Suppes' Theorem For Probability Logic

We develop finitely additive probability logic and prove a theorem of Patrick Suppes that asserts that $\Psi \vdash \phi$ in classical propositional logic if and only if $(\sum \psi \leftarrow \Psi.\; 1 - \mathcal{P} \psi) \geq 1 - \mathcal{P} \phi$ holds for all probabilities $\mathcal{P}$. We also provide a novel dual form of Suppes' Theorem, which holds that $(\sum \phi \leftarrow \Phi.\; \mathcal{P} \phi) \leq \mathcal{P} \psi$ for all probabilities $\mathcal{P}$ if and only $\left(\bigvee \Phi\right) \vdash \psi$ and all of the formulae in $\Phi$ 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 $(\sum \phi \leftarrow \Phi.\; \mathcal{P} \phi) \geq \mathcal{P} \psi$ holds for all probabilities $\mathcal{P}$ if and only if $(\sum \phi \leftarrow \Phi.\; \delta\; \phi) \geq \delta\; \psi$ holds for all binary-valued probabilities $\delta$, along with the dual assertion that $(\sum \phi \leftarrow \Phi. \;\mathcal{P} \phi) \leq \mathcal{P} \psi$ holds for all probabilities $\mathcal{P}$ if and only if $(\sum \phi \leftarrow \Phi.\; \delta\; \phi) \leq \delta\; \psi$ holds for all binary-valued probabilities $\delta$.