An analysis is presented of the emergence of implicational relations within associative memory systems. Implication is first formulated within the framework of Zadeh's theory of approximate reasoning. In this framework, implication is seen to be a fuzzy relation holding between linguistic variables, that is, variables taking linguistic terms (e.g., "young", "very old") as values. The conditional expressions that obtain from this formulation may be naturally cast in terms of vectors and matrices representing the membership functions of the fuzzy sets that, in turn, represent the various linguistic terms and fuzzy relations. The resulting linear algebraic equations are shown to directly correspond to those that specify the operation of certain distributed associative connectionist memory systems. In terms of this correspondence, implication as a fuzzy relation can be seen to arise within the associative memory by means of the operation of standard unsupervised learning procedures. That is, implication emerges as a simple and direct result of experience with instances of events over which the implicational relationship applies. This is illustrated with an example of emergent implication in a natural coarsely coded sensory system. The percepts implied by sensory inputs in this example are seen to exhibit properties that have, in fact, been observed in the system in nature. Thus, the approach appears to have promise for accounting for the induction of implicational structures in cognitive systems.
