Logic is basically the study of valid reasoning. When searching the internet, we use Boolean logic terms, like "and" and "or", to help us find specific web pages that fit in the sets we are interested in. After exploring this form of logic, we will look at logical arguments and how we can determine the validity of a claim or argument.
We can often classify items as belonging to set, like what was covered in the previous lesson. If you went to the library to find a specific book and the librarian asked you to express your search criteria in terms of unions, intersections, and compliments of sets, you would probably look at the librarian like they were crazy. Instead, we typically use words like "and", "or", and "not" to connect our keywords together to form a search. These words, which form the basics of Boolean logic1, are directly related to our set operations that were learned in the previous chapter.
Boolean logic combines multiple statements that are either true or false into an expression that is either true or false.
First, let's look at what makes something a statement. Not all sentences are statements. A statement is a declarative sentence that can be objectively determined to be either true or false, but not both.
Determine if the following are statements or not. If they are statements, declare their truth value. If they are not statements, identify what they are.
Not all logic is Boolean logic. Boolean logic uses multiple conjoined statements to make up a logical thought. While sometimes statements are just simple statements, a simple statement is a statement that doesn't use any connectives. Connectives are words that join one or more sentences together to create a compound statement. Also, sometimes in Boolean logic, we also need to negate a statement. Negations typically use the word "not" and they change the truth value of the statement. It is important to note that a negation is not a connective. We will list it in the table below:
Words that are used to create compound statements:
| Name | Word | Symbol |
|---|---|---|
| Conjunction | "and" | $\wedge$ |
| Disjunction | "or" | $\vee$ |
| Conditional | "If...then..." | $\rightarrow$ |
| Biconditional | "...if and only if..." | $\leftrightarrow$ |
| Negation | "not" | $\sim$ |
Identify the following statements as either simple or compound. If the statement is compound, identify the connective.
Write the negation of the following statements.
In English we could also re-phrase this to say, "It will be cloudy today". While that isn't the same sentence, it has the opposite truth value of what was previously written.
It would sound funny to say "I will not not go to the movies this week." That would be a double negative, thus the two "not's" cancel each other out.
If we refer back to the definition of connectives, we can see the symbols that are given for each connective. It should be no surprise that in Mathematics, we'd rather use symbols than words. After all, Mathematicians prefer numbers and symbols to words.
Allow p = "I like chicken" and q = "I like potatoes." Write the verbal form of the symbolic logic below:
Translate each statement into symbolic notation. Allow p to represent "I like Pepsi" and let q represent "I like Coke".
Words that describe an entire set, such as "all", "every" or "none", are all called Universal Quantifiers because that set could be considered a Universal set. In contrast, words or phrases such as "some", "one", or "at least one", are called Existential Quantifier because they describe the existence of at least one element in a set.
A Universal Quantifier states that an entire set of things share a characteristic.
An Existential Quantifier states that a set contains at least one element.
Identify if the following are Universally or Existentially quantified:
However, when we negate a quantified statement, it isn't as simple as just adding in a "not" to the statement. In order to negate a quantified statement, you'd replace a Universal quantifier with an Existential quantifier and vice-versa.
Negate the quantified statements below: