More Nested and Multiple Assumptions

Let's look at a few more examples of conditional proofs that rely on either nested or serial multiple assumptions.

Consider the argument:

(Q    R),    (Q R)  S        (Q  S)

1.    P  (Q    R)                pr

2.    (Q R)  S                  pr

┌3.        P                     AP

|    ┌4.    Q                   AP

|     |  5.    (Q R)            1, 3  MP

|     |  6.    R                      4,5  MP

|     |  7.    (Q R)            4,6  Conj

|     |  8    S                       2,7  MP

|    9.    (Q  S)               3-8  CP

10.    P  (Q  S)              2-9  CP

When using multiple assumptions, it is not necessary to nest the assumptions.  Using serial multiple assumptions is particularly useful when proving a biconditional.  The general strategy is to prove 2 different conditionals and then use EQUIV to derive the biconditional.  Schematically, here is the template for using 2 separate conditional proofs to establish a biconditional:

 To prove Φ  ↔ Ψ 1. Premises ┌→2. Φ Assumption |      3 ... Derived line |     4 ... Derived line |     5 Ψ Derived line 6. Φ  → Ψ 2-5  CP ┌®7. Ψ Assumption |    8. ... Derived line |    9. ... Derived line |   10. Φ Derived line 11. Φ → Ψ 7-10  CP 12. (Φ  → Ψ)  ● (Φ  → Ψ) 6, 12 Conj 13. Φ  ↔ Ψ 12 ME

Here is an application of the strategy to an argument.  Consider the argument:

(P  ▼ Q)  R,   Q  ~R,   ~Q    P          P   R

1.    (P  ▼ Q)  R                    pr               P   R

2.    Q  ~R                              pr

3.     ~Q    P                            pr

4.    P                                AP

|      5.    P  ▼ Q                       4  ADD

|      6.    R                                1,5  MP

7.    P  R                               4-6  CP

8.    R                                    AP

|     9.    ~~R                                8 DN  (this step is necessary to generate the negation of the consequent of 2)

|    10.  ~Q                                  2,9  MT

|    11.   P                                    3, 11  MP

12.    R                                 8-11  CP

13.    (P  R)  (R  P)          7, 12 Conj

14.    P   R                                13  ME