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