Conditional Random Quantities and Compounds of Conditionals

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH +  P(E|H)H ^c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold.     

Numerical acuity of fish is improved in the presence of moving targets,but only in the subitizing range

There is controversy in comparative psychology about whether on the one hand non-symbolic number estimation of small (≤4) and large numbers involves a single mechanism (an approximate number system), or whether on the other hand enumeration of the numbers 1–4 is accomplished by a separate mechanism, an object tracking system. To date, support for the latter hypothesis has come only from the different ratio-dependency of performance seen in the two numerical ranges, a reading that has been criticized on several grounds. In humans, the two-system hypothesis is supported by evidence showing that manipulation of the physical properties of the stimuli (e.g., the motion of the items) has dissimilar effects on small- and large-number discrimination. In this research, we studied this effect on guppies. Initially, fish were trained to simultaneously discriminate two numerical contrasts having the same easy ratio (0.50): one in the small-number (2 vs. 4) range and one in the large-number (6 vs. 12) range. Half of the fish were presented with moving items; the other half were shown the same stimuli without motion. Fish were then subjected to non-reinforced probe trials in the presence of a more difficult ratio (0.75: 3 vs. 4 and 9 vs. 12). Under both static and moving conditions, the fish significantly discriminated 6 versus 12, but not 9 versus 12 items. As regards small numbers, both groups learned to discriminate a 0.50 ratio, but only fish tested with moving stimuli also discriminated 3 and 4 items. This differential effect suggests that fish may possess two separate systems for small- and large-number discrimination.     

Pacifism and Punishment

This article seeks to expose some of the implications of certain versions of pacifism for matters of criminal punishment, arguing that the plausibility of these versions of pacifism depend on the extent to which their implicit denials of certain central punishment-related concepts are themselves reasonable.