The argument is Invalid
How can we determine this? By use of a logic table shown below
Breakdown of the table
- Column A represents the truth values for logic expression O (either T for true or F for false)
- Column B represents the truth values for logic expression X
- Column C is the truth values for logic expression ~O (this is the complete opposite of what is shown in column A)
- Column D has us compute the first premise's truth values. In this case, no matter what O happens to be, the truth value of ~O > ~O is always true. This is because ~O > ~O is equivalent to ~~O v ~O and that turns into O v ~O. After substitution of truth values, you'd end up with T v F or F v T which is always true. In plain english, it's like saying "I either got the object or I didn't get the object"
- Column E isn't a premise so I didn't mark it in blue. It will help set up the premise in column F. Similar to column D, we use the conditional template. The expression X > O is only false if X is true and O is false. This happens in line 4
- Column F uses column E. Column F is a premise so I marked it in blue. The same rules used prior (with columns D and E) will be used here
- The last column, column G, is the conclusion marked in green. Simply negate or take the opposite of whatever you see in column B (where X is located)
Row 2, where I've marked the premises and conclusion in red, shows us that the argument is invalid. This row is a case where all of the premises are true but the conclusion is false.
In short,
if O = true and X = true
then...
the premise ~O > ~O is true
the premise X > (X > O) is true
but the conclusion ~X is false
This is why the argument is invalid. It is impossible to do a proof derivation of invalid proofs. It's possible your teacher made a typo or s/he is trying to throw a trick question.