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The category of "fireman" does not describe a divine trinity of persons.

Neither does "God" in "Mother of God."

Sure it does. Without adding the full range of confessional qualifiers to the statement, every time it is made, any trinitarian listener is going to understand the term as comprehensive. It's automatic. That's why you make an unqualified statement, to import the whole thing.

Furthermore, when Emily begets Sam, she is his mother in the generative sense. She and the father are both necessary causes for Sam to exist. But for Emily, Sam would not exist. The statement, "But for Mary, God would not exist," is patently false. If you cannot rearrange the terms in both syllogisms in the same way and get the exact same result, the analogy is defective. This comparison between two syllogisms breaks down because the terms are not describing the same category of being.

Peace,

SR