I am trying to prove A ^ B from the premises shown in the screenshot. As you can see in the screenshot I am struggling with the second sub-sub proof. Do you have recommendations for how to continue/finish the proof?

Thank you. enter image description here

Edit You can only use conjunctive, disjunctive, and negation intro/elim and only uses TautCon for DeMorgans.

I updated the picture of the problem. I am only struggling to cite the lines (I think).

  • Are you allowed to use double negation (~~P therefore P) or V-elim (P v Q, ~P, therefore Q) rules? These rules might be called different names depending on your book. – Adam Sharpe Mar 27 '19 at 18:23
  • After step 5 use or-elim. Two cases a) ~~A i.e. A and with premise 1 it's ok. case b) ~B anf thus a contradiction with premise 1 from which A and again it's done. – Mauro ALLEGRANZA Mar 27 '19 at 19:48

Edit You can only use conjunctive, disjunctive, and negation intro/elim and only uses TautCon for DeMorgans.

You also seem to be using rules called "contradiction introduction", and "contradiction elimination". So I suspect what you call "negation elimination" is what is more usually called "double negation elimination".

Anyway, you have the first five lines okay.

 1|  B                 Promise
 2|_ ~((~A & B) v C)   Promise
 3|  ~(~A & B) & ~C    2, Taut Con (DEM)
 4|  ~(~A & B)         3, Conjunctive Elimination
 5|  ~~A v ~B          4, Taut Con (DEM)

Then you should have just used Disjunction Elimination.

 6| |_ ~~A            Assumption
 7| |  A              6, Double Negation Elimination
 8| |  A & B          7, Conjunction Introduction
  | + 
 9| |_ ~B             Assumption
10| |  #              1,9, Contradiction Introduction
11| |  A & B          10, Contradiction Explosion
12| A & B             5,6-8,9-11, Disjunctive Elimination

You can also prove this without invoking de Morgan's Law

You have B as a promise, so you just need to derive A. You may do that by reduction to absurdity.

 1|  B                 Promise
 2|_ ~((~A & B) v C)   Promise
 3|  |_ ~A             Assumption
 4|  |  ...            ...
 5|  |  ...            ...
 6|  |  #              Contradiction Introduction  5
 7|  ~~A               Negation Introduction       3-6
 8|  A                 Double Negation Elimination 7
 9|  A & B
| improve this answer | |
  • Thank you! I now understand the concept! – Graham Streich Mar 28 '19 at 1:38

Proof created with proofs.openlogicproject.org:

enter image description here

The proof assumes that you can use the double-negation rule and the disjuction-elimination rule. (When asking for help with logic problems, it's a good idea to say which rules you're allowed to use since different texts/programs allow different rules that others may not.) Also, lines 6-8 can be simplified by inferring ~~B from B (line 1) if your rules allow it. As far as I could tell, the proof tool I used does not so I had to go about it in a roundabout way.

| improve this answer | |
  • Thank you, I updated the question to show my progress and the rules I can use. I cannot understand your citations (I am very new at this) but use I can use double negation and disjunctive elimination. – Graham Streich Mar 28 '19 at 0:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.