The oracle works with bit strings all of the same length. Each bit position represents a single cocktail.

Each ingredient card is a bit string. Each bit is 1 (opaque) if the ingredient is needed for the cocktail at that bit position.

You take all the ingredients you have and discard their bitstrings ("turn down their cards"). Then you or together the remaining bitstrings of all of the ingredients you don't have (look through their overlapping cards). In other words, not having an ingredient forbids certain cocktails. Any bit positions that remain zero (transparent all the way through to the key card on the back showing the names of the cocktails at each position) are cocktails that you have enough ingredients to make.

Using opacity to represent bitwise "or" is pretty clever. Turning down the ingredient cards you don't have is really clever. I kind of want to make one of these now.