Introduction to Logic: Statements, Symbols, and Operators

For example, if we let: $P$: "Taylor Swift is at the football game" $Q$: "Taylor Swift is on tour"

We can now use mathematical symbols to build more complex ideas without rewriting the sentences every time.

Logical Operators and Compound Statements

When we combine two or more simple statements using logical connectors, we create compound statements. The truth value of the entire compound statement depends on the truth values of its individual parts and the specific operator used to join them.

1. Negation ($\sim$)

Negation is the simplest operator. It reverses the truth value of a statement. If a statement is true, its negation is false, and vice versa. In English, this is usually expressed by adding the word "not."

• Symbol: $\sim$ (a squiggly line).
• Example: If $P$ is "The dog is sleeping," then $\sim P$ is "The dog is not sleeping".

Note on Phrasing: When a negation symbol precedes parentheses, such as $\sim(P \land Q)$, it is translated into English as "It is false that...".

2. Conjunction ($\land$)

A conjunction joins two statements using the word "and".

• Symbol: $\land$.
• English Equivalents: In many contexts, words like "but," "however," or "nevertheless" function as conjunctions because they do not change the underlying truth value of the combined statements.
• Truth Rule: For a conjunction to be true, both individual statements must be true.

3. Disjunction ($\lor$)

A disjunction joins two statements using the word "or".

• Symbol: $\lor$ (resembling an upside-down 'V').
• Inclusive Or: In logic, we use the inclusive or. This means the compound statement is true if the first part is true, the second part is true, or both are true. This is similar to the concept of a "union" in set theory.

Advanced Connectors: Conditional and Biconditional

As we move into more complex reasoning, we encounter two specific types of conditional relationships.

The Conditional ($\to$)

A conditional statement is an "if-then" statement. It is symbolized by an arrow ($\to$).

• Structure: If $P$, then $Q$ (symbolized as $P \to Q$).
• Terminology: The part before the arrow ($P$) is called the antecedent, and the part after the arrow ($Q$) is called the consequent.

The Biconditional ($\leftrightarrow$)

A biconditional statement uses the phrase "if and only if" (sometimes abbreviated as "double iff").

• Symbol: $\leftrightarrow$ (a double-headed arrow).
• Structure: $P \leftrightarrow Q$.
• Truth Rule: A biconditional is true only when both statements have the same truth value (both are true or both are false).

Organizing Complex Statements: Parentheses and Commas

When a compound statement contains multiple connectors, we must use punctuation and grouping symbols to ensure the logic is interpreted correctly.

• Parentheses: Just like in algebra, parentheses tell us which statements to group together first.
• Commas: A comma is often used to indicate where a grouping should occur, especially when there are multiple connectors. For example, in a sentence with three statements, a comma helps clarify whether the first two are grouped together or the last two are.

Pro-Tip: When translating a sentence with a comma into symbolic form, look at which statements are on the same side of the comma. Those statements should be placed inside parentheses in your symbolic expression.

Summary Table of Logical Operators