How useful is intuition according to Torah philosophy?

Our Gemara on Amud Aleph discusses a dispute about two property deeds of sale both with the same date. In such situations where the law is in gridlock, that is there is no reasonable way to resolve a litigation, the Beis Din responds in various ways, depending on factors that we will not discuss in depth right now. The judges may declare that the assets in question be divided equally, which on the one hand is fair but also assures that actual justice, as in the rightful owner getting his due, will not happen. In other cases, the judges might withdraw and say, whoever is stronger should win, כל דאלים גבר. Or, the judges might make a more arbitrary decision, with little evidence, and declare one party the winner. This is known as Shudah Dedayyni. 

There is a discussion amongst the Rishonim as to what is the process for the judges when enacting Shuda Dedayyni. According to Rashi, the judges should base it on some sense of what might be justice, such as hints and impressions about the intent of the person who was selling or gifting the item (see Rashi Kesuvos 85a “Shuda”). However, Tosafos (see Gittin 14b “ukahn”) holds that the judges do not even try to figure out what the original intentions of the party were, instead the judges have the freedom to simply choose whom to give it to.

The responses of Kol D’alim Gavar and Shuda seem to be irresponsible actions on the part of the judiciary. We must understand that this question comes from a perspective that Beis Din is the final and sole word of law. The Jewish court more rightfully sees itself as agents of God, when so authorized. When not authorized, justice goes back to God. Thus, when there is no hope of discerning the claim empirically, the judges trust that somehow God will bring justice about.

Likutei Halachos (Choshen Mishpat 4:2-3) develops this idea further. He says, all possessions are by the will of God and therefore there is a natural, intrinsic force that draws any object to its owner. Therefore, whether via means of a contest of wills or even the intuition of the judges, the draw of the object to its rightful owner can be subtly perceived. (Though he does not make reference to this, perhaps the idea of the apportioning of the land of Israel via lots are based on a similar spiritual dynamic, see Bamidbar 26:56.) What other truths might be subtly perceived if we tuned in?