A Theory of Similarity and Uncertainty
Two sides of uncertainty
Uncertainty may entail at the same time lack of determinacy and imprecise knowledge. Lack of determinacy is an ontological property of the universe we are considering. Imprecise knowledge is an epistemic property of the agents in that universe. A desirable feature of a theory of uncertainty is that both properties should be taken into account and integrated within a unifying framework. A possible route to identifying such a framework is suggested by Henry Kyburg's conception of objective (ontological) probability (see Kyburg, Chapter 2, this volume) and Isaac Levi's view concerning the relative autonomy of cognitive objectives (see Levi, Chapter 3, this volume). Kyburg maintains that 'many people think that the evidence renders certain beliefs irrational' (s. 2.3). He also maintains that 'the issue is important in artificial intelligence for the same reason: are there constraints that degrees of beliefs should satisfy? Or is one coherent distribution as good as another?' (s. 2.3). Finally, Kyburg calls attention to the issue of objectivity in statistical inference: 'if there are no objective constraints, it is hard to know how differences of opinion regarding statistical conclusions can be resolved' (s. 2.3). From the point of view of cognitive commitment, Kyburg's claim about the effectiveness of objectivity constraints is close to Isaac Levi's view concerning the autonomy of cognitive values. According to Levi, the specific features of ampliative inference are a clear instance of cognitive autonomy. Ampliative inference allows new information to be obtained from available evidence by making use of an inferential procedure, that is, through a 'legitimate' reasoning procedure that may require calculations but does not presuppose any increase in direct evidence. This cognitive procedure is 'ampliative' in the sense that the new state of full belief 'is "stronger" than the initial one, so that an agent in the new state believes more than an agent in the initial one' (Levi, 1991, p. 10). It is possible to further explore the characteristics of ampliative inference by a comparison with observation and experiment. For the latter are open cognitive procedures whereby the inquirers do not know in advance the type of new information that will eventually be available. In particular, they will not know whether the new information will be reliable or not. The case of ampliative inference is different, since this type of inference is closely associated with what Levi calls 'doxastic commitment', that is, with the type of commitment whereby 'X is committed to recognize fully the truth of the deductive consequences of what he fully believes' (Levi, 1991, p. 9). There is a sense in which this procedure is not open ended, in so far as inquirers are pre-committing themselves to accept the results of the inferences they have accepted to make.
Kyburg's interest in objective constraints and Levi's emphasis on rational commitment call attention to the domain of intermingled ontological and epistemic conditions from which uncertainty arises. And both Kyburg and Levi acknowledge the need to address uncertainty by explicitly accepting the constraining function of a certain commitment (respectively of the ontological and the epistemic types). As a matter of fact, both lack of determinacy (an ontological condition) and imprecise knowledge (an epistemic condition) reflect the organization of similarity in the relevant domain. This is because individual events appear to be determinate or not depending on whether those events are identified in a more (or less) circumscribed way. The more circumscribed the description of any given event is, the more likely it is that that event will be fully accounted for within its own reference domain. For example, any specific historical occurrence will be subject to indeterminacy if it is considered as a particular instance of some larger class of events (say, an economic disturbance generated within the domain of economic crises). On the other hand, the same historical happening will be increasingly determinate if it is considered as a 'singleton', that is, as a unique occurrence with distinctive features making if different from other occurrences of the same type (say, an economic disturbance taking place at a definite time under given conditions). In this case, identification of causation presupposes ability to reconstruct relevant contexts rather than ability to identify relevant causal laws.2 In short, the identification of deterministic versus non-deterministic events is not independent of the way in which events are described. The degree of precision of any given description is inversely related to the degree to which we may be able to identify general principles explaining specific occurrences. On the other hand, more precise descriptions make theoretical explanations more difficult but are conducive to less imprecise narratives of what generates specific outcomes. In short, there seems to be a trade-off between ontological precision and epistemic precision. The more circumscribed our view of the world is (ontological precision), the less likely we are to explain specific occurrences in terms of general laws; on the other hand, the less circumscribed our view of the world is (ontological imprecision), the more likely we are to make use of law-like explanations.3 As noted by Lotfi A. Zadeh, successful reasoning under uncertainty presupposes a sophisticated interplay of two distinct cognitive capacities: one is the ability to identify 'granular values', that is, intervals in which a certain variable 'is known to lie' (Zadeh, Chapter 6, this volume, s. 6.13); the other is the ability to recognize, for any given object, its corresponding prototype, or 'protoform', that is, an 'abstracted summary' whose 'primary function [...] is to place in evidence the deep semantic structure' of that object (s. 6.14). Any given circumscription presupposes a protoform which in turn is compatible with a greater or smaller interval of possible values depending on its semantic structure. We may conjecture that epistemic precision (the ability to draw conclusions from premises by deliberately following given rules of inference) is more strongly required in the presence of ontological imprecision, as the latter is associated with less clearly marked dividing lines between different objects (or different qualities). At the same time, it is the existence of partial similarities across different objects that makes law-like statements possible. The domain of reasoning under uncertainty coincides with the collection of intermediate situations between ontological precision and epistemic precision.4 Ontological precision makes inferential reasoning increasingly difficult and ultimately impossible, as an increasingly detailed circumscription of objects reduces the scope of similarity and ultimately makes law-like statements impossible. On the other hand, increasing epistemic precision presupposes increasing approximation to Humean regularity and thus also increasing similarity of situations across different contexts.5 This makes identification of distinct objects increasingly difficult and ultimately impossible.6 Uncertainty may be defined as the intersection between a state of the universe and a state of knowledge such that both circumscription and similarity are not complete. This means that objects (or situations) are not singletons, so that categorization is possible. On the other hand, objects (or situations) are only partially similar to one another within the relevant similarity classes. This means that categories may be identified but their membership is subject to a degree of arbitrariness depending on context, agents' epistemic propensities, and so on. The above definition entails that uncertainty may be associated with some degree of imprecision both in the representation (description) of the relevant state of the universe and in the 'rational' understanding of it. It is a hybrid condition in which both circumscription and similarity are 'held back': objects are not unique and irretrievable but similarities among objects are only partial.7 This condition allows human projecti- bility, which would be impossible were objects unique and/or similarities undetectable. At the same time, projectibility is constrained by limited uniformity, which makes surprises possible.