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According to most historiographers of philosophy, the history of the philosophical analysis of scientific explanation began with the publication of ‘Studies in the Logic of Explanation’ in 1948 by Carl Hempel and Paul Oppenheim. In this work, Hempel and Oppenheim propose their deductive-nomological (D-N) model of scientific ex- planation where scientific explanations are considered as being deductive arguments that contain essentially at least one general law in the premises. Later, in 1962, Hem- pel presented his inductive-statistical (I-S) model by which he proposed to analyze the statistical scientific explanations that clearly could not be fitted into the D-N model. (These papers were reprinted in [5].)

Because of his emphasis on the idea that explanations are arguments and his com- mitment to a numerical approach, Hempel’s models perfectly exemplify the deduc- tive-inductive/probabilistic view of intra-theoretic scientific inferences. According to Hempel’s I-S model, the general schema of non-deductive scientific explanations is the following:

Here the first premise is a statistical law asserting that the relative frequency of Gs among Fs is r, r being close to 1, the second stands for b having the property F, and the expression ‘[r]’ next to the double line represents the degree of inductive prob- ability conferred on the conclusion by the premises. Since the law represented by the first premise is not a universal but a statistical law, the argument above is inductive (in Carnap’s sense) rather than deductive.

If we ask, for instance, why John Jones (to use Hempel’s preferred example) re- covered quickly from a streptococcus infection we would have the following argu- ment as the answer:

where F stands for having a streptococcus infection, H for administration of penicil- lin, G for quick recovery, b is John Jones, and r is a number close to 1. Given that penicillin was administrated in John Jones case (Hb) and that most (but not all) strep- tococcus infections clear up quickly when treated with penicillin the argument above constitutes the explanation for John Jones’s quick recovery.

However, it is known that certain strains of streptococcus bacilli are resistant to penicillin. If it turns out that John Jones is infected with such a strain of bacilli, then

Is Plausible Reasoning a Sensible Alternative for Inductive-Statistical Reasoning? 127

the probability of his quick recovery after treatment of penicillin is low. In that case, we could set up the following inductive argument:

J stands for the penicillin-resistant character of the streptococcus infection and r’ is a number close to zero (consequently 1 – r’ is a number close to 1.)

This situation exemplifies what Hempel calls the problem of explanatory or induc- tive ambiguities. In the case of John Jones’s penicillin-resistant infection, we have two inductive arguments where the premises of each argument are logically compati- ble and the conclusion is the same. Nevertheless, in one argument the conclusion is strongly supported by the premises, whereas in the other the premises strongly un- dermine the same conclusion.

In order to solve this sort of problem, Hempel proposed his requirement of maxi- mal specificity, or RMS. It can be explained as follows. Let s be the conjunction of the premises of the argument and k the conjunction of all statements accepted at the given time (called knowledge situation). Then, according to Hempel, “to be rationally acceptable” in that knowledge situation the explanation must meet the following condition: If implies thatbbelongs to a class and that is a subclass of F, then must also imply a statement specifying the statistical probability of G in say, Here, r’ must equal r unless the probability statement just cited is a theorem of mathematical probability theory.

The RMS intends basically to prevent that the property or class F to be used in the explanation of Gb has a subclass whose relative frequency of Gs is different from P(G,F). In order to explain Gb through Fb and a statistical law such as P(G, F) = 0.9, we need to be sure that, for all sets such that the relative frequency of Gs among is the same as that among Fs, that is to say, In other words, in order to be used in an explanation, the class F must be a homogeneous one with respect to G. (All these observations are valid for the new version of the RMS proposed in 1968 and called [5].)

The RMS was proposed of course because of I-S model’s inability to solve the problem of ambiguities. Since the I-S model allows the appearance of ambiguities and gives no adequate treatment for them, without RMS it is simply useless as a model of intra-theoretical scientific inferences. But we can wonder: Is the situation different with the RMS?

First of all, in its new version the I-S model allows us to classify arguments as au- thentic scientific inferences able to be used for explaining or predicting only if they satisfy the RMS. It is not difficult to see that this restriction is too strong to be satis- fied in practical circumstances. Suppose that we know that most streptococcus infec- tions clear up quickly when treated with penicillin, but we do not know whether this statistical law is applicable to all kinds of streptococcus bacillus taken separately (that is, we do not know if the class in question is a homogeneous one). Because of this incompleteness of our knowledge, we are not entitled to use argument (1) to explain

(or predict) the fact that John Jones had (or will have) a quick recovery. Since when making scientific prediction, for example, we have nothing but imprecise and incom- plete knowledge, the degree of knowledge required by the RMS is clearly incompati- ble with actual scientific practice.

Second, the only cases that the RMS succeeds in solving are those that involve class specificity. In other words, the only kind of ambiguity that the RMS prevents consists of that that comes from a conflict arising inside a certain class (that is, a conflict taking place between the class and one of its subclasses.) Suppose that John Jones has contracted HIV. As such, the probability of his quick recovery (from any kind of infection) will be low. But given that he took penicillin and that most strepto- coccus infections clear up quickly when treated with penicillin, we will still have the conclusion that he will recover quickly. Thus an ambiguity will arise. However, as the class of HIV infected people who have an infection does not belong to the class of individuals having a streptococcus infection which were treated with penicillin (and nor vice-versa), the RMS will not be able to solve the conflict.

Third, sometimes the policy of preventing all kinds of contradictions may not be the best one. Suppose that the antibiotic that John Jones used in his treatment belongs to a recently developed kind of antibiotic that its creators guarantee to cure even the known penicillin-resistant infection. The initial statistics showed a 90% of successful cases. Even though this result cannot be considered as definitive (due to the always- small number of cases considered in initial tests), it must be taken into account. Now, given argument (2), the same contradiction will arise. But here we do not know yet which of the two ‘laws’ has priority over the other: maybe the penicillin-resistant bacillus will prove to be resistant even to the new antibiotic or maybe not. Anyway, if we reject the contradiction as the I-S model does and do not allow the use of these inferences, we will loss a possibly relevant part of the total set of information that could be useful or even necessary for other inferences.

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A Nonmonotonic and Paraconsistent Solution