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

We consider the Deduction Theorem 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 previously known statement. However, we argue that one has to manage to establish that a proof exists for the clauses $\phi$ and $\varphi$, i.e., they are known true statements in order to show that $\phi \vdash \varphi$ is provable, and that therefore $\phi \Rightarrow \varphi$ is provable as well. Thus, we are not allowed to assume that the clause $\phi$ or $\varphi$ is a true statement. This 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.

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