By the nature of the following refutation of determinism, it can also be considered a proof (1) of (libertarian-type) volition. It is a refutation of any theory that would deny the existence of non-necessitated events–in the form of fundamental, agent-controlled choices–located in a conceptual consciousness, including incompatibilist determinism and compatibilism.
Here is a short video (by someone else) explaining the general outlines of the argument:
Here is a link to the paper by Harry Binswanger:
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(1) The argument shows that volition is a prerequisite of all knowledge and that the denial of volitional choice implies self-contradiction, thus establishing volitional choice as an axiom of epistemology. Strictly speaking, I think, by Leonard Peikoff’s definitions, this is not a proof, but an axiomatic validation. It establishes the axiom as absolutely and unquestionably true, but does not involve the specific process of logical proof of the positive proposition. The positive proposition is self-evident in each individual’s experience of existence.
Objectivism In Depth 
I would recommend Micheal Mckenna’s reply to Pereboom who says we do not have free will. He argues that either conclusion (that we either have or do not have free will in the basic-desert sense)relies on one’s starting assumption. If true this doesn’t prove that we have free will, only that the debate has reached an impasse.
http://philpapers.org/rec/MCKAHR Here is the link to that paper by Mckenna in Philosophy and Phenomenological Research (2007)
Totally unconvincing. We are born with a deterministic drive to try to learn about the world because it has a survival advantage. We invent mental models and then compare them to what we observe. If we spot a contradiction/falsification, then we must modify our model. The more tests/observations a model passes, the more confidence we have in it and the more we feel it is true. No free will is necessary for any of this.
Of course free will is required. We DON’T have a deterministic drive to learn about the world: note the many people who choose not to learn, but rather to just watch TV and take it easy. How could we invent “mental models” without volition? The universe isn’t causing us to come up with mental models–otherwise the first caveperson would know how to create air conditioning and jumbo jets (because it would be an automatic process). Free will is required to CHOOSE to put forth EFFORT in making correct connections. It’s an active process that requires will and effort; it’s not a passive process.
The video you put in this article says “this video is unavailable” when played.
Yeah, the post is from 2012, and the video was deleted quite some time ago. Not much I can do about it.