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

The problems below contain a bad argument. That means that the conclusion given in the problem is wrong, and we have to prove it.

Problem 1

1. E -> F
2. ~F
---------
* E

Looking at premises 1 and 2, we realize that we can apply Modus Tollens here.

3. ~E          1,2 Modus Tollens

The result, however, contradicts the conclusion (E). The conclusion is invalid; therefore, we have an invalid argument.

1. E -> F
2. ~F
* E
---------
3. ~E          1,2 Modus Tollens

Problem 2

1. G -> H
2. J ^ G
3. L -> K
4. H -> ~K
----------
* L

If we look at premises 1 and 4, we see that the Chain Rule can be applied right away.

5. G -> ~K    1,4 Chain Rule

We know that we eventually have to get to L. There is only one premise containing L: it's premise 3. If we apply Contrapositive to it, we bring ~K to the front, in which case we can use it later together with the result of step 5.

6. ~K -> ~L   3 Contrapositive

Now we can apply the Chain Rule again.

7. G -> ~L    5,6 Chain Rule

In this conditional, ~L is true when G is true. We can extract G from a conjunctive in premise 2 using Simplification:

8. G          3 Simplification

Finally we prove that ~L is true, L is false, and the argument is invalid by Modus Ponens.

9. ~L         8,9 Modus Ponens

Let's retrace our step.

1. G -> H
2. J ^ G
3. L -> K
4. H -> ~K
*  L
----------
5. G -> ~K    1,4 Chain Rule
6. ~K -> ~L   3 Contrapositive
7. G -> ~L    5,6 Chain Rule
8. G          2 Simplification
9. ~L         8,9 Modus Ponens