There's a chapter on this idea in one of the ethics books for my philosophy class: The Fundamentals of Ethics by Russ Shafer Landau.
TL;DR? Altruism exists, or the arguments against it are certainly not strong enough to disprove of its existence.
Think about the question like this:
"I make the claim that no action can be selfless, every action is inherently going to be selfish." This is Psychological Egoism, the idea that what motivates us is our own self-interest, if it is true then altruism (the direct desire to benefit others for their own sake with no ulterior motive) does not exist.
It is easy to think Psychological Egoism is the case, though if it is then such civilian acts of heroism as these: http://www.carnegiehero.org/awardees/carnegie-medal-awardees-june-30-2015/ must therefore not be altruistic and must therefore have an ulterior motive? That seems an odd assertion, does someone who jumps on top of a grenade to save their fellows (such as Ross McGinnis who on December 4th 2007 sacrificed himself to save his squad mates on patrol in Iraq) do this with an ulterior motive?
Here are some of the arguments an egoist might use:
The Argument from Our Strongest Desires
1. Whenever you do something, you are motivated by your strongest desire.
2. Whenever you are motivated by your strongest desire, you are pursuing your self interest.
3. Therefore, whenever you do something, you are pursuing your self-interest.
Let's leave aside premise one, because even if it is true premise two begs the question i.e it presumes the truth of its conclusion. It cannot appeal to fans and opponents of psychological egoism, because if you are a fan then it is inherently right, but if you're an opponent then it is inherently wrong... It is not a neutral statement.
Further just because one has a desire does not define what the desire is for. One might have a desire, but it might be for someone else, the well-being of a friend maybe, and just because one might derive pleasure from providing such a service to someone else does not mean the pleasure being derived was the object of the desire rather just a consequence of it.
The Argument from Expected Benefit
1. Whenever you do something, you expect to be better off as a result.
2. If you expect to be better off as a result of your actions, then you are aiming to promote your self-interest.
3. Therefore, whenever you do something, you are aiming to promote your self-interest.
Premise one may or may not be true (helping someone who fell down even if it means you're late for an important meeting?), but moving on premise two is highly implausible. This is because it seems to be begging the question again, volunteering at homeless shelters might gain something from the activity, but this does not mean they are acting from self-interest.
It is true we can never be sure of our motives however, but this does not mean the egoist must be correct. It seems there are counterexamples. The best way to support the premise is with:
"(G) Whenever you expect your action to result in X, then your aim is to get X" except this is false, a lecturer might have several people fall asleep in class but this does not mean that is their aim.
The Argument from Avoiding Misery
1. If we would never do an action that promised only personal misery, then all of our actions are done in an effort to avoid such misery.
2. We would never do an action that promised only personal misery.
3. Therefore, all of our actions are done in an effort to avoid personal misery - and that is a self-interested motivation.
With premise two, consider a prisoner of conscience who allows themselves to be tortured. Each sacrifice might result in benefits, but this doesn't show egoism to be true. This is because premise one, which is support by the principle "(P) If I would never do an action that promised me only X, then I am always trying to avoid X." is suspect.
Even if we wouldn't do an action if it promised only misery, it doesn't follow that our motivations are always to avoid such misery. If one would never do an action to avoid certain death it doesn't mean that all actions done all the time are done in order to escape certain death.
The Guilty Conscience
Maybe someone does something "altruistic" because they have a guilty conscience? Well if someone were truly good then of course they will be troubled at the thought of doing wrong or doing nothing in the face of wrong. This does not prove their actions are motivated by a desire for a guilt-free conscience. Indeed if one cared not for others then one would never feel guilty about doing nothing...
Expanding the Realm of Self-Interest
Maybe one's well-being depends upon the well-being of others, so to help them is to really help oneself? Except if one's well-being improves after helping another this does not prove that the object of the help was in order to receive this personal gain in well-being. Further this support the idea of altruism not egoism, because if one didn't care about others then they would not suffer when thinking of the misery of others.
Letting the Evidence Decide
Either we let the evidence decide or we don't. If we don't let the evidence decide then egoism is hardly a rational conviction. If we do let the evidence decide then altruism must be true, because people report that the primary motivation of them helping others is out of altruism and not personal gain. Maybe the reports are deceived, either the reporter is lying or they are suffering from self-deception.
The evidence we rely on can be either testimony or behaviour, we can be told what motivates someone and we can see the behaviour they exhibit. But of course we can misinterpret behaviour and people can lie. But the problem for the egoist is they both point to altruism, and if we were to discredit all testimony or reinterpret all evidence of such behaviour then this is hardly an unbiased or rational approach.
