Deduction Theorem: The Problematic Nature of Common Practice in Game Theory
We consider the Deduction Theorem that is used in the literature of game theory to run a purported proof by contradiction. In the context of game theory, it is stated that if we have a proof of $\phi \vdash \varphi$, then we also have a proof of $\phi \Rightarrow \varphi$. Hence, the proof of $\phi \Rightarrow \varphi$ is deduced from a previous known statement. However, we argue that one has to manage to prove that the clauses $\phi$ and $\varphi$ exist, i.e., they are known true statements in order to establish that $\phi \vdash \varphi$ is provable, and that therefore $\phi \Rightarrow \varphi$ is provable as well. Thus, we are only allowed to reason with known true statements, i.e., we are not allowed to assume that $\phi$ or $\varphi$ exist. Doing so, leads immediately to a wrong conclusion. Apart from this, we stress to other facts why the Deduction Theorem is not applicable to run a proof by contradiction. Finally, we present an example from industrial cooperation where the Deduction Theorem is not correctly applied with the consequence that the obtained result contradicts the well-known aggregation issue.