Propositional Logic | Discrete Mathematics | 002 |

Propositional Logic

Propositional logic, also known as propositional calculus or sentential logic, is a branch of logic that deals with the study of propositions and their logical relationships. It focuses on the manipulation and evaluation of logical statements without considering their internal structure or meaning.

In propositional logic, a proposition is a declarative statement that is either true or false, but not both. Propositions are represented by symbols, often using capital letters such as P, Q, or R. Logical operators are used to combine propositions and form more complex statements. Here are some key concepts and operators in propositional logic:

1. Logical Operators:

• Negation (¬): The negation of a proposition negates its truth value. For example, if P is true, then ¬P is false.
• Conjunction (∧): The conjunction of two propositions is true only when both propositions are true. For example, if P is true and Q is true, then P ∧ Q is true.
• Disjunction (∨): The disjunction of two propositions is true if at least one of the propositions is true. For example, if P is true and Q is false, then P ∨ Q is true.
• Implication (→): The implication connects two propositions and asserts that if the first proposition (antecedent) is true, then the second proposition (consequent) must also be true. For example, if P is true and Q is false, then P → Q is false. However, if P is false, the implication is considered true regardless of the truth value of Q.
• Biconditional (↔): The biconditional, also known as the if and only if (iff) statement, asserts that two propositions have the same truth value. It is true when both propositions have the same truth value and false otherwise. For example, if P is true and Q is false, then P ↔ Q is false.
1. Truth Tables: Truth tables are used to systematically list all possible combinations of truth values for propositions and determine the truth value of complex statements. Each row of the truth table represents a combination of truth values, and the final column indicates the truth value of the compound statement.
2. Logical Equivalences: Logical equivalences are statements that assert that two propositions are logically equivalent, meaning they have the same truth value under all circumstances. Examples of logical equivalences include De Morgan's laws, double negation, and the distributive laws.
3. Validity and Satisfiability: In propositional logic, an argument is valid if the conclusion is true whenever all the premises are true. A proposition is satisfiable if there exists at least one assignment of truth values to its variables that makes the proposition true.

Propositional logic provides a formal framework for reasoning and analyzing the validity of arguments. It serves as the foundation for more advanced branches of logic and has applications in various fields, including computer science, mathematics, philosophy, and artificial intelligence.