A large number of indices for presence-absence data that compare two assemblages have been proposed or reinvented. Interpretation of these indices varies across the literature, despite efforts for clarification and unification. Most effort has focused on the mathematics behind the indices, their relationships with diversity, and with each other. At the same time, the following issues have been largely overlooked: (i) requirement that a small re-arrangement of assemblages should only cause a small change in an index, (ii) inferences from the indices about diversity patterns, (iii) inter-dependence of indices based on their information value, (iv) overlap of the ecological phenomena that the indices aim to capture, and (v) incomparability of measures of different phenomena. Neglecting these issues has resulted in the invention or reinvention of indices without increasing their information value, although this value is crucial for correct interpretation of the indices. We offer a framework for pairwise diversity indices that accounts for these issues. We differentiate between statistical and information dependence of indices and show mathematical links between all indices, even those that have not yet been developed. Using linear algebra, we show (1) which set of indices carries complete information on assemblage arrangement, (2) how to calculate any index from two presence-absence indices, which can be used to standardize and compare different indices across the literature, and (3) what can be inferred about diversity phenomena from different informationally independent indices. It is impossible to purify an index of a single biodiversity phenomenon from the effects of other phenomena, because these phenomena inevitably constrain each other. Consequently, many recently proposed indices do not measure the phenomena that they were intended to measure. In contrast, a proper inference can be made by combining classical indices from different, information independent families.

Farmlands occupy more than half the inhabitable Earth, sustaining much of the planet's wildlife. Many farmland species have not been able to adapt to the severely anthropogenically modified environments of modern agricultural lands. The northern lapwing has declined steadily in most countries since the 1980s. Reports of sudden, inexplicable local collapses are reported, where the population is all but gone within a decade, suggest that observed slow regional population declines could just be the result of many small population implosions spread out in time and space. With the northern lapwing as model species, the proposed trap-implosion theory unveils how increases in the proportion of intensively farmed cropland, act as an ecological trap, triggering local population implosions. The simulations show the ratio of high-to-low-quality breeding habitat to be the main trigger. This demonstrates that expansion of modern farmlands alone, rather than loss of natural habitat, may cause population collapse. The rate of habitat-area changes, declines in yearly survival, and breeding success mainly affect the timing of the implosion, but not its pace. Observations of local lapwing-population collapses suggest that the declines of many farmland species, especially farmland bird species, may be the accumulative result of local trap implosions. A conservational focus on farming practices and preserving or improving high-quality habitat may explain why the mechanism of ecological-trap-induced implosions have gone unnoticed, and why farmland-bird conservation measures have largely failed.