Someone has probably thought of this already, but if we defined an integration analogue where larger and larger logarithmic sums cause their exponentiated, etc. value to approach 1 rather than infinity, then we could use it to define a really cool account of logical metaphysics: Each possible state of affairs has an infinitesimal probability, there are infinitely many of them, and their probabilities sum to 1. This probably won’t be exhaustive in some absolute sense, since no formal system is both consistent and complete, but if we define states of affairs as formulas in some consistent language, then why not? We can then assign various differential formulas to different classes of states of affairs.
(That is the context in which this came up. The specific situation is more technically convoluted.)
Someone has probably thought of this already, but if we defined an integration analogue where larger and larger logarithmic sums cause their exponentiated, etc. value to approach 1 rather than infinity, then we could use it to define a really cool account of logical metaphysics: Each possible state of affairs has an infinitesimal probability, there are infinitely many of them, and their probabilities sum to 1. This probably won’t be exhaustive in some absolute sense, since no formal system is both consistent and complete, but if we define states of affairs as formulas in some consistent language, then why not? We can then assign various differential formulas to different classes of states of affairs.
(That is the context in which this came up. The specific situation is more technically convoluted.)