In this work, we present a new structure to realize optical logic operation in a Sagnac interferometer with electro-optical modulation. In the scheme, we divide two counterpropagation signals in a Sagnac loop to two different arms with the electro-optical crystal by using two circulators. Lithium niobate materials whose electro-optical coefficient can be as large as 32.2×10(-12) m/V make up the arms of the waveguides. Using the transfer matrix of the fiber coupler, we analyze the propagation of signals in this system and obtain the transmission characteristic curves and the extinction ratio. The results indicate that this optical switching has a high extinction ratio of about 60 dB and an ultrafast response time of 2.036 ns. In addition, the results reveal that the change of the dephasing between the two input signals and the modification of the modulation voltage added to the electro-optical crystal leads to the change of the extinction ratio. We also conclude that, in cases of the dephasing of two initial input signals Δφ=0, we can obtain the various logical operations, such as the logical operations D=A¯·B, D=A·B¯, C=A+B, and D=A⊕B in ports C and D of the system by adjusting the modulation voltage. When Δφ≠0, we obtain the arithmetic operations D=A+B, C=A⊕B, D=A·B¯, and C=A¯·B in ports C and D. This study is significant for the design of all optical networks by adjusting the modulation voltage.