Large Language Models (LLMs) increasingly serve as reasoning components in modern software systems, yet a model that answers a logical query correctly can contradict itself when the same problem is restated in an equivalent form. Static benchmarks evaluate queries in isolation and therefore fail to expose such inconsistencies. To address this limitation, we present LGMT, an open-source tool that detects reasoning inconsistencies through metamorphic relations grounded in first-order logic. Given a source test case in premise-conclusion form, LGMT generates logically equivalent follow-up cases and checks whether the model's entailment predictions agree across the pair, without ground-truth labels. The tool provides 20 built-in metamorphic relations, supports custom ones verified by the Z3 theorem prover, and runs from a CLI or a web studio, including an offline mock mode requiring no API key. Across six widely used LLMs on a bundled benchmark of 3,091 metamorphic groups, LGMT reveals inconsistency rates ranging from 20.84% to 29.44%. LGMT is publicly available on GitHub at https://github.com/MaxwelsDonc/lgmt-tool, with a demonstration video at https://youtu.be/gou4opptu1U.

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