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
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.
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...
The logical "and" and logical "or" functions are precisely particular cases of the minimum and maximum functions.
The so-called "exclusive or" function is simultaneously a particular case of addition and of subtraction, i.e. addition modulo 2 and subtraction modulo 2 are the same function. Therefore applying the 4 functions minimum, maximum, addition and subtraction to 1-bit numbers produces only 3 functions, AND, OR and XOR.
It makes no sense to search other functions for expressing logical "and" and logical "or" except for minimum and maximum, because this is what they are. The search could only find alternative more complicated ways to express minimum and maximum.
The minimum and maximum functions also work in logic systems with more than 2 values of truth. Also in hardware, analog circuits for minimum and maximum are one of the 2 ways of implementing the logical "and" and "or" functions, the other way being with series and parallel connections.
The main mistake of George Boole was his unsuccessful attempt to find an equivalence between logical "and" and "or" and integer multiplication and addition, because he was not habituated to use minimum and maximum, so he did not see the correct relationships.
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, so there was no need for a long training of how to do computations with them, like for the more complex arithmetic functions.
Most modern instruction-set architectures now include maximum and minimum instructions, together with the addition and subtraction instructions.