Logical Connectives in Propositional Logic: AND, OR, NOT, and More

Explore logical connectives in propositional logic—symbols and words that combine propositions to create compound statements. This guide defines key connectives (negation, conjunction, disjunction, implication, biconditional), explains their truth tables, and illustrates their use in building logical expressions.

Logical Connectives in Propositional Logic

Understanding Propositions

Before diving into logical connectives, let's review propositions. A proposition is a statement that's either true or false (but not both). We use these building blocks to construct more complex logical statements.

Logical Connectives: Joining Propositions

Logical connectives are symbols or words that combine propositions to create compound statements. These connectives define how the truth value of the compound statement depends on the truth values of its constituent propositions.

The Five Basic Logical Connectives

Connective Name Connective Word(s) Symbol
Negation not ¬, ∼, ', -
Conjunction and ∧
Disjunction or ∨
Conditional (Implication) if...then; implies →
Biconditional (Equivalence) if and only if ↔

Detailed Explanation of Each Connective

Negation (¬)

The negation of a proposition p (¬p) is true when p is false and false when p is true.

p ¬p
True False
False True

Conjunction (∧)

The conjunction of propositions p and q (p ∧ q) is true only when both p and q are true.

p q p ∧ q
True True True
True False False
False True False
False False False

Disjunction (∨)

The disjunction of propositions p and q (p ∨ q) is true when at least one of p or q is true.

p q p ∨ q
True True True
True False True
False True True
False False False

Conditional (→)

The conditional statement "if p, then q" (p → q) is false only when p is true and q is false.

p q p → q
True True True
True False False
False True True
False False True

Biconditional (↔)

The biconditional statement "p if and only if q" (p ↔ q) is true when p and q have the same truth value (both true or both false).

p q p ↔ q
True True True
True False False
False True False
False False True

Important Notes on Logical Connectives

  • Digital Electronics Equivalents: Negation corresponds to NOT, conjunction to AND, disjunction to OR, and biconditional to XNOR.
  • Order of Operations: Logical connectives have an order of operations (precedence) that needs to be followed when evaluating complex expressions.
  • Commutative and Associative Properties: Negation, conjunction, disjunction, and biconditional operations are both commutative and associative. The conditional operation is neither commutative nor associative.

Conclusion

Logical connectives are the glue that holds together propositions, allowing us to build complex logical statements. Understanding their meanings and how they affect truth values is essential for logical reasoning.