What is Buddhist doctrine on the question of counterfactual definiteness? Counterfactual definiteness is, "is the ability to speak "meaningfully" of the definiteness of the results of measurements that have not been performed."

The classic question to illustrate is, "When a tree falls in a lonely forest, and no animal is near by to hear it, does it make a sound? Why?"

Another classic example is Einstein asking Bohr whether he really believed that, "the moon does not exist if nobody is looking at it?"

What does Buddhism have to say about these questions if anything at all? 

The standard procedure for understanding anatta is to investigate and look for the self and upon not finding it concluding that the self does not exist as we imagined. However, doesn't this procedure implicitly rely upon the supposition that existing things can be found if one investigates and looks for them? That which can't be observed must not truly exist?

Couldn't someone relying upon counterfactual definiteness just say that even though we can't observe the self that it still truly exists just like an unobserved tree in the forest that falls still makes a real sound?
What does Buddhism have to say about these questions if anything at all?

That's easy to answer: the only good use of mano ("mind") for anybody who want to stop dukkha is the yoniso manasikara ("appropriate attention"), which is, in one word, phrasing sati (awareness, mindfulness) and what you do with sati, in terms of mano. This is the first step of the path (after, of course, hearing the doctrine to remember) -- the famous "proper attention" on the source and ending of whatever is dukkha, from the reply "you need a good friend and proper attention" of the buddha, when he was asked what is required to reach nibanna.

Also, The standard procedure for understanding anatta has nothing to do with "investigate and look for the self and upon not finding it concluding that the self does not exist as we imagined." Leave drawing conclusion to the toxic puthujjanas (wordlings): the logicians, the daydreamers, the creators of views, the people who love to argue about views.

Use mano only for ascertainment, for attestation, for witnessing, of the source and fall of phenomena, of vedana (feelings), of sanna (perceptions) and so on -- and of course, for the ascertainment of the failure or the success, from the activities you have done so far, of your practice (then you pursue successful activities and you abandon failing activities).

Here is what happens when a puthujjana uses mano for something else than yoniso manasikara:


  Ayoniso-manasikara Sutta: Inappropriate Attention (SN 9.11)
  
  I have heard that on one occasion a certain monk was dwelling among the Kosalans in a forest thicket. Now at that time, he spent the day's abiding thinking evil, unskillful thoughts: i.e., thoughts of sensuality, thoughts of ill will, thoughts of doing harm.
  
  Then the devata inhabiting the forest thicket, feeling sympathy for the monk, desiring his benefit, desiring to bring him to his senses, approached him and addressed him with this verse:
  
  From inappropriate attention
  you're being chewed by your thoughts.
  Relinquishing what's inappropriate,
      contemplate
      appropriately.
  
  Keeping your mind on the Teacher,
  the Dhamma, the Sangha, your virtues
  you will arrive at
      joy,
      rapture,
      pleasure
      without doubt.
  
  Then, saturated
      with joy,
  you will put an end
  to suffering &amp; stress.
  
  The monk, chastened by the devata, came to his senses.


Here is the proper use of mano:


Natumha Sutta -- Not Yours (SN 12.37)
Channa Sutta -- With Channa (SN 35.87)


Use mano for striving to find the sources of what comes to be, and the ending of what came to be -- of how things arise and how they pass away (and not "see things as they really are", contrary to the worst translation ever, created by a puthujjana craving to cram the word "real").

Now, tHe standard procedure to "understand anatta" is the one to become an arahant -- and the standard procedure to become an arahant is for the citta to have samadhi, and once the citta has stable samadhi, to use sati to focus on anatta through the sequence anicca, dukkha, anatta -- or to just hear the discourse by a Buddha which will be about this sequence anyway.

Since the bikkhus  do not know what is anicca, but are already more or less good at getting the citta in samadhi by being good ascetics, they once more rely on a discourse, this time directly from the Buddha, like the second one (Anatta-lakkhana-sutta -- The Not-self Characteristic) where as usual the five aggregates that any human knows and experiences are anatta -- so no need to go deep looking for anatta.

Now atta is  only a word, so what experience is there behind this word?
Well as usual the answer is given, and it is:


  This is mine, this is I, this is my self


... and what does this apply to? The 5 aggregates -- which is the sakya-ditthi (i.e. the view that the aggregates are self) and the only ditthi (view) that you must care about and reject, once you want to stop dukkha --- which of course has nothing do with eternalism or annihilism, and the fantasy of some puthujjanas that the dhamma is some middle way between the two, because those people love to (very dubiously) tack their favorite  words like "existence", "real", "self" on their experience and these words are even more meaningless when they are used to talk about the Buddha.

The (non-ascetic) lay people who stop being puthujjanas do so by listening to the Buddha, and they become stream-enterers. After that the remaining work no longer concerns views. Instead the remaining task is to destroy various lusts, and the energy spent fueling those lusts. A puthujjana is done with views as as soon as the puthujjana become a sotapana.