Here’s a way to capture lexical threshold utilitarianism with a separable theory and while avoiding Pascalian fanaticism, with a negative threshold t−<0 and a positive threshold t+ > 0:
σ(∑iui)+∑iI(ui≥t+)−∑iI(ui≤t−)
The first term is just standard utilitarianism, but squashed with a function σ:R→R into an interval of length at most 1.
The second/middle sum is the number of individuals (or experiences or person-moments) with welfare at least t+, which we add to the first term. Any change in number past this threshold dominates the first term.
The third/last sum is the number of individuals with welfare at most t−, which we subtract from the rest. Any change in number past this threshold dominates the first term.
Either of the second or third term can be omitted.
We could require t−≤ui≤t+ for all i, although this isn’t necessary.
More thresholds could be used, as in this comment: we would apply σ to the whole expression above, and then add new terms like the second and/or the third, with thresholds t++>t+ and t−−<t−, and repeat as necessary.
Here’s a way to capture lexical threshold utilitarianism with a separable theory and while avoiding Pascalian fanaticism, with a negative threshold t−<0 and a positive threshold t+ > 0:
The first term is just standard utilitarianism, but squashed with a function σ:R→R into an interval of length at most 1.
The second/middle sum is the number of individuals (or experiences or person-moments) with welfare at least t+, which we add to the first term. Any change in number past this threshold dominates the first term.
The third/last sum is the number of individuals with welfare at most t−, which we subtract from the rest. Any change in number past this threshold dominates the first term.
Either of the second or third term can be omitted.
We could require t−≤ui≤t+ for all i, although this isn’t necessary.
More thresholds could be used, as in this comment: we would apply σ to the whole expression above, and then add new terms like the second and/or the third, with thresholds t++>t+ and t−−<t−, and repeat as necessary.