Model
As is widely known, a wide range of physical phenomena (gravity, light, sound, etc) obey an inverse-square law, i.e. . This is justified by arguments from vector calculus or, informally, by considering the surface area of a sphere expanding out from a central point. As described in previous work, these arguments also generalize to moral obligation: an event's moral signifiance to you diminishes as you move further away from it, following an inverse-square law. Along with being theoretically justified (moral events, after all, take place in 3D space), this has been shown to accurately predict human behaviour.
Previous work has, however, failed to describe a precise model for moral obligation based on this principle. In this paper, we both precisely specify the inverse-square relationship involved and estimate the physical constants involved using data from Peter Singer's Drowning Child experiment.
As is trivially obvious to anyone at all trained in physics, a person's moral obligation to rectify a particular situation is measured in joules () - intuitively, this is the amount of work they are morally required to do to help. We thus write the inverse-square law as , where is morally required work, is the distance from the relevant moral actor to the situation in metres, is the base moral significance of the event, and is a new physical constant quantifying the permittivity of space to morality fields. We also introduce a new base unit, (singers), named after Peter Singer, for measuring the base moral significance of a situation. It can be shown by dimensional analysis that the units of are . One (1) singer is defined as the moral significance of one child drowning in one standard-sized pond. We believe that determining the moral significance of other possible scenarios is a simple matter of empirical philosophysics and leave this to later works.
In Singer's experiment, it was shown that you are morally obligated to save a child which is drowning in a pond, assuming the child is relatively close to you (about 10 metres). Assuming a standard-sized child of standard Earth gravity of , and a lifting distance of , this corresponds to of work done. We thus obtain a value for of . We anticipate that further work will provide independent confirmation of this result and more precise values for .
Applications
Our work provides a novel solution to the trolley problem. While we have not yet determined the moral significance of a person being killed by a runaway trolley, we need only consider the ratio of moral obligations to determine the correct choice. While no scale is provided in the above illustration, we can assume that the person on the top track is about three metres from the moral actor controlling the switch and the centre of morality in the bottom group about five, based on average population heights. Thus, the ratio of moral obligations is , using a linear approximation for the importance of the 5 people. Since this means that our moral obligation to the nearest person has a much smaller size, the lowest-free-energy/most moral solution is to divert the trolley to the top track, killing 1 person, at least assuming there is no energy cost to pulling the lever. Depending on the exact value of in this case, the extra obligation to the distal group may not outweigh the energy costs of pulling the lever, making it morally permissible to stand aside. Another interesting result is that as distance from the trolley increases, the resulting ratio converges to the value provided by spatially invariant ethics, although the absolute size of the obligation declines rapidly.
Nonlinear Ethics
During our research we encountered the problem that "one death is a tragedy, but a million deaths is a statistic". This is of course a violation of the principle of superposition, and shows the necessity of more sophisticated models of ethics than previous naive linear models at large scales. Using mathematics, we derived a model in which the total moral significance of moral events of significance happening to different people is rescaled using the natural choice for such a function, , and a scaling constant . We determined a value for using the assumption that linear ethics holds at small scales and that marginal moral significance turns negative at 150 as previously proposed. This can be expressed mathematically in the following way: This gives . An interesting consequence of this model is that while linear ethics predicts human extinction would be very bad, we determine that it is in fact a morally irrelevant problem due to the exponential decline in significance with number of people involved in the infinite limit. This handily models the behaviour of human moral agents working in AI.
Conclusion
Our combined theory (nonlinear, spatially varying ethics) lends significant predictive power to ethical reasoning. We demonstrate a satisfying solution to the longstanding trolley problem and unify previous work under a simple and applicable model. Further research is required to determine more exact empirical values of our introduced physical constants and to determine the absolute moral significance of many more scenarios than we have covered. Another interesting research direction may be to find an ethical analogue to special relativity, to explain observations indicating that moral obligation may decline as speed away from a situation increases.