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By brainwashed I mean indoctrinated with some political, religious or other type of propaganda.
I assume that you can't be aware of this, because the lack of that awareness is the point of propaganda - to make you believe that something is true/false etc. But there must be a logical way to determine if you are brainwashed. Like "your" arguments following certain patterns maybe?
As Schroedinger's Cat pointed out, there's no clear dividing line between "brainwashing" and the normal processes of growing up in or assimilating into a culture or social group. Even in the case of deliberate, forced procedures, it can be unclear -- for example, the US military has recruits do drills and engage in synchronized practices that we know to affect thoughts, feelings, and beliefs, but most people wouldn't call that brainwashing. However, you could look into military training practices, as well as techniques used by abusive cults, and see if you've been subjected to those at any point in your life. That might answer your question.
Another way would be to think about the criteria you would use to describe a thoroughly un-brainwashed person, and work from there. For example, do you believe that you should be able to cite a preponderance of evidence for all of your beliefs? That you should have a clear and volitional explanation for your life choices? You can then work toward those goals, questioning or changing your beliefs as necessary. Be aware that hardly anyone really meets these criteria! We all absorb cultural values and prejudices, use unconscious heuristics to make decisions, and make choices that we rationalize later. But you could certainly work on moving in a more rationalist direction if you want to.
I think there is a strong argument that everyone is brainwashed, in the sense of having accepted ideas and principles that they cannot provide a-priori justification for. It is something of the definition of a worldview, which we all need to survive.
Even if you take it to the perspective of "have we had our worldview changed by external sources", the answer is also yes, for many people. We have all had a moment of realisation that something we had read or heard is important and significant, and we change our understanding - slowly or quickly - towards an acceptance of that.
The only real definition of brainwashing is whether we have had our worldview changed against our will. That is a far more difficult question. Does watching TV constitute "against our will"? Or reading controversial things on the internet that challenge us?
It is a good question, but I think there needs to be much clearer definition before a clear answer can be arrived at.
This is just lay theorizing, but I would try and find a measure of my attitudes at a certain point in time (e.g., an old journal entry, a blog post, an essay) and compare the reasoning in that to current reasoning. If I identify huge changes in personal ideology between that old bit of material and current thinking, I should perhaps ask a third-party to attempt to objectively determine whether that discrepancy reflects a natural maturation process or something more perverse.