So, another one of my writing where I write to understand things better myself. Anyways, today I will clarify what is the traditional square of opposition is, and how it is useful (in exams for 10 marks questions).
What is a traditional square of opposition?
I hope you know what are A, E, I, and O propositions, because if you don't...then it's gonna be tough. Moving on, the actual square of opposition looks like this - 
So this is the square. There are A E I O at the 4 corners are they are joined as follows:
1) A to O is joined by a bi-directional arrow.
2) E to I is joined by a bi-directional arrow.
3) A to I is joined by a bi-directional arrow.
4) E to O is joined by a bi-directional arrow.
BUT
5) A to E is joined by a uni-directional arrow.
6) I to O is joined by a uni-directional arrow.
With bi-directional arrows, you can travel in both directions, but with uni-directional, as the name suggests, you can travel in only one direction. Meaning you can go from A to O and back from O to A, but you can go from A to E, but not E to A.
When to use this thing?
This ancient tablature is used to determine/find out the truth value of unknown propositions from known ones. For example, you might be given the truth value of an E-statement and be told to infer the truth value of all other propositions, this square can help you there.
Ex - If "All socialists are pacifists" is true, what is the truth value of "Some socialists are non-pacifists"? [5 marks] (2017 Logic Question Paper)
How to use this thing?
Let's start with an example and a few rules to go along. Let's assume that you are given - It is false that some A are B, determine the truth value of all other propositions.
We know "Some A are B" is an I proposition. Therefore the question is saying I is false, determine the truth value of A, E, and O from the given data that I is false.
Now, before we start using this square of opposition, here are some simple ways in which this operates:
1) Diagonal lines always have opposite truth values. Meaning, if A is False, then O has to be True, and if A is True then O has to be False. Because diagonals are connected by bi-directional arrows, so knowing the truth value of one diagonal will give us the truth value of the other. If A is false then O is true or if O is False then A is True.
2)Truth flows downwards and falsity flows upwards. You can imagine this as a water pump fixed at the top of your roof, it takes bad (false) water from the ground upwards to the roof and then purifies and send the purified good (true) water downwards to your house as drinking water. This rule says that the True values can move downwards only and the false values can move upwards only. Meaning, if A is true, then I will be True (the True value of A traveled downwards to I), and if E is true then O will be true in the same way. BUT. If A is false, then we cannot tell if I is false too, because the false value of A doesn't travel downwards (false flows upwards only). But if it was given that I is false, then we can say A is false too, because Falsity of I now flowed upwards to A.
Now back to our example, we are given that it is false that some A are B, i.e. I proposition is False.
1) If I proposition is false, then by rule 1, rule of diagonal, we know that E proposition will be opposite truth value of I, therefore E will be True.
2) Now by rule 2, we know truth flows downwards, therefore the True value of E will travel down and thus O will be True too.
3) Again by rule 1, rule of diagonal, we know if O is True, then A will have the opposite truth value of O, therefore as O is true therefore A is False.
Thus we found all the truth values of A, E, and O just by the truth value of I proposition through the square of opposition.
Let's take another example -
It is false that no A is B. Find the truth value of all other propositions.
Here we can see "No A is B" is an E proposition, therefore we are given a false E-statement, and asked to find out the rest truth values.
1) We by rule 1, know if E is false then I will be opposite, i.e. I will be True.
2) But after that we are stuck, as I is true and truth flows downwards therefore we cannot say if A is True or False. We cannot use the bi-directional arrow connecting A and I unless they are following rule 2.
3) We also cannot go from E to O, because only truth flows downwards and as O is below E, and falsity cannot flow downwards, we cannot say if O is false or true.
4) Lastly because A and E are connected by a uni-directional arrow going from A to E, so we cannot use E to go to find truth value of A either.
Therefore all we can find out is that I proposition will be false. That's all. Nothing more can be found. Yes, sometimes this happens, sometimes enough data is not provided to determine the truth or falsity of all statements.
Solving a question paper question.
Let's go back to the 2017 question.
Q. If "All socialists are pacifists" is true, what is the truth value of "Some socialists are non-pacifists"?
Ans- "All socialists are pacifists" is an A statement, therefore we are given that A proposition is true.
Now we can see that "Some socialists are non-pacifists" is an O statement. So we have to find the truth value of an O statement, given that A statement is true.
Now in the square of opposition, we mark A as true. 
Now by the rule of contradictory statements (1st rule/rule of diagonal), we know diagonals have opposite truth values. Therefore as A is True, therefore O will be false.
BOOM. 5 marks secured in the exam. As simple as that.
Let's also take the next part of the question just to complete the set.
Q. If "All socialists are pacifists" is true, what is the truth value of "No pacifists are socialists."?
Ans - Here to we are given that the A proposition, "All socialists are pacifists" is true, and we have to find truth value of "No pacifists are socialists."
Now "No pacifists are socialists" is not having the same subject and predicate combination as our A statement, therefore, by the rules of conversion, we can say that "No pacifists are socialists" is the same as "No socialists are pacifists".
Therefore, from a true A proposition, "All socialists are pacifists", we have to find the truth value of an E-statement "No socialists are pacifists".
We again mark the A in the square of opposition as true just as we did before.
1) If A is true then by the rule of sub-alternation (rule 2) we know truth flows downwards, therefore I will be True too.
2) If I is true, then, by the rule of contradictory statements (rule 1/rule of diagonals) E will be false.
Therefore the E-statement that "No socialist are pacifists" is false and therefore the equivalent of the statement that we get from conversion, "No pacifists are socialists" is false too.
Boom. Another 5 marks secured. Easy 10 marks.
Ok, that's all. I hope that helped. Ask any doubts or point out any mistakes if you can find them.
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