With that interpretation, your argument can be translated as such:

A) Each fool a friend of someone S1
B) Each fool is a foe of someone S2
C) No foes of someone are also a friend to them

The conclusions:
D) No fools are a foe to anyone
E) No fools are a friend to anyone

I gather this is the intended interpretation of your argument. If not, explain how it isn't.

The conclusion doesn't follow from the premises; all that is necessary is that each friend S1 of each fool is not also a foe S2 to that fool. S1 =/= S2. Feel free to substitute cats and dogs into the argument form and derive nonsense.

It isn't true that C) No friends are foes. I'm a friend to some people and I'm a foe to others. The premise is false and therefore any conclusions drawn from it are unsound.

I don't think you realize how absolutely massive and hot the sun is compared to anything we could lob into it. Even if it was done for the specific purpose of destroying it, I don't think humanity could muster more than a petty insult to the sun.

Then my mistake; 'can' is an ambiguous word.

I think this is what tim wood referred to so I'll leave him as the accepted answer. It's true that ejecting directly opposite from base gets you there quickest, and that's what I calculated, albeit instead comparing with vectors that rotate your motion vector exactly 45 degrees which involves different numbers like square roots of 2.

If depression is caused in part by guilt as you seem to say it is, then it isn't hard to think of a guilty thing one could do intentionally which would lead to depression. If one can become depressed as a result of guilty behavior, then much more so as a result of continued guilty behavior. Along that reasoning, it would be likely that taking responsibility to cease guilty behavior would mitigate depressive feelings. Of course, that isn't to say that erroneous feelings of guilt can be dealt with in the same way.

Right, brain disorders are in another league.

Right about chronic conditions. There may be a component of morality to some depression and anxiety though. I overcame my chronic depression and anxiety by taking responsibility, and I have even received a tentative diagnosis of schizophrenia before and I have all but overcome all of my troubles without any real medication or treatment. I still experience anxiety but I have been overcoming it with hard work.

Also, your charge was faulty; my post was mainly about the benefits of treatment, not personal responsibility, although your charge about my views happens to be correct.

You do not have to be on medication forever to recover from disorders. For example, the standard treatment for anxiety disorder is to put the patient on medication for a year and they generally get better afterward. Studies show that depression can also be improved by diet for instance, which is one way that psychiatrists can help and inform patients. Sometimes just taking responsibility for something isn't enough and treatment can help. You may learn interesting things in psychology, and it isn't all about disorders. There is also the field of positive psychology for preventing disorders.

The fool can be a good friend, or he can be a good foe. That doesn't mean he is either of them. Since he need not be either of them in your scenario, he isn't paradoxically simultaneously both.

Actually, I just did the math and it looks like by ejecting perpendicularly your final velocity ends up being 4x, compared to if you just cancel out your velocity you end up with a velocity of 6.24x. Even though by ejecting perpendicularly you never negate any of your velocity in any single shot. How about that. :D

The question only asked how to get to base the quickest. Maybe home base has a circular magnet that slows you down as you go through it. :) Or you can shoot the alien with the fifth projectile at lower energy.

@Sir2u Good job. This way you never negate any component of your velocity vector at any time. Each ejection turns you (roughly over) 45 degrees toward the opposite direction. All momentum is conserved.

@tim wood That doesn't count, since it's only obvious to some people.

Sorry but this is trivial, comparable to concern about scratching yourself in the presence of oxygen-producing green plants since it will contaminate them with dead skin cells. :)

How about if you wanted to state that a set contains at least one member? The only sensible way to write it is ∃x∈S. That is a case of a standalone quantifier. That isn't to say anything about it being a predicate, but it could be extended for that case.

Next, how would you state that a set is empty in predicate logic?

Existence is a predicate because this statement makes complete sense:

∀x∈S[¬∃x]

The universal quantifier does not require existence. This statement says that for every member of S, it doesn't exist. Since there are no members of S, we can assign any attribute we want.

The only case where the above statement makes sense of course is when S is the empty set. If it had members, it would be false. And it works for the empty set because, again, the universal quantifier does not require existence.

It's only contradictory if you choose both simultaneously. The reason for each choice is the same: you have the upper hand around the fool, whether friend or foe or a neutral party.

In both cases the genius is the worse pick because he has the upper hand. However, if we pick the fool we have the upper hand - in that case, we become the genius.