Proof by contradiction in Game Theory: is this paper correct? I have doubts whether this
paper is sound and correct in its very basic setting. It discusses the relationship between $\varphi\vdash\psi$ and $\varphi\Rightarrow\psi$ and denies it in general, i.e. whenever $\varphi$ and $\psi$ might not be true. The paper goes a bit beyond what I (and as the author points out, many other researchers) generally trust (and use) about logic in connection with game theory. I'm also interested in a summary on what it really says beyond what I have just put in the body of this question.
I'm interested this in connection with the Maschler's bargaining set $\cal M$. It is a set defined by boolean combinations of linear inequalities, and I believe it goes like this: $x\in \cal M$ if there is no justified objection at $x$. I have obtained by this reasoning $x\in{\cal M} \Longleftrightarrow x\in X \land A\Longrightarrow B$ where the statement $A$ says that there is a objection at $x$ and $B$ that there is a counterobjection at $x$.
Now the author claims this is not sound and that it should read $x\in{\cal M} \Longleftrightarrow x\in X \land A\land B.$ This is very weird for me, anyone agrees with the author ?
 A: The paper in question looks quite confused to me.
A: By virtue of the authority granted to them by StackExchange, some members declare the Discussion Paper MPRA Paper 66637 as "confusing" or "utter nonsense" without answering the underlying question of how one should conduct a clean proof by contradiction based on the Deduction Theorem when the provability or deducibility of the clauses $\phi$ and $\varphi$ are assumed in order to establish that $\phi \vdash \varphi$ is provable/deducible. How can one assure under these circumstances that $\phi$ and $\varphi$ are consistent and logically independent clauses, and that this leads to a correct logical conclusion within a model? How can we avoid the Principle of Explosion also subsumed under "ex falso sequitur quodlibet",~i.e., from falsehood anything follows? I am wondering what is on this line of reasoning "confusing" or "utter nonsense"?
The answer of @Andreas Blass as well as the comments fail to explain these crucial issues. I invite those members to provide the answer. As long as they could not succeed, I regard my objections against this proof technique as not refuted.
