Generalizations will be expressed symbolically using quantifiers, operations that take predicates as input and yield sentences as output. More specifically, we will use two universal quantifiers both written using the symbol ∀ (for all). The sentences form using these quantifiers will be called universals. The two quantifiers are the restricted universal quantifier, which applies to a pair of predicates to form a sentence, and the unrestricted universal quantifier, which applies to a single predicate. We will apply quantifiers only to abstracts. Since any pair of abstracts can be written in the form λx (...x...) and λx (---x---) using the same variable, we can abbreviate universal sentences as (∀x:...x...) ---x--- and ∀x ---x---. These may be put into English notation as Everything, x, such that ρx is such that θx and Everything, x, is such that θx. (Here the word thing is used as a dummy restriction that merely provides a hook for the relative clause.) The component expressions ...x... and ---x---, the restricting and quantified formulas of the universal, will not ordinarily be sentences in the strictest sense because they will contain free occurrences of the variable x. (The exceptions are the bodies of vacuous abstracts expressing predicates with a constant value.) Such expressions are included in the broader class of formulas, among which sentences are distinguished as closed formulas. Terms, too, can be classified as open or closed. A restricted universal says that the extension of the first predicate to which it is applied, the restricting predicate, is included in the extension of the second, the quantified predicate—i.e., it says that the second expresses a property that is at least as general as that expressed by the first. The unrestricted quantifier says that the quantified predicate to which it applies is universal, that it is a predicate that expresses a fully general property. An unrestricted universal sentence can be restated as a restricted universal whose domain predicate is universal, and a restricted universal can be restated as an unrestricted universal provided we make the attribute predicate conditional on the domain predicate.
An English generalization may be analyzed symbolically by using restricting and quantified predicates that capture its domain and attribute. If its domain consists of all reference values, an unrestricted universal may be used and we need only capture its attribute. In an affirmative generalization, the attribute predicate will be the quantified predicate of the English generalization while in a negative generalization it will be the negation of the quantified predicate. A formula applying the restricting predicate can be formed from the class indicator C by using the form x is a C, adding negation if the generalization is complementary. (However, we start with x is a person in the case of everyone and no one.) Bounds and exceptions may be captured by conjoining predicates of the same form, negated in the case of exceptions. The phrase all and only is used to express a conjunction of affirmative direct and negative complementary generalizations; but a generalization of this sort can be analyzed also by an unrestricted universal applying to a biconditional predicate because the two generalizations it implies can be expressed using an if-conditional and an only-if-conditional, respectively.