Abstract
Our objective in this section is to establish a prima facie case for the appropriateness of assessing the soundness or rationality of deductive inferences in terms of a new requirement or criterion of rationality beyond the usual truth-conditional criterion: that it should be impossible for the premises of an inference to be true while its conclusion is false. The proposed supplementary criterion results when the words ‘probable’ and ‘improbable’ are substituted for ‘true’ and ‘false’, respectively, in the truth-conditional criterion, yielding the probabilistic soundness criterion: it should be impossible for the premises of an inference to be probable while its conclusion is improbable. This formulation is vague and we shall want to clarify it later, but our present concern is with the legitimacy of demanding that deductive inferences satisfy something like this requirement if they are to be regarded as ‘rational’.
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Notes
Note the explicit temporal aspect of the situation described, where one ‘premise’ was known prior to the second’s being learned. Temporal aspects of reasoning will not be considered in Chapters 1—III, but will be taken up in Chapter IV, especially in IV.9 and IV.1o.
Entropic uncertainty (see, e.g. Khinchin [35]) will be only occasionally relevant to the concerns of this book (for instance as it can be used to describe the effects of information-acquisition and its ‘inverse’, probability mixing, as in Section 2), and this type of uncertainty is obviously very different from probability of falsity. In fact, maximum probability of falsity entails minimum entropic uncertainty.
Brian Ellis [17] and [18] has also independently developed a probabilistic theory of the logic of conditionals which in many respects parallels the present theory. A fundamental difference between Ellis’ approach and the present one is that he treats probability as a ‘concept of truth’. Limitations of space preclude a detailed comparison with the present theory.
Probably the most radical implication of the present approach is that we are no longer able to give a uniform ‘semantics’ for arbitrary iterations of compounding by conditionalization, or of forming other sentential compounds with conditional constituents (see Section 8, where these matters are discussed in some detail). Lewis [40] has taken this implication in particular as showing that the present approach makes too radical a departure from orthodox theory.
We will attempt so far as possible in this work to sidestep problems having to do with defining p (A ⇒ B) when p (A) equals O. In earlier papers [l] and [2] I made a conventional stipulation that p(A ⇒ B)=1 when p(A)=0, but here we have preferred to leave the `zero antecedent probability case’ an open problem, and have tried to indicate to what extent we may expect further developments in the probabilistic logic of conditionals to depend on that special case.
The argument of this paragraph can be made entirely rigorous so far as it applies to the theory formulated in [1] and [2], where p (A ⇒ B) is defined to be 1 when p(A) is O. In this case p(A amp;B) is always a function of p(A) and p(A and B), and the argument shows that this would entail that t (A ⇒ B) should be a function of t (A) and t (A and B). In this case a ‘triviality argument’ paralleling the one immediately following would show that for any A and probability function p, either p (A) =0 or p(A)=1: i.e., the only possible probability values would be truth values.
There is an exception in the case of the self-contradictory (A and —A), or more generally any combination A and B where A and B are represented as disjoint. These are of course propositions with probability 0.
oltan Domotor in private conversation has suggested dealing with the zero antecedent probability problem by representing probabilities in a non-Archimedean ordered vector space (which would allow some probabilities to be ‘incomparably small’ with respect to others). The details of this intriguing suggestion remain to be worked out.
We will be able to make only incidental remarks in this work about `conditional related’ connectives like ‘even if’, ‘only if’, and `unless’. The most natural reading of ‘A, even if B’ is as a conjunction, “A, and if B then A”. Thus, ‘even if’ includes ‘if’, but is much stronger. On the truth-conaitional analysis, “A, even if B” comes out equivalent to A, but this depends on the material conditional fallacy that A entails “if B then A”. Note that our probabilistic ‘semantics’ does not allow us to attach a probability to even-if constructions, since these are conjunctions with conditional conjuncts, which we will show in Section 8 to involve difficulties for probability.
nes first reaction to this example is that it is a special instance of the fallacy of material implication to infer —AFB from A. That is, one is apt to interpret a speaker who makes an assertion of the form “if it is the case that if A then B, then B” as saying no more than A. However, there are reasons for thinking the compound in question is more nearly equivalent to A V B, in which case the fallacy is more complicated.
he intuitive inconsistency of the conditionals AFB and A=B is assumed in Lewis Carroll’s intriguing `barbershop paradox’ [14] (not to be confused with the ‘paradox’ of the barber who shaves all men in town who don’t shave themselves). Assuming our probabilistic interpretation of conditionals, the paradox is not resolved by pointing out that AFB and A.*.—B are not inconsistent, for probabilistically they are inconsistent. However, other features of Carroll’s paradoxical argument, and in particular its being of the Reductio ad Absurdum form involving counterfactuals, put it beyond the reach of the present analysis.
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Adams, E.W. (1975). The Indicative Conditional. In: The Logic of Conditionals. Synthese Library, vol 86. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7622-2_1
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