LOGIC MAIN PAGE

LOGICAL OPERATORS

Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator

RULES OF LOGIC

Chain Rule
Contrapositive
DeMorgan's Law
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion
Simplification

VALIDITY PROOFS

2-step
3-step
4-step
5-step or more

Validity Proof Problems (3 steps)

The problems below can be solved in three steps.

Problem 1

1. C -> ~(D v E)
2. C
--------
* ~D

We have a conditional in premise 1 and its antecedent in premise 2. This immediately prompts us to use Modus Ponens.

3. ~(D v E)   1,2 Modus Ponens

Now we can apply DeMorgan's Law to this new expression.

4. ~D ^ ~E    3 DeMorgan's Law

We have to prove that ~D is true, and we see that ~D is part of the conjunction in step 4. We can apply Simplification to separate it out.

5. ~D         4 Simplification

We have gotten to the conclusion.

1. C -> ~(D v E)
2. C
*  ~D
--------
3. ~(D v E)   1,2 Modus Ponens
4. ~D ^ ~E    3 DeMorgan's Law
5. ~D         4 Simplification

Problem 2

1. A -> B
2. B XOR C
3. C ^ D
--------
* ~A

We can easily separate C out of premise 3 using Simplification. We could do so with D as well, but we are not going to need it anywhere.

4. C          3 Simplification

Premise 2 has an exclusive disjunction, and we have just proved that one of its disjuncts, namely C, is true. By using Mutual Exclusion, we can prove that B is therefore false.

5. ~B         2,4 Mutual Exclusion

Now let's look at premise 1. We can apply Modus Tollens here.

6. ~A         1,5 Modus Tollens

We have gotten to the conclusion.

1. A -> B
2. B XOR C
3. C ^ D
*  ~A
--------
4. C          3 Simplification
5. ~B         2,4 Mutual Exclusion
6. ~A         1,5 Modus Tollens