On the Logic of High Probability |
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Adams (Ernest) |

Source: Journal of Philosophical Logic, Vol. 15, No. 3 (Aug., 1986), pp. 255-279 |

Paper - Abstract |

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__Author’s Introduction__

- What we are concerned with are vague logical propositions such as might be expressed in English by saying "It is not possible for both 'it will rain' and 'it will not rain' to be highly probable, though it is possible for neither of them to be." Vague as such claims are, there are a number of reasons for taking an interest in them. One is the practical reason that high probabilities are often the best we can expect 'in the circumstances' i.e., assuming that David Hume was right, as concerns future contingencies. We would like to be able to study everyday life reasoning that is based on premises of this sort, which are not 'given' as certainties but only as 'reasonable probabilities'. Furthermore, it is argued in Adams (1983) that important types of such reasoning are
*enthymematic*in the sense that they tacitly assume not that certain things 'are the case', but rather that certain things 'aren't too improbable', and we would like to be able to study this systematically. Finally, there is a curious coincidence that requires explanation. One of David Lewis's calculi of counterfactual conditionals ("Lewis (David) - Counterfactuals", 1973) yields precisely the same logic of truth for disjunctions^{1}of conditionals as the present theory yields for the logic of their high probabilities. We wish to inquire into the significance of the apparent intersubstitutability of truth and high probability in the two theories. - The remaining sections deal with the following topics.
- Section 2 sets forth the formal language of the logic of high probability, which is based on a relational 'metalinguistic high probability predicate', H(x,y), which is intended to express the fact that the ordinary language conditional with antecedent x and consequent y is highly probable. This section also formulates a system of four axioms and two axiom schemas for the theory of this relation.
- Section 3 states the first two of four 'meta-metatheoretic Hauptsatze' of the theory, and comments on their application to probabilistic enthymemes. These theorems state fundamental inequalities involving probabilities applying to the terms that are related by H(x, y), first when they enter into theorems of the theory, and second when they enter into non-theorems. These inequalities provide the rationale for calling our system a logic of high probability.
- Section 4 states the third meta-metatheorem, which gives a kind of truth-table decision method for determining theoremhood for purely universal sentences in the language of the theory. This section also discusses corollaries to the decision method which show that in special cases the non-material conditional behaves like the material conditional, although it does not do so always and in a certain sense it is not reducible to the unconditional.
- Section 5 will state and discuss the significance of our final key meta-metatheorem which shows the equivalence, in a certain sense, of our present theory and one of Lewis
^{2}'s systems of counterfactual logic. The status of the Law of the Conditional Excluded Middle will be important in our consideration of the significance of this equivalence.

- Most of the mathematical propositions to be discussed in this paper are easy consequence of related results already published. An Appendix cites the relevant results and comments on how they may be adapted to the present study, though detailed proofs will not be given.

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