Abstract
In our 2007 paper David and I studied consequence relations that correspond to conditional probability functions above thresholds, the probabilistic consequence relations. We showed that system O is a probabilistically sound system of Horn rules for the probabilistic consequence, and we conjectured that O might also provide a complete axiomatization of the set of finite premised Horn rules for probabilistic consequence relations. In a 2009 paper Paris and Simmonds provided a mathematically complex way to characterize all of the sound finite-premised Horn rules for the probabilistic consequence relations, and they established that the rules derivable from system O are insufficient. In this paper I provide a brief accounts of system O and the probabilistic consequence relations. I then show that O together with the probabilistically sound (Non-Horn) rule known as Negation Rationality implies an additional systematic collection of sound Horn rules for probabilistic consequence relations. I call O together with these new Horn rules ‘O+’. Whether O+ is enough to capture all probabilistically sound finite premised Horn rules remains an open question.
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Notes
- 1.
That is, \(p\) satisfies the usual classical probability axioms on sentence of a language for sentential logic: (1) \(p(a) \ge 0\), (2) if \({\vert }\!\!- a\) (i.e. if a is a tautology), then \(p(a) = 1\), (3) if \({\vert }\!\!- \lnot (a\wedge b)\), then \(p(a\vee b) = p(a) + p(b)\); and conditional probability is defined as \(p(a~{\vert }~b) = p(a\wedge b)/p(b)\) whenever \(p(b) > 0\). All of the other usual probabilistic rules follow from these axioms.
- 2.
- 3.
That is, any set of sound rules for ProbCRs that are in Horn rule form will be satisfied by some relations \({\vert }\!\!\sim \) on all pairs of sentences that are not in ProbCRs. A rule is in Horn rule form just when it is of form, “if \(a_{1} {\vert }\!\!\sim x_{1}, \ldots , a_{n} {\vert }\!\!\sim x_{n}\) , then \(b {\vert }\!\!\sim y\)” (with at most a finite number of premise conditions of form \(a_{1} {\vert }\!\!\sim x_{1}, \ldots , a_{n} {\vert }\!\!\sim x_{n})\), and perhaps also containing side conditions about logical entailments among sentences.
References
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Hawthorne, J. (2014). New Horn Rules for Probabilistic Consequence: Is \(\mathrm{O}{+}\) Enough?. In: Hansson, S. (eds) David Makinson on Classical Methods for Non-Classical Problems. Outstanding Contributions to Logic, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7759-0_9
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