Neural Tensor Networks (NTNs), which are structured to encode the degree of relationship among pairs of entities, are used in Logic Tensor Networks (LTNs) to facilitate Statistical Relational Learning (SRL) in first-order logic. In this paper, we propose Randomly Weighted Tensor Networks (RWTNs), which incorporate randomly drawn, untrained tensors into an NTN encoder network with a trained decoder network. We show that RWTNs meet or surpass the performance of traditionally trained LTNs for Semantic Image Interpretation (SII) tasks that have been used as a representative example of how LTNs utilize reasoning over first-order logic to exceed the performance of solely data-driven methods. We demonstrate that RWTNs outperform LTNs for the detection of the relevant part-of relations between objects, and we show that RWTNs can achieve similar performance as LTNs for object classification while using fewer parameters for learning. Furthermore, we demonstrate that because the randomized weights do not depend on the data, several decoder networks can share a single NTN, giving RWTNs a unique economy of spatial scale for simultaneous classification tasks.