MHB Solving Inference Exercise: ¬t∨w

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The discussion revolves around the inference exercise involving the premises {¬p→r∧¬s, t→s, u→¬p, ¬w, u∨w} and the conclusion ¬t∨w. The user presents a series of logical steps leading to the conclusion but believes they have made an error. Participants clarify that the conclusion reached at step 12 is indeed correct, suggesting that the user may have overcomplicated the proof process. Additionally, there is a discussion on the reuse of premises after proving contradictions, confirming that previously used premises can be utilized again. The conversation emphasizes the importance of clarity when referencing less common logical laws in proofs.
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Given the following premises: {¬p→r∧¬s, t→s, u→¬p, ¬w, u∨w}
The conclusion is said to be: ¬t∨w
Here are my steps. My conclusion is different from the supposed one, therefore I would appreciate it if any of you can point out my error.

Thank You.

1
¬p→(r∧¬s)
Premise

2
p∨(r∧¬s)
Implication law: 1

3
(p∨r)∧(p∨¬s)
Distributivity: 2

4
(p∨¬s)
Simplification: 3

5
t→s
Premise

6
¬t∨s
Implication law: 5

7
p∨¬t
Resolution: 4 & 6

8
u→¬p
Premise

9
¬u∨¬p
Implication law: 8

10
¬t∨¬u
Resolution: 7 &9

11
u∨w
Premise

12
¬t∨w
Resolution: 10 & 11

13
¬w
Premise

14
¬t
Disjunctive Syllogism: 12 & 13 AND Conclusion
 
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Hey Quintessential!

Isn't step 12 the conclusion that you're supposed to reach (and prove)?
I think you are done at step 12.
You can always find more conclusions if you want to, but I presume that is not what the problem statement asks.
 
Makes sense, thanks!
I was confused because for some of the exercises, I couldn't reach the right conclusion.

For example, the following's conclusion should be t, Given: {p∨q, q→r, p∧s→t, ¬r, ¬q→u∧s}

Yet I find q∨t.
That said. If you prove a contradiction between two premises, and thus dismiss it with either Resolution or Disjunctive Syllogism, is that premise void of future use? In other words, can I reuse previously used premises when stuck?1
p∨q
Premise

2
q→r
Premise

3
¬q∨r
Implication law: 2

4
p∨r
Resolution: 1 & 3

5
p∧s→t
Premise

6
¬p∨(¬s∨t)
Implication law and Associativity: 5

7
r∨(¬s∨t)
Resolution: 4 & 6

8
¬r
Premise

9
¬s∨t
Disjunctive Syllogism: 7 & 8

10
¬q→(u∧s)
Premise

11
q∨(u∧s)
Implication law: 10

12
(q∨u)∧(q∨s)
Distributivity: 11

13
q∨s
Simplification: 12

14
q∨t
Resolution: 9 & 13
 
Quintessential said:
Makes sense, thanks!
I was confused because for some of the exercises, I couldn't reach the right conclusion.

For example, the following's conclusion should be t, Given: {p∨q, q→r, p∧s→t, ¬r, ¬q→u∧s}

Yet I find q∨t.

Perhaps you can combine 3 and 8 to find ¬q?
That said. If you prove a contradiction between two premises, and thus dismiss it with either Resolution or Disjunctive Syllogism, is that premise void of future use? In other words, can I reuse previously used premises when stuck?

Err... you can use any premisse or previous step as often as you like.
 
I like Serena said:
Err... you can use any premisse or previous step as often as you like.

And here I thought, that was a big no no.
Thanks again!
 
Quintessential said:
Given the following premises: {¬p→r∧¬s, t→s, u→¬p, ¬w, u∨w}
The conclusion is said to be: ¬t∨w
Here are my steps. My conclusion is different from the supposed one, therefore I would appreciate it if any of you can point out my error.

Thank You.

1
¬p→(r∧¬s)
Premise

2
p∨(r∧¬s)
Implication law: 1

3
(p∨r)∧(p∨¬s)
Distributivity: 2

4
(p∨¬s)
Simplification: 3

5
t→s
Premise

6
¬t∨s
Implication law: 5

7
p∨¬t
Resolution: 4 & 6

8
u→¬p
Premise

9
¬u∨¬p
Implication law: 8

10
¬t∨¬u
Resolution: 7 &9

11
u∨w
Premise

12
¬t∨w
Resolution: 10 & 11

13
¬w
Premise

14
¬t
Disjunctive Syllogism: 12 & 13 AND Conclusion

Your proof is icorrect you need one more step to complete it

And another thing.

If you mention in your proof a law that is not commonly used( like the law resolution in your proof) you have to state clearly how that law works
 
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