I want to built a unit of information, somewhat like a qubit, but it should encode besides the real-valued 'value' also the measure of 'uncertainty' (inverse of confidence) and measure of 'contradiction' (impossibilities). (All real-valued, and uncertainty+contradiction≤100%)
I've looked into Łukasiewicz logic, different generalizations of many-valued logics, fuzzy logic and Bochvar's internal three-valued logic.
Fuzzy logic (more specifically, the multiplication variant) comes closest to how true and false (and in between) should interact with each other (i.e. multiplicative T-norms). Bochvar's comes closest to how I want to propagate either uncertainty or contradiction (value 'I', and when both are 100%).
Then I found Subjective logic that almost precisely encodes how I would implement value and uncertainty on two dimensions. (Although I'm looking for an objective logic, not a subjective one.)
The problem is still that of how to combine both uncertainty and contradiction.
Then I found Bayesian probabilities, which I have a hard time understanding how it works (the wiki page contains no math at all).
I did some thought experiments as to how I would numerically implement my logic. My logic will be encoded in a triple [v, u, c] standing for Value, Uncertainty and Contradiction.
With these values I want to perform the following 6 operations: NOT, AND, Un-AND, OR, Un-OR, and XOR.
Examples:
First, define: x AND y = x * y = z, x Un-AND y = x / y = z, and none of the values can exceed the closed range $[0, 1]$.
Also, define: 'value' is the probability that you would observe this value as true, and false otherwise. This is distinct from 'uncertainty' because the latter encodes that you don't know something about 'value'. An uncertainty of 100% means you don't know if the value would be observed true 10% of the time or 50% or with any other probability.
[½, 0, 0] And [½, 0, 0]=[¼, 0, 0]because when the inputs are both true 50% of the time, then you can make 4 combinations, of which only one results in true, hence 25%.[¼, 0, 0] Un-And [½, 0, 0]=[½, 0, 0]because of the rule above.[0, 0, 0] Un-And [0, 0, 0]=[~, 1, 0](read: "Uncertain") because every possible value (except impossible values) when AND-ed with false, produces false.[1, 0, 0] Un-And [0, 0, 0]=[~, 0, 1](read: "Contradiction") because there exists no value that, when AND-ed with false, produces true.
Going a bit more complex one can show that,
[~, 1, 0] Un-And [½, 0, 0]=[~, ½, ½]. Of all the possible inputs on the left, half of them $(½..1]$ would be too high, producing a contradiction, and the other half $[0..½]$ when Un-ANDed by $½$, produce all the values from $0..1$, so uncertainty is maximum.
Problems arise when the output contains a non-uniform range or a range not spanning the full $0..1$. For example:
[½, 0, 0] Un-And [~, 1, 0]=[ , , ½]. Of all the possible inputs on the right, half of them $[0..½)$ produce a contradiction, which is fine. The othe half $[½..1]$ produces a range spanning $[½..1]$ which is not linear (it's a segment of the reciprocal curve).
I'm uncertain as to how to encode the above result into my unit of information. Perhaps my triple [v, u, c] is inadequate or perhaps v and u should be an intricate probability distribution functions.
I feel like I opened a can of worms that I can't manage myself, but at the same time, I feel it ought to have been opened by mathematicians before, but I can't find evidence of it. Do you guys know if such a logic exists, or how I can fix my problem?