Compared to the traditional probabilistic-statistical view of non-deductive intra- theoretical scientific inferences, our proposal’s main shift can be summarized as fol- lows. First, we import from AI some techniques often used there in connection to nonmonotonic reasoning to express non-deductive scientific inferences. Second, in order to prevent the appearance of ambiguities we provide a mechanism by which
exceptions to laws can be represented. This mechanism has two main advantages over Hempel’s RMS: it can prevent the class specificity ambiguities without rejecting both arguments (as Hempel’s does), being also able to treat properly those cases of ambi- guity that do not involve class specificity (that remained uncovered by Hempel’s system.) Finally, in order to consider the cases where the ambiguities are due to the very nature of the knowledge to be formalized and, as such, cannot be prevented, we supply a paraconsistent apparatus by which those ambiguities can be tolerated and sensibly reasoned out, without trivializing the whole set of conclusions. Conse-
Is Plausible Reasoning a Sensible Alternative for Inductive-Statistical Reasoning? 129
quently, even in the presence of contradictions we can make use of all available in- formation, achieving just the reasonable conclusions.
Our proposal takes the form of a logical system consisting of two different logics, organically connected, which are intended to operate in two different levels of reason- ing. At one level, the nonmonotonic one, there is a logic able to perform non- deductive inferences. This logic is conceived in a style which resembles Reiter’s default logic [13], but incorporates a very important distinction: it is able to generate extensions including contradictory conclusions obtained through the use of default rules. By this reason, it is called Inconsistent Default Logic (IDL) [1], [11]. The con- clusions achieved by means of nonmonotonic inferences do not have the same epis- temic status of the ones deductively derived from known facts and assumed princi- ples. They are taken as just plausible (in opposition to certain, as far as the theory and the observations are themselves so taken). In order to explicitly recognize this epis- temic fact, and thus make it formally treatable in the reasoning system, they are marked with a modal sign (?), where means is plausible.” In this way, differ- ently from traditional nonmonotonic logics, IDL is able to distinguish revisable for- mulae obtained though nonmonotonic inferences from non-refutable ones, deduc- tively obtained.
At the second level, operates a deductive logic. But here again, not a classic one, but one able to properly treat and make sensible deductions in the theory that comes out from the first level, even if it is inconsistent, as it may very well be. This feature makes this logic a paraconsistent one, but, again, not one of the already existent para- consistent logics, as the ones proposed by da Costa and his collaborators [3], but one specially conceived to reason properly under the given circumstances. It is called the
Logic of Epistemic Inconsistencies (LEI) [1], [11]. In this logic, a distinction is made between strong contradictions, a contradiction involving at least one occurrence of deductive, non-revisable knowledge, from weak contradictions, of the form
which involves just plausible conclusions. This second kind of contradictions are well tolerated and do not trivialize the theory, as the first kind still do, just as in clas- sical logic.
The general schema of an IDL default rule is which can be read as can be nonmonotonically inferred from unless Adopting Reiter’s terminology, represents the prerequisite, the consequent, and the negation of the justification, here called exception. One important difference between Reiter’s logic and ours is that the consistency of the consequent does not need to be explicitely stated in it is internally guaranteed by the definition of extension. Translating Reiter’s normal and semi-normal defaults to our notation, for example, would produce respectively and where is an abbreviation for Other difference is that the consequent is added to the extension necessarily with the plausi- bility mark ? attached to it. The definition of IDL theory is identical to default logic’s one. Above it follows the definition of IDL extension.
Let S be a set of closed formulae and <W, D> a closed IDL theory. is the smallest set satisfying the following conditions:
(i) (ii) (iii)
If then
If and is ?-consistent, then
A set of formulae E is an extensionof <W, D> iff that is, iff E is a fixed point of the operator
The symbol refers to the inferential relation defined by the deductive and para- consistent logic LEI, according to which weak inconsistencies do not trivialize the theory. Similarly, the expression “?-consistent” refers to consistency or non- trivialization under such relation. Above we show the axiomatic of LEI. Latin letters represent ?-free formulae and ~ is a derived operator defined as follows:
where is any atomic ?-free formula. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
where t is free for x in
where x is not free in where x is a varying object.
where t is free for x in
where x is not free in
where ? is a varying object.
Axiom schema 24, which is a weaker version of the reductio ad absurdum axiom, is the key of LEI’s paraconsistency. By restricting its use only to situations where B is ?-free, it prevents that from weak contradictions we derive everything; at the same time that allows ?-free formulae to have a classical behaviour. Axiom schema 27 is another important one in LEI’s axiomatic. It allows for the internalization and exter- nalization of ? and ¬ with respect to each other and represents, in our view, one of the main differences between the notions of possibility and plausibility. The varying object restriction present in some axiom schemas is needed to guarantee the univer- sality of the deduction theorem. For more details about LEI’s axiomatic (and seman- tics) see [11].
