Propositional Logic Rules

Propositional logic rules are key tools for reasoning and drawing conclusions. They help us understand how statements relate to each other, making it easier to analyze arguments and solve problems in Formal Logic I. Here are the essential rules to know.

1. Modus Ponens

   • If "P implies Q" (P → Q) and "P" is true, then "Q" must also be true.
   • It is a fundamental rule of inference in propositional logic.
   • Often used in arguments to derive conclusions from given premises.

2. Modus Tollens

   • If "P implies Q" (P → Q) and "Q" is false, then "P" must also be false.
   • This rule allows for the negation of the antecedent based on the falsity of the consequent.
   • Essential for reasoning about implications and their contrapositive forms.

3. Disjunctive Syllogism

   • If "P or Q" (P ∨ Q) is true and "P" is false, then "Q" must be true.
   • It helps eliminate one possibility in a disjunction to affirm the other.
   • Useful in decision-making processes and logical deductions.

4. Conjunction

   • If "P" is true and "Q" is true, then "P and Q" (P ∧ Q) is true.
   • This rule combines two true statements into a single compound statement.
   • Important for constructing complex arguments from simpler propositions.

5. Simplification

   • If "P and Q" (P ∧ Q) is true, then "P" is true and "Q" is true individually.
   • It allows for the extraction of individual components from a conjunction.
   • Useful for breaking down complex statements into manageable parts.

6. Addition

   • If "P" is true, then "P or Q" (P ∨ Q) is also true for any proposition Q.
   • This rule allows for the introduction of new possibilities based on a true statement.
   • Important for expanding the scope of logical arguments.

7. Double Negation

   • If "not not P" (¬¬P) is equivalent to "P."
   • This principle asserts that negating a negation returns to the original proposition.
   • Fundamental in simplifying expressions and understanding logical equivalences.

8. Hypothetical Syllogism

   • If "P implies Q" (P → Q) and "Q implies R" (Q → R), then "P implies R" (P → R).
   • This rule allows for chaining implications to derive new conclusions.
   • Essential for constructing multi-step logical arguments.

9. Constructive Dilemma

   • If "P or Q" (P ∨ Q) is true, and both "P implies R" (P → R) and "Q implies S" (Q → S) are true, then "R or S" (R ∨ S) is true.
   • This rule combines disjunctions and implications to derive new conclusions.
   • Useful for evaluating multiple scenarios and their outcomes.

10. De Morgan's Laws

    • The laws state that "not (P and Q)" (¬(P ∧ Q)) is equivalent to "not P or not Q" (¬P ∨ ¬Q), and "not (P or Q)" (¬(P ∨ Q)) is equivalent to "not P and not Q" (¬P ∧ ¬Q).
    • These laws provide a way to transform negations of conjunctions and disjunctions.
    • Important for simplifying logical expressions and understanding the relationships between propositions.