Optical logical operations demonstrate the key role of optical digital computing, which can perform general-purpose calculations and possess fast processing speed, low crosstalk, and high throughput. The logic states usually refer to linear momentums that are distinguished by intensity distributions, which blur the discrimination boundary and limit its sustainable applications. Here, we introduce orbital angular momentum (OAM) mode logical operations performed by optical diffractive neural networks (ODNNs). Using the OAM mode as a logic state not only can improve the parallel processing ability but also enhance the logic distinction and robustness of logical gates owing to the mode infinity and orthogonality. ODNN combining scalar diffraction theory and deep learning technology is designed to independently manipulate the mode and spatial position of multiple OAM modes, which allows for complex multilight modulation functions to respond to logic inputs. We show that few-layer ODNNs successfully implement the logical operations of AND, OR, NOT, NAND, and NOR in simulations. The logic units of XNOR and XOR are obtained by cascading the basic logical gates of AND, OR, and NOT, which can further constitute logical half-adder gates. Our demonstrations may provide a new avenue for optical logical operations and are expected to promote the practical application of optical digital computing.