Turning back to the problem of inductive inconsistencies, as Hempel himself ac- knowledged [5], the appearance of ambiguities is an inevitable phenomenon when we deal with non-deductive inferences. Surprisingly enough, all cases of inductive ambi- guities identified by Hempel are not due to this suggested connection between ambi-
Is Plausible Reasoning a Sensible Alternative for Inductive-Statistical Reasoning? 131
guity and induction, but to the incapacity of his probabilistic approach to represent properly the situations in question.
Consider again John Jones’s example. The situation exposed in section 2 can be formalized in IDL as follows:
Here (3) is a default schema that says that if someone has a streptococcus infection and was treated with penicillin, then it is plausible that it will have a quick recovery unless it is verified that the streptococcus is a penicillin resistant one. (4) states that John Jones has a streptococcus infection and that he took penicillin. Given
and as the IDL-theory, we will have as the only extension of <W,D>, where is the set of all formulae that can be inferred from A through
Suppose now that we have got the new information that John Jones’s streptococ- cus is a penicillin resistant one. We represent this through the following formula:
Like in Hempel’s formalism, if someone is infected with a penicillin-resistant ba- cillus, it is not plausible that he will have a quick recovery after the treatment of peni- cillin (unless we know that he will recover quickly). This can be represented by the following default schema:
Given and as our new IDL-
theory, we will have as the only extension of <W’,D’>. Since in Hempel’s approach there is no connection between laws (1) and (2), the conclusion of has no effect upon the old conclusion Gb. Here however it is being represented the priority that we know law (5) must have over law (3): the clauses Jx in the exception of (3) and Jx in the prerequisite of (5) taken together mean that if (5) can be used for inferring, for example (3) cannot be used for infer- ring Gb?. So, if after using law (3) we get new information that enable us to use law (5), since in the light of the new state of knowledge law (3)’s utilization is not possi- ble, we have to give up the previous conclusion got from this law. So, since we are no longer entitled to infer Gb? from (3). The only plausible fact that we can conclude from (3) and (5) is As such, in contrast to Hempel’s approach, we do not have the undesirable consequence that it is plausible (or in Hempel’s approach, high probable) that John will quickly recover and it is plausible that he will not.
As we have said, in this specific case we know that law (5) has a kind of priority over law (3), in the sense that if (5) holds, (3) does not hold. Like we did in Section 2, suppose now 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 but due to the always-small number of initial cases, this result cannot be con- sidered as definitive. Even so, we can set up the following tentative law:
Here H’ stands for administration of the new kind of antibiotic. To complete the formalization we have the two following formulae:
Given and
(laws (5) and (6)) as our new IDL-theory, we have that the extension of <W”,D”> is
In this case, we do not know which of the two ‘laws’ has priority over the other. Maybe the penicillin-resistant bacillus will prove to be resistant even to the new anti- biotic or maybe not. Instead of rejecting both conclusions, as I-S model with its RMS would do, we defend that a better solution is to keep reasoning even in the presence of such ambiguity, but without allowing that we deduce everything from it. Formally this is possible because of the restriction imposed by the already shown LEI’s axiom of non-contradiction. However, if a modification that resolves the conflict is made in the set of facts (a change in (5) representing the definitive success of the new kind of penicillin, for example) the IDL’s nonmonotonic inferential mechanism will update the extension and exclude one of the two contradictory conclusions.
Finally, the HIV example can be easily solved in the following way.
Here A stands for having contracted HIV and I for having an infection. The solu- tion is similar to our first example. Since (7) has priority over (3’), we will be able to conclude only ¬Gb? and consequently the ambiguity will not arise.
We have shown therefore that our formalism solves the three problems identified in Hempel’s I-S model. One consideration remains to be done. Hempel’s main inten- tion with the introduction of the I-S model was to analyze the scientific inferences which contain statistical laws. At a first glance, it is quite fair to conclude that in those cases where something akin to a relative frequency is involved, a qualitative approach like ours will not have the same representative power as a quantitative one. However, there are several ways we can “turn” our qualitative approach into a quanti- tative one in such a way as to represent how much plausible a formula is. For in- stance, we could drop axiom 29 as to allow the weakening of the “degree of plausibil- ity” of formulae: would represent the highest plausibility status a formulae may have, which could be weakened by additional ?’s. In this way, a default could repre- sent the statistical probability of a law by changing the quantity of ?’s attached to its conclusion. A somehow inverse path could also be undertaken. In LEI’s semantic, it is used a Kripke possible worlds structure (in our case we call them plausible worlds)
Is Plausible Reasoning a Sensible Alternative for Inductive-Statistical Reasoning? 133
to evaluate sentences in such a way that is true iff is true in at least one plausible world [11]. We could define then as as as
as
where p and q are different ?-free atomic formulae, and so on, in such a way that the index n at the abbreviation says in how many plausible worlds is true.
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