Luce's Coffee Cups

This is a thought experiment that shows that the Indifference relation is not transitive

Suppose we have 402 cups of coffee:

  • $c_0$ with 0 grains of coffee
  • $c_1$ with 1 grain of coffee, $c_0 \ I \ c_1$
  • $c_2$ with 2 grains of coffee, $c_1 \ I \ c_2$
  • ...
  • $c_{401}$ with 401 grain of coffee, $c_{400} \ I \ c_{401}$
  • luce-1.png

But we cannot say that there's no difference in $c_0$ and $c_{401}$

  • $c_0 \ \not I \ c_{401}$
  • luce-2.png


It shows that $I$ is not transitive:

  • $a \ I \ b, b \ I \ c \not \Rightarrow a \ I \ c$


The reason for it is that there's a sensitivity threshold that is not taken into account


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