Predicate Logic

From Wolf:

1. A mathematical variable is a symbol (or combination of symbols like ) that stands for an unspecified number or other object. The collections of objects from which any particular variable can take its values is called the domain or universe of that variable. Variables with the same domain are said to be of the same sort.
2. Two symbols, called quantifiers, stand for the following words:
   1. for ``for all'' or ``for every'' or ``for any''. Universal quantifier.
   2. for ``there exists'' or ``there is'' or ``for some''. Existential quantifier.
3. The quantifiers are used in symbolic mathematical language as follows: If P is any statement, and x is any mathematical variable (not necessarily a real number variable), then and are also statements. In this case we say that x is quantified.
4. A mathematical variable occurring in a symbolic statement is called free if it is unquantified and bound if it is quantified. If a statement has no free variables it's called closed. Otherwise it's called a predicate, an open sentence, an open statement, or a propositional function.
5. A statement of the form is defined to be true provided P(x) is true for each particular value of x from its domain. Similarly, is defined to be true provided P(x) is true for at least one value of x from that domain.
6. For any statement P(x), is logically equivalent to . Also, is logically equivalent to .
7. Let P be any statement, x be any mathematical variable, and t any term that denotes a set. (So t could be a single letter standing for a set, or a more complicated expression like .) Then
   1. is an abbreviation for .
   2. is an abbreviation for .