

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 (4 steps) The problems below can be solved in four steps.
Problem 1 1. ~A > D At first nothing seems to work, but if we look carefully at premise 1 and apply Contrapositive to it, we'll get something resembling the answer. Let's try it. 4. ~D > A 1 Contrapositive Immediately after this we can connect premises 4 and 3 using the Chain Rule. 5. ~D > B 4,3 Chain Rule We're getting closer, but we're not there yet. We still need to somehow connect ~D with ~C. We've just proved ~D > B, and premise 2 connects C and B. Let's explore that. If we apply Contrapositive here, we then might connect premise 5 with the result. 6. B > ~C 2 Contrapositive It's working! Now all we need is to use the Chain Rule on premises 5 and 6. 7. ~D > ~C 5,6 Chain Rule We've solved the problem. 1. ~A > D
Problem 2 1. W ^ X The first thing we can do immediately is apply Simplification to premise 1. 3. W 1
Simplification Looking at premise 2, we realize that its first part is false since we have just proven X. Therefore, by Disjunctive Inference, we conclude that Y is true. 5. Y 2,4 Disjunctive Inference We know that W is true (step 3), and we have just found out that Y is true as well. We can connect them by Conjunctive Addition. 6. Y ^ W 5,3 Conjunctive Addition And that's our conclusion. 1. W ^ X
