accolade comments on Trying to Try - Less Wrong
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Assuming freedom of will in the first place, why should you not be able to choose to try harder? Doesn't that just mean allocating more effort to the activity at hand?
Did you mean to ask "Can you choose to do better than your best?" ? That would indeed seem similar to the doubtable idea of selecting beliefs arbitrarily. By definition of "best", you can not do better than it. But that can be 'circumvented' by introducing different points in time: Let's say at t=1 your muscle capacity enables you to lift up to 10 kg. You can not actually choose to lift more. You can try, but would fail. But you can choose to do weight training, with the effect that until t=2 you have raised your lifting power to 20 kg. So you can do better (at t=2) than your best (at t=1).
But Eliezer's point was a different one, to my understanding: He suggested that when you say (and more or less believe) that you "try your best", you are wrong automatically. (But only lying to the extent of your awareness of this wrongness.) Because you do better when setting out to "succeed" instead of to "try"; because these different mindsets influence your chances of success.
About belief choice: Believing is not a simply choosable action like any other. But I can imagine ways to alter one's own beliefs (indirectly), at least in theory:
Self-fulfilling prophecy effects, when you are aware of them, create cases where you may be able to select your belief. Quoting Henry Ford:
If you believe this quote, then you can select whether to believe in yourself, since you know you will be right either way.
(More examples of manipulating one's own beliefs, there in the form of "expectancy", can be found under "Optimizing Optimism" in How to Beat Procrastination. You can also Google "change beliefs" for self-help approaches to the question. Beware of pseudoscience, though.)