### Unit 2: The Logic of Quantified Statements

In the previous unit, we discussed the logical analysis of compound statements, including those comprised of simple statements joined by OR, AND, NOT, etc. operators. This analysis provides us with a better understanding of human reasoning, but cannot be used to determine the validity of the majority of mathematical situations. In some cases, it becomes necessary to separate statements into parts, just as we parse sentences in order to facilitate comprehension.

In this unit, we will learn to analyze and understand the special roles that keywords and predicates (like "for all" and "some") play − exercises that constitute the foundation of predicate calculus.

**Completing this unit should take you approximately 8 hours.**

Upon successful completion of this unit, you will be able to:

- form predicate calculus statements using universal and existential quantifiers;
- translate to and from English to predicate calculus statements;
- apply AND, OR, NOT, => to statements containing universal and existential quantifiers;
- apply predicate calculus rules to transform statements that contain both universal and existential quantifiers; and
- prove predicate calculus statements using logic rules of inference.

### 2.1: Quantified Statements

Read Section 3 through 3.1 on pages 11 and 12. This section 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, like 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.

Read Sections 3.2 on page 12 and 3.6 on pages 14 and 15. This section 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 section gives important examples of using logic to represent statements in mathematics. As you read this text, 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.

### 2.2: Operating on Quantified Statements

Read Sections 3.3 and 3.4 on page 13. We have seen the use of quantifiers in representing statements in languages like English, mathematics, and programming. The next step is to see how statements with quantifiers can be combined and transformed using logic rules. This section looks at statements that have both types of quantifiers (universal and existential) used together. It also looks at the order in which the quantifiers appear.

Read example 15 on page Lo-19. Here you read about the use of quantifiers together with the use of logic operations, such as AND, OR.

As you study and read, you should THINK about the material, both on what it means in relation to what you already know and how it relates to other topics you have studied. To help you think about the material, you should look at the exercises, even if the instructions don't explicitly state this.

Read Section 3.5 on page 14 on the use of quantifiers with negation.

Read example 9 and example 10 on pages Lo-14 and Lo-15. Here is where you will have to apply some of your self-learning skills: you should review what a contrapositive, converse, and inverse are, and use what you have learned about how quantifiers and negation affect one another.

Given a statement in logic, as you have seen with the propositional logic, we can negate it. We can do the same for a statement in predicate logic. Then, once we negate it, how can we rewrite the negated statement as a logically equivalent statement, so that the negation applies to the parts of the statement, rather than the entire statement? Why do we care? Because by doing so, we often simplify the statement or put it into a more convenient form for a particular purpose. In effect, we can study ways to translate from logic to logic in order to obtain a statement that is more convenient for what we might need.

### 2.3: Statements Containing Multiple Quantifiers

Read from example 11 up to Exercises for Section 2 on pages Lo-16 to Lo-19. The examples work with multiple quantifiers and pertain to translation between logic and math and English domains.

The direction of the translation from formal to informal language is typically done to communicate a result obtained from logic, back to the language in which the problem was specified.

Translation from a natural language to logic, is done to apply logic to a problem in some other discipline or everyday task (like history, science, making a decision, or debating). Furthermore, in applying logic, we also find it useful to translate from logic to logic (to transform a statement into a more convenient form), and to translate from a natural language to a natural language to simplify translation to logic.

Read this text to better understand the definition of a limit and primarily to illustrate the use of the notation from this unit.

Limits are studied in continuous mathematics, such as the differential and integral calculus, and in analysis. Discrete mathematics (which is studied in discrete structures) provides the concepts that are used in defining topics in continuous mathematics.

### Unit 2 Assessment

Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You can take this assessment as many times as you want, whenever you want.

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