Significant Inferences : Preliminary Report

We explore the possibility of a logic where a conclusion substantially improves over its premise(s): Specifically, we intend to rule out inference steps such that the premise conveys more information, in a simpler form, than the conclusion does. In fact, most reasoning formalisms, among them classical logic, come with means for generating disjunctive or conditional information in a fairly arbitrary way. The basic principle for drawing disjunctive information is disjunctive weakening, which allows for deriving φ ∨ ψ from φ (for any ψ). Thus, given that “Nancy is married to Ron”, disjunctive weakening makes us infer that “Nancy is married to Ron or Monica is married to Bill”. Although the latter propositions may still be seemingly related, one should not forget (1) that any arbitrary proposition can serve as the additional disjunct, eg. “Nancy is married to Ron or the Queen of England is bald”, and (2) that this can be iterated so that real information is buried in the generated disjunction among irrelevant propositions. What is the point in inferring such disjunctive formulas? Similar phenomena can be traced back to conditionalization, which allows for deriving ψ → φ from φ (for any ψ). As a result, we propose a natural deduction system along the intuitions sketched above. 1 MOTIVATION Perhaps the most salient feature of reasoning is to make explicit what is only implicit. Accordingly, the less obvious a (correct) conclusion is, the more valuable is any reasoning by which that conclusion is drawn. In a sense, some conclusions may then not be worth inferring. Such a point of view is latent in relevance logics [Anderson and Belnap,1975] because they reject certain conclusions that weaken some premise in a peculiar way, as happens with arbitrary conditionalization for instance, i.e., p ` q → p (where p and q are propositional symbols). In this respect, relevance logics pave the way to a logic where only conclusions worth expressing are drawn. However, relevance logics stop short of fully achieving this idea. This can be viewed from relevantly valid schemes such as p ` p∨q. Indeed, one may regard p ∨ q as a dubious weakening of p, as much as q → p is. In fact, we regard the scheme p ` p ∨ q as drawing a conclusion which fails to be significant in view of its premise. What do we mean by significant here? We mean that the conclusion is not worth stating when its premise is stated: Once it is clear that “the winning ticket is number 36”, there is no point in making the inquiry whether “the winning ticket is number 17 or 36”. Much as it would make no sense to investigate iteratively about the fact that “the winning ticket is number n1 or. . . or nk or 36” for k increasing. It does not even matter that any ni be 36 as well: What is the point of stating that “the winning ticket is number 36 or the winning ticket is number 36” when it has already been stated that “the winning ticket is number 36”? That is, even the restricted form p ` p ∨ p is a particular form of dubious weakening. Hence, we want to explore the possibility of a logic where a conclusion substantially improves over its premise(s): Specifically, we intend to rule out inference steps such that the premise conveys more information, in a simpler form, than the conclusion does. Indeed, we are not aiming at ignoring a conclusion that is less informative than its premises, provided that the conclusion has a simpler form. For instance, p ∨ q, p→ ¬r, s↔ r,¬p→ (q → ¬s) ` ¬s is an inference whose conclusion ¬s does not exhaust all the information about p, q, r, s that the premises provide but is much easier to grasp. 2 TOWARDS SIGNIFICANT INFERENCE In contrast to a proof system such as resolution [Robinson,1965] that has a single inference rule (ignoring factorization), natural deduction [Gentzen,1935] is traditionally a suitable framework for analyzing inferences. So let us consider the matter of conditionalization and disjunctive weakening from the perspective of natural deduction. Conditionalization consists of turning an inference of ψ from some premises including φ into a proof of φ → ψ. In extensional logics, the notion of being a premise is fairly liberal so that it is not required that φ actually serves for deriving ψ. Intensional logics such as relevance logics insist on φ being actually used to infer ψ: Relevance is then a necessary condition for φ → ψ to be derived. We adopt the same criterion because it matches also our idea of significance: on the one hand, φ → ψ is certainly significant whenever ψ is derived by means of φ, and, on the other hand, if ψ is derived independently of φ, then φ→ ψ is both less simple and less insightful than ψ itself. In natural deduction, generalizing this leads us to requiring that the so-called “auxiliary assumptions” have to take part in the derivation of the conclusion that is declared to rely on them. Disjunctive weakening consists of concluding φ ∨ ψ from φ (or similarly from ψ). We have argued above that this is never justified on its own, although it is sometimes useful as a device for inferring intermediate conclusions, as needed for reasoning by cases (whose principle is that, given φ ∨ ψ, if χ is concluded from φ and if χ is concluded from ψ, then χ is inferred). So, weakening is needed for combining the conclusions obtained in each case: p ∨ q, p→ r, q → s ` r ∨ s . In fact, we permit disjunctive weakening only for deriving conclusions drawn by reasoning by cases (see below). Technically, banning disjunctive weakening while restricting conditionalization as just indicated could simply yield a subsystem of a relevance logic. However, we ban disjunctive weakening according to our intuitions about significant conclusions and these intuitions are different from those behind relevance logics, in particular, they make us depart from the relevance view against disjunctive syllogism (viz. φ∨ψ,¬φ ` ψ). A well-taken objection by the relevantists is that if φ ∨ ψ holds because φ does then applying disjunctive syllogism is flawed: There is a contradiction between φ and ¬φ but ψ is irrelevant in the matter. We agree that disjunctive syllogism is inappropriate in such a case but we contend that it is a valuable pattern when no contradiction is involved. Indeed, our intuitions about φ∨ψ are that the disjunction is really an alternative between φ and ψ so that it actually is about φ as well as about ψ. Accordingly, it is not about just φ or about just ψ and this warrants that if either case is denied then the other must hold. As appears from the preceding discussion, significant inference is paraconsistent (i.e., the so-called ex falso φ,¬φ ` ψ does not hold). The intuitive notion of a significant conclusion appeals for paraconsistency in at least two different ways. One is that nothing is more informative or substantially simpler than φ∧¬φ. Another is that, should paraconsistency be ruled out, then everything would be concluded from a contradiction (meaning that everything is significant, which is antinomic). The issue of paraconsistency naturally leads us to that of analyzing the standard deduction of the ex falso: