### Abstract

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 $\delta$.

### License

BSD License

### Topics

- Logic/General logic/Classical propositional logic
- Mathematics/Probability theory

### Related publications

- Suppes, P. (1966). Probabilistic Inference and the Concept of Total Evidence. Studies in Logic and the Foundations of Mathematics, 49–65. https://doi.org/10.1016/s0049-237x(08)71662-8

### Session Suppes_Theorem

- Probability_Logic
- Suppes_Theorem

### Depends on

- Class-based Classical Propositional Logic

### Used by

- A Sound and Complete Calculus for Probability Inequalities

### Auto-related entries

- A Sound and Complete Calculus for Probability Inequalities
- Propositional Proof Systems
- Abstract Completeness

## Cite

×

Suppes_Theorem-AFP

    @article{Suppes_Theorem-AFP,
      author  = {Matthew Doty},
      title   = {Suppes' Theorem For Probability Logic},
      journal = {Archive of Formal Proofs},
      month   = {January},
      year    = {2023},
      note    = {\url{https://isa-afp.org/entries/Suppes_Theorem.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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