# A Counterexample to Theorems of Cox and Fine

```@inproceedings{Halpern1996ACT,
title={A Counterexample to Theorems of Cox and Fine},
author={Joseph Y. Halpern},
booktitle={AAAI/IAAI, Vol. 2},
year={1996}
}```
• Joseph Y. Halpern
• Published in AAAI/IAAI, Vol. 2 4 August 1996
• Mathematics, Computer Science
Cox's well-known theorem justifying the use of probability is shown not to hold in finite domains. The counterexample also suggests that Cox's assumptions are insufficient to prove the result even in infinite domains. The same counterexample is used to disprove a result of Fine on comparative conditional probability.
106 Citations
Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural.
On the Correctness and Reasonableness of Cox's Theorem for Finite Domains
• Paul Snow
• Mathematics, Computer Science
Comput. Intell.
• 1998
Although Halpern questioned whether the assumption in Cox's Theorem is reasonable for finite sets of sentences, it supports features that distinguish Cox's work from other, more restrictive motivations of probabilism.
Cox's Theorem Revisited
Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural.
The Cox Theorem: Unknowns And Plausible Value
• Mathematics
• 2006
We give a proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of
The Disappearance of Equation Fifteen , a Richard Cox Mystery
Halpern has retracted an earlier claim that Cox’s Theorem is deductively unsound, but he has renewed and amplified his objections to the reasonableness of the theorem for finite domains. His new
Bridging the intuition gap in Cox's theorem: A Jaynesian argument for universality
• Computer Science, Mathematics
Int. J. Approx. Reason.
• 2017
This work formalizes an invariance principle implicit in the work of Jaynes and uses it to construct a class of elementary examples and derives conditions as Cox's theorem and eliminates the need for ad hoc assumptions.
The assumptions needed to prove Cox's Theorem are discussed and examined. Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and,
Associativity and normative credal probability
• Paul Snow
• Computer Science, Medicine
IEEE Trans. Syst. Man Cybern. Part B
• 2002
A discrete counterpart of Cox's Theorem can be assembled from results that have been in the literature since 1959, and so may be more transparently applied to finite domains and discrete beliefs.
The philosophical significance of Cox's theorem
• M. Colyvan
• Computer Science, Mathematics
Int. J. Approx. Reason.
• 2004
The logical assumptions of Cox's theorem are examined and it is shown how these impinge on the philosophical conclusions thought to be supported by the theorem.
Constructing a logic of plausible inference: a guide to Cox's theorem
• K. Horn
• Mathematics, Computer Science
Int. J. Approx. Reason.
• 2003
The theorem states that any system for plausible reasoning that satisfies certain qualitative requirements intended to ensure consistency with classical deductive logic and correspondence with commonsense reasoning is isomorphic to probability theory.

## References

SHOWING 1-10 OF 26 REFERENCES
A summary of a new normative theory of probabilistic logic
A new axiomatization of a probabilistic logic theory is presented which avoids a finite additivity axiom, yet which retains many useful inference rules.
An axiomatic framework for belief updates
It is shown that belief updates in a probabilistic context must be equal to some monotonic transformation of a likelihood ratio.
Theories of Probability
A theory of probability will be taken to be an axiom system that probabilities must satisfy, together with rules for constructing and interpreting probabilities. A person using the theory will
A Framework for Comparing Alternative Formalisms for Plausible Reasoning
• Computer Science
AAAI
• 1986
It is demonstrated that the logical relationship can facilitate the identification of differences among alternative plausible reasoning methodologies and make use of the relationship to examine popular non-probabilistic strategies.
The logical view of conditioning and its application to possibility and evidence theories
• Computer Science, Mathematics
Int. J. Approx. Reason.
• 1990
This paper is both a survey of works pertaining to the introduction of conditioning relations in logic and a discussion about how these conditioning relations leave some room for non-monotonicity and might be useful in formalizing the concept of production rules in expert systems.
Theories of Probability
My title is intended to recall Terence Fine's excellent survey, Theories of Probability [1973]. I shall consider some developments that have occurred in the intervening years, and try to place some
An inquiry into computer understanding
This paper shows that the difficulties McDermott described are a result of insisting on using logic as the language of commonsense reasoning, and if (Bayesian) probability is used, none of the technical difficulties found in using logic arise.
The Uncertain Reasoner's Companion: A Mathematical Perspective
Introduction 1. Motivation 2. Belief as probability 3. Justifying belief as probability 4. Dempster-Shafer belief 5. Truth-functional belief 6. Inference processes 7. Principles of uncertain
Plausibility Measures: A User's Guide
• Mathematics, Computer Science
UAI
• 1995
This paper examines a new approach to modeling uncertainty based on plausibility measures, where a plausibility measure just associates with an event its plausibility, an element is some partially ordered set, and examines their "algebraic properties", analogues to the use of + and × in probability theory.
Modeling Belief in Dynamic Systems, Part I: Foundations
• Mathematics, Computer Science
Artif. Intell.
• 1997
A framework in which knowledge, plausibility, and time are talked about, which extends the framework of Halpern and Fagin for modeling knowledge in multi-agent systems and examines the problem of “minimal change”.