Is the process of inferring facts using neural networks a unique form of reasoning? Is there really a different type of reasoning separate and distinct from deduction and induction? Does there exist a single fundamental form of inference for reasoning symbolically, qualitatively, quantitatively, possibilistically (about "fuzzy" concepts), and probabilistically? YES, it is called abduction. This paper presents abduction and abductory induction. Abduction not only classifies the distinct type of reasoning performed when neural networks are applied, but gives a logical framework for expanding current neural network research to include network concepts not constrained by neuron analogies. These networks are called abductive networks. In describing abductive networks, this paper unveils the true source of the "power" of networks of functional elements. A practical machine learning tool for synthesizing abductive networks from databases of examples, called the Abductory Induction Mechanism (AIMTM), is also presented.