Step 3 Tutorial
Keep in mind that this tutorial will be different than the proceeding tutorials. For step 3 there are many possible different proofs for step 3 problems. So, this tutorial will guide you to see just one possible proof.
In working on step 3 proofs, most important is to do as many proof steps as possible and make sure that each one is a correct application of the nine rules.
Suppose we had the proof below to work on. We know what we need to do. We have to find a valid logical trail of reasoning from a line 6 on until we reach the conclusion. How do we even start? If you have prepared well with steps 1 and 2, you should see some fairly easy steps to do. Note the highlighting.
1. ~D ⊃ M ------ Hint
2. ~D
3. E • I ------------ Hint
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. ??
So suppose we did both of the suggested hint steps above. We would have:
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. ??
Remember that we get to create lines 6 and 7 because they are conclusions of premises we have by rules that are valid.
But what next? Once we get started, we look at the lines we have produced and see if one or both work with any of the premises above. Line 7 does not look like it works with the premises above. But how about line 6?
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z) ----- Hint
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp. -------- Hint
7. M (1)(2) MP
8. ??
So suppose we applied the suggeted hint to create line 8.
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP
9. ??
Now what? Can we now use line 8? Yes, notice line 5?
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP ---- Hint
9. ??
So suppose we apply the suggested hint to create line 9.
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP
9. K ⊃ ~D (8)(5) HS
Notice it seems we are getting close to the conclusion. Line 9 is half so to speak of the conclusion -- (K ⊃~D) • (~Z ⊃ M). Is there any way to get the second half (~Z ⊃ M)? Yes.
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP
9. K ⊃ ~D (8)(5) HS
Ok, let's apply hint #4. Now we are almost done.
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP
9. K ⊃ ~D (8)(5) HS
10. ~Z ⊃ M (5)(1) HS
Now we have both parts of the conclusion. How can we finish?
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP
9. K ⊃ ~D (8)(5) HS -- Hint
10. ~Z ⊃ M (5)(1) HS -- Hint
Because doing this will produce the conclusion, we are done. Notice that line 7 ended up being an extra step. The step is correct but we could have made the proof more elegant by eliminating it. But you do not normally know what steps are crucial until you are finished. Most important is that all the steps are correct applications of the rules.
1. ~D ⊃ M
2. ~D
3. E • I
4. E ⊃ (K ⊃ ~Z)
5. ~Z ⊃ ~D / ∴ (K ⊃~D) • (~Z ⊃ M)
6. E (3) Simp.
7. M (1)(2) MP
8. K ⊃ ~Z (4)(6) MP
9. K ⊃ ~D (8)(5) HS
10. ~Z ⊃ M (5)(1) HS
11. (K ⊃~D) • (~Z ⊃ M) (9)(10) Conj.
Premise 3 is perfect for applying the Simplification rule.
(3) E • I
/ ∴ (6) E
p • q
/ ∴ p
Premises 1 and 2 are perfect for applying the MP rule.
1. ~D ⊃ M
2. ~D / ∴ (7) M
p ⊃ q
p / ∴ q
Line 6 will work perfect with line 4 and the rule MP.
(6) E ⊃ (K ⊃ ~Z)
(4) E / ∴ (8) K ⊃ ~Z
p ⊃ q
p / ∴ q
Line 5 will work with line 8 as follows:
(8) K ⊃ ~Z
(5) ~Z ⊃ ~D / ∴ (9) K ⊃ ~D HS
p ⊃ q
q ⊃ r / ∴ p ⊃ r
Line 5 will work perfectly with line 1 as follows:
(5) ~Z ⊃ ~D
(1) ~D ⊃ M / ∴ (10) ~Z ⊃ M HS
p ⊃ q
q ⊃ r / ∴ p ⊃ r
We can combine lines 9 and 10 by Conjunction.
(9) K ⊃ ~D
(10) ~Z ⊃ M / ∴ (11) (K ⊃~D) • (~Z ⊃ M) Conj.
p
q / ∴ p • q