I might be being an idiot here, but I can't for the life of me figure out how the following inference is disallowed in the Barker-Plummer/Barwise/Etchemendy textbook:
1. Ex(Rax)
2. ExEx (Rxx) 1, E introduction
This is clearly invalid, because there are anti-symmetric relations (note that if it were valid one can prove ExEx (Rxx) from ExEy (Rxy), and that ExEx(Rxx) has the same truth conditions as Ex(Rxx)).
Intuitively, Existential Introduction should be restricted so that one cannot replace a name or eigenvariable with a variable that is already bound in the sentence. But I can't find this restriction in Barker-Plummer/Barwise/Etchemendy. My friend couldn't find it in the old red Mates book either, so we think it's not unlikely that we're both missing something obvious.
I turned to the soundness proof in Barker-Plummer/Barwise/Etchemendy and they leave the case of Existential Introduction as an excercise to the reader. It has a little star next to it showing that it is a difficult problem. I'm wondering if it's impossible. But, again, it's more likely that I'm missing something, so if anyone who teaches from the book could take a look my introductory logic students would appreciate it.
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