It seems he's questioning the classic formal logic's source of validity when it comes to any kind of logical forms having a universal quantified physical premise with its conclusion, such as the universal proposition Everyday the sun rises in the east on earth. Just look out of your window, there's no such logic printed in the sky. So from what source can we be sure it was the case long ago when there were no living beings? And from what source can we be sure it'll continue to be so in future without exceptions? Maybe the author is even skeptical about modus tollens or do not see deductive inference rules as mere formal rules. Quoting from your above texts "How can one know what holds in all situations?", so once the author starts to doubt certain logical entailment validity applied elsewhere under deductive formal logic, eventually one will doubt any universal logical consequence therein which may lead to outright skepticism about deductive reasoning and formalism of math.
Graham Priest is known for his defense of dialetheism and non-classical logics such as paraconsistent logic and the non-being Asian metaphysical logic. So maybe he's stimulating his readers to think about these assumed undisputable contingent laws/logicformal logic and thus hints there're other types of logics in application. However without access to his full text, these are just what I can conceive of...