Certain objects essential to the act of divination by which in early times the Hebrews sought to learn the will of God. They seem to have opposite meanings, as "yes" and "no," "light" and "darkness." Scholars are not agreed as to the exact meaning of either word (1 Sam. 28:6). The Urim and Thummim were borne on the breastplate of judgement (Ex. 28:30). It is probable that they were in some way connected with the sacred lot (Lev. 8:8; Num. 27:21; Deut. 33:8; 1 Sam. 14:41). From Ezra 2:63; Neh. 7:65, it seems that they were in disuse at that time.
Some have suggested that the Urim and Thummim were white and black stones, respectively - the white meaning 'yes' and the black meaning 'no.' Neither stone was likely black and neither stone was likely white. Both stones were most likely made of onyx or agate, and so would look more like a cameo. If one were ground so only the white was left (if that's even possible without making the stone unusably thin) the other might have been similarly processed to be reddish but not black.
While any dispute was probably stated in Boolean algebra, some have suggested that questions could have been more complex. It is possible that some decision making significance was placed on the six tribes inscribed on either stone (twelve names in all). Nobody has ever given anything beyond speculation on this subject, and it not likely that anything conclusive is possible given the information we have.
For example, it is possible that the high priest selected the stone with the disputant's tribe inscribed on it and somehow discerned an answer, not necessarily even Boolean, from that stone alone. We're talking about cases in which the legal arguments from the Torah, etc.. have been exhausted without a solution. Some combination of the tribe and the gem selected, in a still unknown process (throwing in the air and catching, for example), - might have tipped the scale of the argument one way or the other. There are myriads of such speculations.