assumption procedures Philosophy 200 assumption procedures

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

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

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.

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.

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)

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.

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).

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

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.

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.

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

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.

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

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

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

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

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

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

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

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

Indirect Proof

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

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.

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

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.

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. …

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 3-4 RAA. We already had ~Q, so this doesn’t do us any good.

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

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

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)

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 3-5 RAA QED

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