Logical connectives are the building blocks of propositional logic, allowing us to combine simple statements into complex ones. They include negation, conjunction, disjunction, conditional, and biconditional, each with its own symbol and rules for determining truth values.

Truth tables are a key tool for understanding how logical connectives work. They show all possible combinations of truth values for propositions and help us determine the truth value of compound statements, making complex logical relationships easier to grasp.

Logical Connectives

Negation and Conjunction

• Negation represented by the symbol $\neg$ or $\sim$
  • Negates or reverses the truth value of a proposition
  • Example: If $P$ is true, then $\neg P$ is false
• Conjunction represented by the symbol $\wedge$ or $&$
  • Combines two propositions with "and"
  • The conjunction is true only when both propositions are true
  • Example: $P \wedge Q$ is true if and only if both $P$ and $Q$ are true

Disjunction, Conditional, and Biconditional

• Disjunction represented by the symbol $\vee$
  • Combines two propositions with "or"
  • The disjunction is true when at least one of the propositions is true
  • Example: $P \vee Q$ is true if $P$ is true, $Q$ is true, or both are true
• Conditional represented by the symbol $\rightarrow$ or $\supset$
  • Represents an "if-then" statement
  • The conditional is false only when the antecedent (left side) is true and the consequent (right side) is false
  • Example: $P \rightarrow Q$ is read as "if $P$, then $Q$"
• Biconditional represented by the symbol $\leftrightarrow$ or $\equiv$
  • Represents an "if and only if" statement
  • The biconditional is true when both propositions have the same truth value (both true or both false)
  • Example: $P \leftrightarrow Q$ is true if $P$ and $Q$ are both true or both false

Truth and Logical Connectives

Truth-Functional Connectives

• Truth-functional connective a logical operator whose truth value is determined solely by the truth values of its component propositions
• The truth value of a compound proposition depends only on the truth values of its constituent propositions
• All the logical connectives (negation, conjunction, disjunction, conditional, biconditional) are truth-functional
• Example: The truth value of $P \wedge Q$ depends only on the truth values of $P$ and $Q$

Truth Tables

• Truth table a table that shows all possible combinations of truth values for a set of propositions and the resulting truth value of a compound proposition
• Used to determine the truth value of a compound proposition for all possible truth value assignments of its component propositions
• Each row in a truth table represents a unique combination of truth values for the propositions
• The number of rows in a truth table is $2^n$, where $n$ is the number of distinct propositions
• Example: The truth table for $P \wedge Q$ has 4 rows ($2^2$) and shows that the conjunction is true only when both $P$ and $Q$ are true