Deduction Theorem: The Problematic Nature of Common Practice in Game Theory
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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 ϕφ, then we also have a proof of ϕφ. Hence, the proof of ϕφ is deduced from a previous known statement. However, we argue that one has to manage to prove that the clauses ϕ and φ exist, i.e., they are known true statements in order to establish that ϕφ is provable, and that therefore ϕφ is provable as well. Thus, we are only allowed to reason with known true statements, i.e., we are not allowed to assume that ϕ or φ 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.
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