2.1: Quantified Statements
Read Section 3 through 3.1 on pages 11 and 12. This reading introduces the universal quantifier and the existential quantifier. Logic without quantifiers is called propositional calculus; logic with quantifiers is called the (first order) predicate calculus.
Read Section 2: "Predicate Logic" up to and including Definition 4 on pages Lo-12 and Lo-13. As you read this text, consider that statements in the first order predicate calculus, or for us, simply, the predicate calculus, involve variables that can take on values from a set in a reference domain. We interpret the statement by introducing a domain of discourse or reference domain that the symbols (statements and operators), the rules, and variables represent or refer to. This is essentially what we do when we translate from English to logic. In other words, translation is using one domain, e.g. logic, to represent another domain, and setting up an association between symbols in one to those of the other. Just keep in mind, that the variables in a predicate calculus statement take on the values from a particular set, for example, the set of all boys in Chicago or the set of positive integers.
For an understanding of the universal quantifier, study Definition 4, on page Lo-12, which is critically important for the study of logic, science and mathematics. This definition also defines the existential quantifier. These two quantifiers are often used together, as the examples in the next subunits will show.
Read Sections 3.2 on page 12 and 3.6 on pages 14 and 15. This reading also pertains to the topic in subunit 2.1.2. This reading shows how logic (a formal language) can be used to describe sets (another formal language).
Read Definition 5 up to and including example 8 on pages Lo-13 and Lo-14. This reading also applies to the topic for Subunit 2.1.2 of this course. This reading gives important examples of using logic to represent statements in mathematics. As you read this text, please keep in mind that formal language includes logic, binary functions, sets, and programming languages. Informal language includes English and other natural languages.
In our study of logic, our primary interest is translating between logic and English (or natural languages). In addition, you will find it useful to translate between informal (that is, natural) languages. For example, suppose you have an English statement that you find difficult to translate to logic. Rewriting or translating the statement to an equivalent English statement usually makes the translation to logic easy. Translating between informal or natural languages also occurs when we translate from one natural language to another, for example, from English to Spanish.