I had some trouble understanding the whole enemies analogy for explaining statement analysis. I understood pretty easily that it mattered what order the for all and there exist clauses were placed in, but I didn't understand how that related to the idea of an enemy choosing. For that reason, I initially couldn't do that part of the past test where we had to analyze a lengthy statement and its negation and choose which one was correct.
How I've come to understand it (and I'm not sure if this is correct) is equating "there exists X" to "I choose X" and "for all Y" to "my enemy chooses Y". If my interpretation is correct, here's why: for all clauses are selected by the enemy because proving a proof entails showing that it holds true not in any case, but in the worst case. We can rely on our enemy – the devil's advocate who will choose the worst case in order to prove us wrong – to reliably represent the for all clause. By contrast, we are allowed to choose there exist variables because there is a lot more flexibility in what those variables can be (since there only needs to exist one example). So, we needn't an enemy to choose the worst case for that.
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