PHI 2100: Introduction to Formal Logic

Rules of Inference and Replacement

1. The first 8 of the rules below are rules of inference, i.e., they "work" in "one direction only." They are not, in other words, equivalence rules. All of them except "addition" and "simplification" must utilize more than one line.

2. The second set of rules are rules of replacement, i.e., they are equivalence rules. They apply to an entire line or only to a part of any individual line of a proof.

Rules of Inference

1. Modus Ponens (MP) p --> q

p/ˆq

2. Modus Tollens (MT) p --> q

-q/ˆˆp

3. Hypothetical Syllogism (HS) p --> q

q --> r/ˆp --> r

4. Disjunctive Syllogism (DS) p v q

-p/ˆq and p v q

-q/ˆp

5. Constructive Dilemma (CD) (p --> q) @ (r --> s)

p v r/ˆq v s

6. Conjunction (Conj) p

q/ˆp @ q

7. Simplification (Simp) p @ q/ˆp and p @ q/ˆq

8. Addition (Add) p/ˆp v q

Rules of Replacement (Equivalence Rules)

9. DeMorgan's Theorems (DeM) -(p @ q) <--> (-p v -q)

-(p v q) <--> (-p @ -q)

10. Distribution (Dist) [p v (q @ r)] <--> [(p v q) @ (p v r)]

[p @ (q v r)] <--> [(p @ q) v (p @ r)]

11. Exportation (Exp) [p --> (q --> r)] <--> [(p @ q) --> r]

12. Contraposition (Contra) (p --> q) <--> (-q --> -r)

13. Commutation (Com) (p v q) <--> (q v p) and

(p @ q) <--> (q @ p)

14. Tautology (Taut) p <--> (p @ p) and

p <--> (p v p)

15. Equivalence (Equiv) (p <--> q) <--> [(p --> q) @ (q --> p)]

(p <--> q) <--> [(p @ q) v (-p @ -q)]

16. Material Implication (Impl) (p --> q) <--> (-p v q)

17. Association (Assoc) [p v (q v r)] <--> [(p v q) v r]

[p @ (q @ r)] <--> [(p @ q) @ r]

18. Double Negation (DN) p <--> --p