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This paper is devoted to the development of procedures for synthesizing threshold-logic elements which can be employed in redundant networks based on the von Neumann multiplexing principle. A major result is the statement of rules for synthesizing elements whose outputs are correct despite the presence of e or fewer errors on the input lines. Several theorems are proved and used as the basis for establishing synthesis rules for four canonical redundant threshold element specifications. Some of these canonical elements are optimum in the sense that they require the minimum sum of weights or minimum sum of input bundle sizes.