Two layer neural networks are similar to traditional decision making techniques. In the following passage, Franklin describes a method of multiplying weights by individual positive and negative factors, and using a threshold to determine the decision. (What is missing from this method is a formal way of adjusting the weights depending on how well they work for past decisions.)
Some theoretical justification for such methods is given in the marvelously titled 1979 paper "The robust beauty of improper linear models in decision making" by Robyn Dawes.
I'm curious to see if any divination techniques were, under the surface, actually performing such a weighing of pros and cons. I haven't come across any yet, but it doesn't seem that large a leap to make; the mathematical complexity of astrology or geomancy was greater than this.
"In the affair of so much importance to you, wherein you ask my advice, I cannot for want of sufficient premises, advise you what to determine, but if you please I will tell you how. When those difficult cases occur, they are difficult, chiefly because while we have them under consideration, all the reasons pro and con are not present to the mind at the same time, but sometimes one set present themselves, and at other times another, the first being out of sight. Hence the various purposes or inclinations that alternatively prevail, and the uncertainty that perplexes us.
To get over this, my way is to divide half a sheet of paper by a line into two columns; writing over the one Pro, and over the other Con. Then, during three or four days of consideration, I put down under the different heads short hints of the different motives, that at different times occur to me, for or against the measure.
When I have thus got them all together in one view, I endeavor to estimate their respective weights; and where I find two, one on each side, that seem equal, I strike them both out. If I find a reason pro equal to some two reasons con, I strike out the three . . . and thus proceeding I find at length where the balance lies; and if, after a day or two of further consideration, nothing new that is of importance occurs on either side, I come to a determination accordingly.
And, though the weight of reasons cannot be taken with the precision of algebraic quantities, yet when each is thus considered, separately and comparatively, and the whole lies before me, I think I can judge better, and am less liable to make a rash step, and in fact I have found great advantage from this kind of equation." -- Benjamin Franklin, letter to Joseph Priestly