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That makes sense, thank you for your great reply. As someone without much mathematical ropes, this might also be related to non-integer algebra and the distinc

That makes sense, thank you for your great reply. As someone without much mathematical ropes, this might also be related to non-integer algebra and the distinction of "smooth" (edit: or differentiable) and discontinuous functions, no? > The functions maximum and minimum are arithmetic functions as important as addition and subtraction. The only reason for which they have not been counted among the basic arithmetic functions is because when computed by a human they required much less effort than even addition

adrian_b 3y ago
As I have mentioned, in the logic systems with 3 or more discrete values of truth (e.g. false, unknown and true, or false, unlikely, unknown, likely and true) and also in the logic systems with a continuous range of truth values (e.g. those based on probabilities), to the base logic functions AND, OR and NOT, correspond the base logic functions minimum, maximum and inversion a.k.a. reflection functions (the last may be e.g. integer negation when the truth values are symmetric around zero, or it may be e.g. 1-x when the truth values are between 0 and 1). A set with minimum and maximum operations has a special algebraic structure named distributive lattice, which is a structure as useful and frequently encountered as structures like the algebraic groups and rings that characterize addition-like and multiplication-like operations.
- cubefox 3y ago
  It's interesting that maximum/minimum for disjunction/conjunction doesn't apply for probabilities, except when the probabilities "overlap as much as possible" (when at least one implies the other). And when the probabilities overlap as little as possible, then the formulas are: P(a or b) = min(1, P(A)+P(B)) P(a and b) = max(0, P(a)+P(b)-1) I guess there is some lesson about logic and truth here, but I can't quite see it...

adrian_b 3y ago

As I have mentioned, in the logic systems with 3 or more discrete values of truth (e.g. false, unknown and true, or false, unlikely, unknown, likely and true) and also in the logic systems with a continuous range of truth values (e.g. those based on probabilities), to the base logic functions AND, OR and NOT, correspond the base logic functions minimum, maximum and inversion a.k.a. reflection functions (the last may be e.g. integer negation when the truth values are symmetric around zero, or it may be e.g. 1-x when the truth values are between 0 and 1). A set with minimum and maximum operations has a special algebraic structure named distributive lattice, which is a structure as useful and frequently encountered as structures like the algebraic groups and rings that characterize addition-like and multiplication-like operations.

cubefox 3y ago
It's interesting that maximum/minimum for disjunction/conjunction doesn't apply for probabilities, except when the probabilities "overlap as much as possible" (when at least one implies the other). And when the probabilities overlap as little as possible, then the formulas are: P(a or b) = min(1, P(A)+P(B)) P(a and b) = max(0, P(a)+P(b)-1) I guess there is some lesson about logic and truth here, but I can't quite see it...