Rules of Inference

Rules of Inference are key tools in Formal Logic II, helping us draw valid conclusions from given statements. These rules, like Modus Ponens and Modus Tollens, guide logical reasoning and strengthen our ability to analyze arguments effectively.

1. Modus Ponens

   • If P implies Q, and P is true, then Q must also be true.
   • Form: If P → Q, P, therefore Q.
   • Essential for establishing direct conclusions from conditional statements.

2. Modus Tollens

   • If P implies Q, and Q is false, then P must also be false.
   • Form: If P → Q, ¬Q, therefore ¬P.
   • Useful for disproving hypotheses based on the failure of their consequences.

3. Hypothetical Syllogism

   • If P implies Q and Q implies R, then P implies R.
   • Form: If P → Q, Q → R, therefore P → R.
   • Allows chaining of implications to derive new conclusions.

4. Disjunctive Syllogism

   • If either P or Q is true, and P is false, then Q must be true.
   • Form: P ∨ Q, ¬P, therefore Q.
   • Effective for eliminating possibilities to reach a conclusion.

5. Conjunction

   • If both P and Q are true, then the conjunction P and Q is also true.
   • Form: P, Q, therefore P ∧ Q.
   • Fundamental for combining statements to form a compound statement.

6. Simplification

   • If P and Q are both true, then P is true.
   • Form: P ∧ Q, therefore P.
   • Useful for extracting individual components from a conjunction.

7. Addition

   • If P is true, then the disjunction P or Q is also true.
   • Form: P, therefore P ∨ Q.
   • Allows for the introduction of additional possibilities without negating the original statement.

8. Constructive Dilemma

   • If P implies Q and R implies S, and either P or R is true, then either Q or S must be true.
   • Form: (P → Q) ∧ (R → S), P ∨ R, therefore Q ∨ S.
   • Facilitates decision-making between two conditional scenarios.

9. Destructive Dilemma

   • If P implies Q and R implies S, and both Q and S are false, then both P and R must be false.
   • Form: (P → Q) ∧ (R → S), ¬Q ∧ ¬S, therefore ¬P ∧ ¬R.
   • Useful for eliminating options based on the failure of their consequences.

10. Resolution

    • If P or Q is true, and ¬P is true, then Q must be true.
    • Form: P ∨ Q, ¬P, therefore Q.
    • A powerful method for deriving conclusions from conflicting statements.