I need help with this proof for my philosophy class. This proof is supposed to be done via indirect proof or conditional proof, so it is supposed to use AIP and IP or ACP and CP to derive the conclusion! This is an assignment that is submitted through Aplia, so I need it to be precise and the assignment is due tonight, so I would like it done before tonight please!
This is the key:
The * = dot
The > = horseshoe
The v = wedge
The ≡ = triple bar
The ~ = tilde
The statement after the / (slash) is the conclusion that I need.
The problem is:
1. R ≡ ~O
/ (~T > M) v (T > ~H)
To prove the statement (~T > M) v (T > ~H) using indirect proof or conditional proof, we will assume the negation of the conclusion and derive a contradiction. Here's a step-by-step breakdown of the proof:
1. R ≡ ~O Premise
1.1. ~R ≡ O Equivalence of 1 (Double Negation)
1.2. ~O ≡ R Equivalence of 1 (Commutation)
2. Assume ~(~T > M) v (T > ~H) Assumption for Indirect Proof
3. ~(~T > M) ∧ ~(T > ~H) De Morgan's Law from 2
4. ~(~T > M) Simplification from 3
4.1. ~~~T ≡ M Equivalence of 4 (Double Negation)
4.2. ~~T ≡ M Double Negation of 4.1
4.3. T ≡ M Equivalence of 4.2 (Double Negation)
5. ~(T > ~H) Simplification from 3
5.1. ~T v ~H Material Implication of 5
5.2. ~H Disjunction Elimination from 5.1
5.3. ~~~H Double Negation of 5.2
5.4. ~~H ≡ H Equivalence of 5.3 (Double Negation)
5.5. H Equivalence of 5.4 (Double Negation)
6. T ≡ M ∧ H Conjunction of 4 and 5
7. T Simplification of 6
8. M Simplification of 6
9. ~T > M Conditional Proof (Assume ~T and derive M)
9.1. ~M ∧ ~~T Conjunction of 8 and 7
9.2. ~M Simplification of 9.1
9.3. ~~M > ~T Conditional Proof (Assume ~~M and derive ~T)
9.3.1. ~~T v ~T Material Implication of 9
9.3.2. ~T Disjunction Elimination of 9.3.1
9.3.3. ~M ∧ ~T Conjunction of 9.2 and 9.3.2
9.3.4. ~T ∧ ~M Commutation of 9.3.3
9.4. ~~M > ~T Conditional Proof (Assume ~~M and derive ~T)
9.4.1. ~T ∧ ~M Commutation of 9.3.4
9.5. ~T > M Conditional Proof (Assume ~T and derive M)
9.5.1. ~~~T ≡ M Equivalence of 9.4.1 (Double Negation)
9.5.2. ~T ≡ M Double Negation of 9.5.1
10. (~T > M) v (T > ~H) Proof by Contradiction (Assume the negation of the conclusion)
Since we have now derived the negation of the conclusion, we have reached a contradiction. Therefore, the original statement (~T > M) v (T > ~H) must be true.
To solve this proof, we will use conditional proof (CP) and indirect proof (IP). Let's break it down step by step:
1. R ≡ ~O / (~T > M) v (T > ~H)
Step 1 gives us the premise: R is equivalent to ~O.
Now, we need to derive the conclusion: (~T > M) v (T > ~H) using CP and IP.
First, let's assume the negation of the conclusion ~[(~T > M) v (T > ~H)]. This will lead us to a contradiction, which would then allow us to derive the original conclusion.
2. Assume ~(~T > M) v (T > ~H)
Next, we will apply De Morgan's Law to the negation of the conclusion in Step 2.
3. ~(~T > M) ∧ ~(T > ~H) (De Morgan)
Now, let's break down the two separate negations using conjunction (∧).
4. ~(~T > M) (Simplification from Step 3)
5. ~(T > ~H) (Simplification from Step 3)
The negation in Step 4 can be further simplified using conditional (~ > ≡).
6. T > ~M (Double Negation from Step 4)
Now, let's apply indirect proof (IP) to derive a contradiction and reach the original conclusion.
7. Assume ~[(~T > M) v (T > ~H)] / (~T > M) v (T > ~H)
To derive a contradiction, we need to consider two cases: (~T > M) and (T > ~H).
8. Case 1: Assume ~T > M / (~T > M) v (T > ~H)
Using CP, let's assume ~T > M.
9. ~T (Assumption)
10. ~T > ~H (Modus Ponens from Step 9 and given premise R ≡ ~O)
11. ~H (Modus Ponens from Step 10)
12. T > ~H (Implication from Step 11)
At this point, we have derived the right side of the original conclusion (T > ~H).
13. (~T > M) v (T > ~H) (Disjunction Introduction from Step 12)
Now let's move to the second case.
14. Case 2: Assume T > ~H / (~T > M) v (T > ~H)
Using CP, let's assume T > ~H.
15. T (Assumption)
16. T > ~M (Modus Ponens from Step 15 and given premise R ≡ ~O)
17. ~M (Modus Ponens from Step 16)
18. ~T > M (Implication from Step 17)
At this point, we have derived the left side of the original conclusion (~T > M).
19. (~T > M) v (T > ~H) (Disjunction Introduction from Step 18)
Now, we have derived both sides of the original conclusion. Using disjunction elimination (∨E) on Steps 13 and 19, we can conclude the original conclusion.
20. (~T > M) v (T > ~H) (Disjunction Elimination from Steps 13 and 19)
By reaching this conclusion, we have arrived at a contradiction to our assumption in Step 7.
21. Contradiction (~[(~T > M) v (T > ~H)] > (~T > M) v (T > ~H))
Since we have derived a contradiction based on our assumption ~[(~T > M) v (T > ~H)], we can conclude that the original conclusion (~T > M) v (T > ~H) must be true.
Therefore, using the methods of conditional proof (CP) and indirect proof (IP), we have proven that the conclusion (~T > M) v (T > ~H) follows from the given premise R ≡ ~O.