We investigated a possible use of the magnonic interferometric switches in multi-valued logic circuits. The switch is a three-terminal device consisting of two spin channels where input, control, and output signals are spin waves. Signal modulation is achieved via the interference between the source and gate spin waves. We report experimental data on a micrometer scale prototype based on the Y3Fe2(FeO4)3 structure. The output characteristics are measured at different angles of the bias magnetic field. The On/Off ratio of the prototype exceeds 13 dB at room temperature. Experimental data are complemented by the theoretical analysis and the results of micro magnetic simulations showing spin wave propagation in a micrometer size magnetic junction. We also present the results of numerical modeling illustrating the operation of a nanometer-size switch consisting of just 20 spins in the source-drain channel. The utilization of spin wave interference as a switching mechanism makes it possible to build nanometer-scale logic gates, and minimize energy per operation, which is limited only by the noise margin. The utilization of phase in addition to amplitude for information encoding offers an innovative route towards multi-state logic circuits. We describe possible implementation of the three-value logic circuits based on the magnonic interferometric switches. The advantages and shortcomings inherent in interferometric switches are also discussed.