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When multiple knowledge statements linked with "OR" logic have to be proved and verified, the existing solution is the proof technique by Cramer et al , in which multiple proof and verification protocols are performed in parallel. It is pointed out in this paper that the highly strong soundness in Cramer et al's solution is only theoretically valuable and not needed in many applications, which suffer from its low efficiency. So batch proof and verification, a cryptographic tool to improve efficiency by reducing strength of soundness with only theoretic value to a practical level, is employed to modify Cramer et al's solution. More precisely, the multiple instances of proof and verification in Cramer et al's solution are batched in this paper. As existing batch ZK (zero knowledge) proof and verification techniques can only deal with proof and verification of statements linked with AND logic, a new batch proof and verification technique called batch ZK proof and verification of OR logic is designed to overcome this limit. It is the first batch cryptographic technique to handle multiple statements linked with OR logic. This new technique employs only one single modified proof and verification protocol to prove and verify multiple knowledge statements linked with OR logic, so greatly improves efficiency of the proof and verification technique by Cramer et al . It is the most efficient solution to the question raised by Cramer et al . Strength of soundness of the new proof and verification protocol is reduced to a level still strong enough for most practical applications. Namely, an appropriate trade-off is made between soundness and efficiency to design a more practical solution to ZK proof and verification of OR Logic. The new scheme is analysed in a security model called van Σ protocol, an extension of Σ protocol used in the technique by Cramer et al . The analysis illustrates that satisfactory security and performance are achieved in the new scheme.