can expand existential quantifications into alternations and universal quantifications into conjunctions(mol140)

It thus appears that quantification could be dispensed with altogether in favor of truth functions if we were willing to agree for all purposes on a fixed and finite and listed universe. However, we are unwilling; it is convenient to allow for variations in the choice of universe. This is convenient not only because philosophers disagree regarding the limits of reality, but also because some logical arguments can be simplified by deliberately limiting the universe of discourse to animals or to persons for the space of the problem in hand. For most problems, moreover, the relevant universe comprises objects which we are in no position to list. In many problems, the universe even comprises infinitely many objects; e.g., the integers. Thus it is that quantification is here to stay(mol140)

What makes ontological questions meaningless when taken absolutely is not universality but circularity(or53)

One can effect the reduction of one ontology to another with help of a proxy function: a function mapping the one universe into part or all of another(or55)

There is more to be said of a theory, ontologically, than just saying what objects, if any, the theory requires; we can also ask what various universes would be severally sufficient. The specific objects required, if any, are the objects common to all those universes(or96)

The universe of discourse may conveniently be varied application to application; so formulas are counted valid only if true under all interpretations of 'F', 'G', etc., in all non-empty universes. The empty universe is profitably excepted because some formulas fail for it which hold generally elsewhere. The question whether a formula also holds for the empty universe is easily settled, when desired, by a separate test; for all Boolean equations hold true for the empty universe, and accordingly any truth function of them can be tested by 'evaluation'(slp41)

Interpretation of a formula consists in choosing a universe as range of values of 'x', 'y', etc., choosing truth values for 'p', 'q', etc., choosing specific objects of the universe for any free variables, and deciding what objects (or pairs, etc.) the predicates 'F', 'G', etc., are to be true of. A formula is valid if true under all interpretations with non-empty universes(slp43)

In the empty universe all universal quantifications are there true and all existential ones false(slp43)

We can blithely apply quantification theory without regard to the limits or extravagance of our universe of discourse, but we must take care that something exist in it. A basic technique in quantification theory is transformation of a formula in such a way as to bring all its quantifiers out to the beginning (prenexing) or, alternatively, to drive every quantifier in so that it governs only clauses in which its variable recurs(purifying). The transformations depend on eight familiar equivalences, called the rules of passage. Four of these eight fail for the empty universe(slp278)