Truth tables are essential tools in logic, helping us analyze and understand compound propositions. They show all possible combinations of truth values for simple propositions, allowing us to evaluate complex statements systematically.

Constructing truth tables involves determining the number of simple propositions, calculating rows, and filling in truth values. By following these steps, we can evaluate compound propositions and gain insights into their logical structure and meaning.

Truth Tables

Components of a Truth Table

• Simple proposition represents a declarative sentence that can be either true or false
• Truth table row shows all possible combinations of truth values for the simple propositions in a compound proposition
  • Each row represents a distinct truth value assignment (a set of truth values assigned to each simple proposition)
• Truth table column lists the truth values for a specific simple proposition or the compound proposition across all possible truth value assignments
  • One column for each simple proposition and one for the compound proposition
• Truth value assignment specifies the truth value (true or false) for each simple proposition in a given row of the truth table
  • For example, in a truth table with two simple propositions $p$ and $q$, one possible truth value assignment is $p = \text{true}$ and $q = \text{false}$

Constructing Truth Tables

• Determine the number of simple propositions in the compound proposition
  • Each simple proposition will have its own column in the truth table
• Calculate the number of rows needed in the truth table using the formula $2^n$, where $n$ is the number of simple propositions
  • For example, a compound proposition with 3 simple propositions will have $2^3 = 8$ rows in its truth table
• Fill in the truth values for each simple proposition column, alternating between true and false
  • For the leftmost column, alternate every row (true, false, true, false, ...)
  • For the next column, alternate every two rows (true, true, false, false, ...)
  • Continue this pattern, doubling the number of rows before alternating for each subsequent column
• Evaluate the truth value of the compound proposition for each row using the truth values assigned to the simple propositions and the logical connectives

Logical Connectives

Types of Logical Connectives

• Logical connective is a symbol or word used to combine simple propositions into compound propositions
  • Common logical connectives include negation ($\neg$), conjunction ($\wedge$), disjunction ($\vee$), conditional ($\rightarrow$), and biconditional ($\leftrightarrow$)
• Truth-functional analysis determines the truth value of a compound proposition based on the truth values of its constituent simple propositions and the logical connectives used
  • Each logical connective has a specific truth function that determines the truth value of the compound proposition
  • For example, a conjunction ($p \wedge q$) is true only when both $p$ and $q$ are true, and false in all other cases

Evaluating Compound Propositions

• Identify the main logical connective in the compound proposition
  • If there are multiple connectives, use parentheses to determine the order of operations
• Evaluate the truth value of the compound proposition for each row of the truth table using the truth function of the main logical connective
  • For negation ($\neg p$), the truth value is the opposite of the truth value of $p$
  • For conjunction ($p \wedge q$), the compound proposition is true only when both $p$ and $q$ are true
  • For disjunction ($p \vee q$), the compound proposition is true when at least one of $p$ or $q$ is true
  • For conditional ($p \rightarrow q$), the compound proposition is false only when $p$ is true and $q$ is false
  • For biconditional ($p \leftrightarrow q$), the compound proposition is true when $p$ and $q$ have the same truth value