Point-free notation is used here mainly as a device for obscuring notation. A function "f x g" pairs up f and g and applies them separately to a tuple of argum

Point-free notation is used here mainly as a device for obscuring notation. A function "f x g" pairs up f and g and applies them separately to a tuple of arguments: (f x g)(a, b) = (f(a), g(b)). The commutative diagram becomes clearer when we label the points: not x not (a, b) ---------> (not(a), not(b)) | | and | | or V V and(a,b) ---------> not(and(a, b)) = or(not(a), not(b)) not (This is the same as what user lmm wrote in prose in https://news.ycombinator.com/item?id=23203478. https://news.ycombinator.com/item?id=23203478.) As for what "path-space" notation means, even the author appears confused, since they claim that "and[not] <=> or" corresponds to "If you flip the input and output bits of an `and` function, ..." corresponds to "not(and(a, b)) = or(not(a), not(b))", but this clearly isn't the case: flipping both the inputs and outputs of `and` to get `or` would be "not(and(not(a), not(b))) = or(a, b)".