Deduction Theorem: The Problematic Nature of Common Practice in Game Theory

This paper is almost entirely incomprehensible to me, and I have a background in formal logic. It might be that there is some error in the proofs that this author is referring to, but the paper does not make any clear point about what the problem is. There is a lot of unusual use of language (talk of propositional formulas "existing" and things like only being allowed to make deductions where the formulas are "previously known to be true").

Even in the introduction there are statements which do not make sense, like the statement that you are required to show that A and B are true in order to conclude that A logically entails B. This is simply not the case, since it is very well possible for A to be a formula which is always false (unsatisfiable/inconsistent), in which case A entails ANY statement, including B.