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I need help using the rules of implication/inference to construct a proof for the following argument:

1.(A ∨ B) ⊃ (C ∨ D)

  1. C ⊃ E

  2. C ∨ ~F

  3. A ● ~E

  4. F ∨ (D ⊃ Z)

.: Z

KEY:

  • Tilde (~) forms negations (“not,” “it is not the case that”).

  • Dot (●) forms conjunctions ( “and” “also”)

  • Wedge (∨) forms disjunctions (“or,” “unless”).

  • Horseshoe (⊃) forms conditionals (“if . . . then,” “only if,” etc.).

  • Triple bar (≡) forms biconditionals (“if and only if,” etc.).

5
  • please i need help
    – max
    Aug 1 '20 at 19:05
  • @ClydeFrog please help me i don't even know where to start
    – max
    Aug 1 '20 at 19:26
  • @cookiemonster help me somebody oh god im gonna fail highschool
    – max
    Aug 1 '20 at 20:11
  • 1
    What is to be proven? Is all of #5 the intended conclusion? Or is the conclusion only the statement Z? Or what? Aug 2 '20 at 1:29
  • the conclusion is just Z, 5. is the last premise @MarkAndrews
    – max
    Aug 4 '20 at 21:59
2

Perhaps, a possible proof (using Fitch-style natural deduction system) could be:

enter image description here

1

First, let's examine the premises.

|  1. (A ∨ B) ⊃ (C ∨ D)
|  2. C ⊃ E
|  3. C ∨ ~F
|  4. A ● ~E
|_ 5. F ∨ (D ⊃ Z)

The fourth premise is a conjunction. Start by eliminating that. Simplification is trivial but quite often useful.

|  6. ~E              ●e 4
|  7. A               ●e 4

The first premise is a conditional whose antecedent is a disjunction of A. We can now introduce that disjunction so we may eliminate that conjunction. See, it has become useful already.

|  8. A ∨ B           ∨i 7
|  9. C ∨ D           ⊃e 8,1

Now you have three disjunctions, C ∨ ~F, F ∨ (D ⊃ Z), and C v D. Some nested disjunction eliminations are indicated. As we have to start somewhere, let us start with the latest. This makes use of the second premise and the ~E we derived earlier.

|  |_ 10. C
|  |  11. E           ⊃e 10,2
|  |  12. #           ~e 11,6
|  |  13. Z           x  12
|  14. C ⊃ Z          ⊃i 10-13
|  |_ 15. D
|  |   :
|  |  26. Z           ... 
|  27. D ⊃ Z          ⊃i 15-26          
|  28. Z              ∨e 9,16,27

Thus do we derive Z from the derivations for C ∨ D , C ⊃ Z, and D ⊃ Z.

But wait! We, or rather, you still need to actually derive D ⊃ Z. Well we do have two conjunctions, premises 3 and 5 that have yet to be eliminated. So, ... do that there.

3
  • I prefer your proposed solution, @Graham Kemp. Are you thinking along these lines for the derivation of D ⊃ Z? Derivation
    – F. Zer
    Aug 6 '20 at 13:40
  • If you are allowed to use Disjunctive Syllogism, yes. Aug 6 '20 at 23:23
  • Perfect. Thank you.
    – F. Zer
    Aug 6 '20 at 23:37

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