Suppes' Theorem For Probability Logic

We develop finitely additive probability logic and prove a theorem of Patrick Suppes that asserts that $\mathrm{\Psi }⊢\varphi$ in classical propositional logic if and only if $(\sum \psi \leftarrow \mathrm{\Psi }.1-\mathcal{P}\psi )\ge 1-\mathcal{P}\varphi$ holds for all probabilities $\mathcal{P}$. We also provide a novel dual form of Suppes' Theorem, which holds that $(\sum \varphi \leftarrow \mathrm{\Phi }.\mathcal{P}\varphi )\le \mathcal{P}\psi$ for all probabilities $\mathcal{P}$ if and only $(\bigvee \mathrm{\Phi })⊢\psi$ and all of the formulae in $\mathrm{\Phi }$ are logically exclusive from one another. Our proofs use Maximally Consistent Sets, and as a consequence, we obtain two collapse theorems. In particular, we show $(\sum \varphi \leftarrow \mathrm{\Phi }.\mathcal{P}\varphi )\ge \mathcal{P}\psi$ holds for all probabilities $\mathcal{P}$ if and only if $(\sum \varphi \leftarrow \mathrm{\Phi }.\delta \varphi )\ge \delta \psi$ holds for all binary-valued probabilities $\delta$, along with the dual assertion that $(\sum \varphi \leftarrow \mathrm{\Phi }.\mathcal{P}\varphi )\le \mathcal{P}\psi$ holds for all probabilities $\mathcal{P}$ if and only if $(\sum \varphi \leftarrow \mathrm{\Phi }.\delta \varphi )\le \delta \psi$ holds for all binary-valued probabilities $\delta$.