Non Rigorous Reasoning Unjustified Assumptions Poorly Defined Terms Unsound Reasoning Invalid applications of formalisms

At least four types of non rigorous reasoning can be found in Penrose's two books. First, he makes unjustified assumptions. He frequently admits he is making an assumption, but then goes on to make deductions from it. Second, he uses terms in an idiosyncratic manner with no precise definitions, for example "genuine understanding". Again, Penrose admits his lack of precise definitions, but this still can lead to unsound reasoning. For example, he uses the word "awareness" in more than one sense. Third, there are several examples of unsound reasoning. For example, he will list four alternatives to some complex issue, when logically there are many more possibilities. Finally, he introduces all sorts of formalisms but then reasons with them in invalid ways. This is very reminiscent of Russell's paradox in set theory. Thus just as we now believe it is invalid to reason logically about certain sorts of infinite sets, the same applies to most of Penrose's arguments.