Admittedly most of this post goes over my head. But could you explain why you want logical correlation to be a metric? Statistical correlation measures (where the original “correlation” intuition probably comes from) are usually positive, negative, or neutral.
In a simple case, neutrality between two events A and B can indicate that the two values are statistically independent. And perfect positive correlation either means that both values always co-occur, i.e. P(A iff B)=1, or that at least one event implies the other. For perfect negative correlation that would be either P(A iff B)=0, or alternatively at least one event implying the negation of the other. These would not form a metric. Though they tend to satisfy properties like cor(A, B)=cor(B, A), cor(A, not B)=cor(not A, B), cor(A, B)=cor(not A, not B), cor(A, B)=-cor(A, not B), cor(A, A)=maximum, cor(A, not A)=minimum.
Though it’s possible that (some of) these assumptions wouldn’t have a correspondence for “logical correlation”.
Admittedly most of this post goes over my head. But could you explain why you want logical correlation to be a metric? Statistical correlation measures (where the original “correlation” intuition probably comes from) are usually positive, negative, or neutral.
In a simple case, neutrality between two events A and B can indicate that the two values are statistically independent. And perfect positive correlation either means that both values always co-occur, i.e. P(A iff B)=1, or that at least one event implies the other. For perfect negative correlation that would be either P(A iff B)=0, or alternatively at least one event implying the negation of the other. These would not form a metric. Though they tend to satisfy properties like cor(A, B)=cor(B, A), cor(A, not B)=cor(not A, B), cor(A, B)=cor(not A, not B), cor(A, B)=-cor(A, not B), cor(A, A)=maximum, cor(A, not A)=minimum.
Though it’s possible that (some of) these assumptions wouldn’t have a correspondence for “logical correlation”.