1. assumption procedures
Philosophy 200
assumption procedures

2. Making Assumptions
Students often struggle with the idea of making assumptions during a proof.

3. Making Assumptions
Students often struggle with the idea of making assumptions during a proof.
Consider, however, that a proof itself begins with assumptions.

4. Making Assumptions
Students often struggle with the idea of making assumptions during a proof.
Consider, however, that a proof itself begins with assumptions.
When we test an argument for validity, we are seeing if we can end up with the conclusion if we assume that the premises are all true.

5. Validity as Conditional Proof
In other words, when we do a proof, we demonstrate that IF the premises are true, THEN the conclusion must be.

6. Validity as Conditional Proof
In other words, when we do a proof, we demonstrate that IF the premises are true, THEN the conclusion must be.
Any valid argument would make the below conditional a tautology:
(Prem) · (Prem)…  (Conc)

7. Conditional Proof
If we need to prove that a conditional is true during a proof, then we must show we can get the consequent whenever we assume the antecedent is true.

8. Conditional Proof
If we need to prove that a conditional is true during a proof, then we must show we can get the consequent whenever we assume the antecedent is true.
So the first step in a conditional proof is to introduce an assumption (which is always the antecedent of the conditional you want to prove).

9. Discharging assumptions
Whenever you introduce an assumption, you must discharge it before the proof can be complete.

10. Discharging assumptions
Whenever you introduce an assumption, you must discharge it before the proof can be complete.
To discharge an assumption is to finish the purpose for which you introduced the assumption.

11. Discharging assumptions
Whenever you introduce an assumption, you must discharge it before the proof can be complete.
To discharge an assumption is to finish the purpose for which you introduced the assumption.
A proof with undischarged assumptions will not demonstrate that the original argument was valid, but rather that the original argument PLUS assumptions is valid.

12. Assumption procedures
Think of an assumption procedure as a proof within a proof.

13. Assumption procedures
Think of an assumption procedure as a proof within a proof.
Whatever is true before making the assumption is true after making it, so any line can be used in the assumption section, but lines within the assumption section cannot be used outside of the assumption section because they all rely on the assumption.

14. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R)

15. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp

16. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp |5. Q 3,4 MP

17. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp |5. Q 3,4 MP |6. ~Q v ~S 2, DeM

18. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp |5. Q 3,4 MP |6. ~Q v ~S 2, DeM |7. ~S 5,6 DS

19. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp |5. Q 3,4 MP |6. ~Q v ~S 2, DeM |7. ~S 5,6 DS |8. (R  S) 1, simp

20. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp |5. Q 3,4 MP |6. ~Q v ~S 2, DeM |7. ~S 5,6 DS |8. (R  S) 1, simp |9. ~R 7,8 MT

21. Conditional Proof Example
1. (P  Q) · (R  S) premise 2. ~(Q · S) premise / P  ~R |3. P assumption for CP (~R) |4. (P  Q) 1, simp |5. Q 3,4 MP |6. ~Q v ~S 2, DeM |7. ~S 5,6 DS |8. (R  S) 1, simp |9. ~R 7,8 MT 10. P  ~R 3-9 CP

22. Indirect Proof

23. Indirect Proof
Indirect proof is also known as ‘reductio ad absurdum’ or ‘proof by contradiction’.

24. Indirect Proof
Indirect proof is also known as ‘reductio ad absurdum’ or ‘proof by contradiction’.
The idea is that if we assume a thing is true, and that assumption leads to a contradiction, we can conclude that thing that we assumed must be false after all.

25. Procedure
When initiating an indirect proof, assume the negation of the thing that you want to end up with.

26. Procedure
When initiating an indirect proof, assume the negation of the thing that you want to end up with.
When you get a contradiction (anything of the form P · ~P), you can discharge the assumption.

27. Example of what most people try to do at first:
P · Q prem
~Q prem / S
| 3. Q assumption for RAA
| 4. Q · ~Q 2,3 Conj. (contra)
5. …

28. Example of what most people try to do at first:
P · Q prem
~Q prem / S
| 3. Q assumption for RAA
| 4. Q · ~Q 2,3 Conj. (contra)
5. ~Q RAA.
We already had ~Q, so this doesn’t do us any good.

29. Example of what to do: P · Q prem ~Q prem / S
| 3. ~S assumption for RAA

30. Example of what to do: P · Q prem ~Q prem / S
| 3. ~S assumption for RAA
| 4. Q 1, Simp.

31. Example of what to do: P · Q prem ~Q prem / S
| 3. ~S assumption for RAA
| 4. Q 1, Simp.
| 5. Q · ~Q 2,4 Conj. (contra)

32. Example of what to do: P · Q prem ~Q prem / S
| 3. ~S assumption for RAA
| 4. Q 1, Simp.
| 5. Q · ~Q 2,4 Conj. (contra)
6. S RAA
QED

33. MT by indirect P  Q prem 2. ~Q prem / ~P |3. P assumption for RAA
|4. Q 1,3 MP
|5. Q · ~Q 2,4 Conj. (contra)
6. ~P
QED