

Sample sentences AND operator IF/THEN operator NOT operator OR operator XOR operator Chain Rule Conjunctive Addition Contrapositive DeMorgan's Law Disjunctive Addition Disjunctive Inference Disjunctive Infer. (XOR) Double Negation Modus Ponens Modus Tollens Mutual Exclusion Simplification 2step 3step 4step 5step or more Bad Argument 
Validity Proof Problems (5 steps or more) The problems below can be solved in five or more steps.
Problem 1 1. ~X > ~Y Looking at premise 2, we realize that we can apply Simplification. Here is what happens: 3. Y 2
Simplification The conclusion has Z and X, and we have Z already. How about getting X out of premise 1? If we look carefully at the consequent of the condition in premise 1, and then at step 3, we realize that we can use Modus Tollens here. 5. ~(~X) 1,3 Modus Tollens This looks very much like Double Negation. We apply it here. 6. X 5 Double Negation We finally got X, but the conclusion requires X and Z together. We can join them using Conjunctive Addition. 7. X ^ Z 6,4 Conjunctive Addition We are done. 1. ~X > ~Y
Problem 2 1. (W ^ N) > M Premise 2 looks rather complicated, so we can apply DeMorgan's Law here to simplify it. 3. N ^ ~M 2 DeMorgan's Law Now we can use Simplification to separate out both conjuncts from the result. 4. N 3
Simplification In step 5, we have the opposite of the consequent in premise 1. We can apply Modus Tollens to it. 6. ~(W ^ N) 1,5 Modus Tollens Again, this is a good candidate for DeMorgan's Law. 7. ~W v ~N 6 DeMorgan's Law In step 4, we proved that N is true, so we can use Disjunctive Inference as our last step: 8. ~W 7,4 Disjunctive Inference We are done. 1. (W ^ N) > M
