Despite remarkable progress in multi-modal AI research, there is a salient domain in which modern AI continues to lag considerably behind even human children: the reliable deployment of logical operators. Here, we examine three forms of logical operators: relations, negations, and discrete numbers. We asked human respondents (N=178 in total) to evaluate images generated by a state-of-the-art image-generating AI (DALL-E 3) prompted with these `logical probes', and find that none reliably produce human agreement scores greater than 50\%. The negation probes and numbers (beyond 3) fail most frequently. In a 4th experiment, we assess a `grounded diffusion' pipeline that leverages targeted prompt engineering and structured intermediate representations for greater compositional control, but find its performance is judged even worse than that of DALL-E 3 across prompts. To provide further clarity on potential sources of success and failure in these text-to-image systems, we supplement our 4 core experiments with multiple auxiliary analyses and schematic diagrams, directly quantifying, for example, the relationship between the N-gram frequency of relational prompts and the average match to generated images; the success rates for 3 different prompt modification strategies in the rendering of negation prompts; and the scalar variability / ratio dependence (`approximate numeracy') of prompts involving integers. We conclude by discussing the limitations inherent to `grounded' multimodal learning systems whose grounding relies heavily on vector-based semantics (e.g. DALL-E 3), or under-specified syntactical constraints (e.g. `grounded diffusion'), and propose minimal modifications (inspired by development, based in imagery) that could help to bridge the lingering compositional gap between scale and structure. All data and code is available at https://github.com/ColinConwell/T2I-Probology