As with standard similarity judgements, with likelihood it is essential to bear in mind the plurality of 'orders' in which a reasonable judgement may be expressed. In short, what is likely under certain conditions (that is, under a given order of likelihood) may turn out to be utterly unlikely if different conditions are considered. For example, we may distinguish between the orders of likelihood associated with different time horizons, and assess the likelihood of any given situation (that is, its greater or smaller closeness to certainty) for different time scales.^{18} Figure 5.5 shows the implications of different orders of likelihood for what concerns our ability to assess the overall likelihood of any given situation.

Figure 5.5 shows two different orders of likelihood for the short term (A) and the long term (B) respectively, and calls attention to the fact that the same situation may be associated with different likelihoods depending on which order is considered. Clearly, likelihood can be the same only under exceptional circumstances, such as those associated with crossover points A and B. In this case, a crossover point may be interpreted as corresponding to a situation in which two different ontologies coincide at a given point of time. If there are two separate orders of likelihood for the short term and the long term respectively, a crossover point would be a situation at which the circumscription set appropriate for the short term coincides with the circumscription set suitable for long-term analysis: for example, a set of events that can be envisaged only if we take a long-term point of view (say, a slow-moving geological process)

Figure 5.5 Short-term and long-term orders of likelihood

may also be of short-term relevance if it brings about sudden disruption within a narrow time interval.

The previous discussion of similarity orders is relevant for this case too. For we can directly compare any two situations if those situations have at least some common value for some of their features or if those situations are associated with the same value on different orders of similarity. In particular, short-term and long-term likelihoods may be directly compared if one or the other of the two following conditions is met: (i) at least one feature is common to the short-term and the long-term orders of likelihood; (ii) a situation has the same similarity value on the short-term and the long-term orders of likelihood. The two above conditions are clearly different. In case (i), different time horizons would give rise to the same set of features; in case (ii), a given situation would take the same likelihood value on different time horizons. The two cases are in fact opposed to one another. For (i) entails that different processes (long-term and short-term) bring about the same situation, whereas (ii) entails that the same situation is equally likely independently of which time horizon is considered. The former statement concerns ontology (that is, the processes by which events are generated from one another); the latter statement concerns the epistemic domain (that is, the criteria by which similarity conditions are assessed across different situations). In general, conditions (i) and (ii) will not be satisfied, and uncertainty cannot be assessed across different orders of likelihood. However, the previous argument suggests a set of practical criteria that may be useful in evaluating more common situations, which are in a sense intermediate with respect to the two polar cases.

Any given situation would have to be assessed differently depending on whether we are considering short-term or long-term orders of likelihood, and we may expect likelihood to be different in the two cases. In Figure 5.6, a specific situation s* must be separately assessed for short-term likelihood (A) and long-term likelihood (B), respectively. Clearly s* may be associated with low or high likelihood on the long-term likelihood order, and we may conjecture that the likelihood change may be associated with a change in the way s* is described in the two cases. For example, a change in the similarity features that are being considered is likely to change our perception of the likelihood of future events relatively to some (known) event in the past. The same argument applies to the short-term order of likelihood. Here, the expansion (or contraction) in the number of similarity features may significantly influence the overall likelihood judgement for s*.

More formally, let s* be associated with similarity features (r_{1}, r_{2} ,..., r_{k} ) on the assumption that all (partial) similarity features a_{r}(r = 1, ..., k) take values above the corresponding significance thresholds a * (that is, a_{r} > a_{r}*). We know that overall similarity a may be assessed by combining partial similarity features according to the following criterion: a = k, where k is the number of partial similarity features for which a_{r} > a*. The above argument entails that the likelihood of s* may increase or decrease depending on changes in the number of significant similarity features. This condition draws us back to the discussion in sections 5.1 and 5.2. For a change of k may be associated with a change of either the circumscription set or the epistemic set. In the former case (that is, in the case of switch to a different ontology), it is likely that less circumscribed descriptions would be associated with an increase in the number of similarity features for which a_{r} > a* (and

Figure 5.6 Short-term likelihood and long-term likelihood for situation s vice versa if more circumscribed descriptions are considered). In the latter case (that is, in the case of a switch to a different set of categories), it is likely that more comprehensive descriptions would allow identification of partial similarities across a wider range of phenomena than is the case with descriptions of a less comprehensive type (and vice versa). Figure 5.6 highlights the fact that what is apparently the same situation s* may be associated with different likelihood values, and that this may happen both for any given order of likelihood and across different likelihood orders. We may conjecture that likelihood changes for any given order of likelihood may be associated either with ontology change or with epistemic change (or both). On the other hand, changes of likelihood across different likelihood orders would be primarily associated with a switch from one particular ontology to another. This latter condition makes cross-order comparisons especially difficult, given that we would have to assess the likelihood of what are in fact two different situations, one grounded in the short term and the other in the long term. It is in view of this difficulty that crossover points may be of special interest. For it is precisely at those points that the long-term and the short-term perspectives come to coincide. Crossover points highlight intersections between different ontologies and draw attention to ways in which we can move from one ontology to another. In particular, any given crossover point highlights circumscriptions relevant to both a short-term ontology and a longterm ontology. The fact that only at those points may likelihood be the same on different likelihood orders suggests that, apart from those exceptional cases, uncertainty assessment should be separately carried out for different orders of likelihood. However, awareness that crossover points are possible recommends prudence in likelihood assessment. In particular, we may be justified in having more confidence in a statement that is equally likely on different orders of likelihood. For crossover points strengthen the weight of our assessment and make it especially relevant to decision both when a given situation is to be avoided and when it is to be actively promoted.^{19}