The sentence; -I am also Jewish,- is not grammatically incorrect, but it is syntactically ambiguous in that, when the sentence syands by itself, -one does not know whether the logical connector; -also, -refers to the subject I, -or to some embedded additional object noun, (such as Buddhist, or biped, for example.)
And yet, the embedded ambiguity is the truth, and the essence of the problem, I think.