# Many-valued inference possible: extant?

• I
Gold Member
Would it be possible to have a type of implication relation that weakens the truth value in a many-valued logic? For instance, a first attempt:
(1) define ⇒k, 0<k<1, ⇒1 ≡ ordinary ⇒ , so that, if the V(.) is the valuation (the assignment of truth value), then A⇒kB gives V(B) = k*V(A).
(2) The rule [A⇒kB &B⇒mC] ⇒ [A⇒(k*m)C] would hold instead of transitivity.
(3) [A⇒kB]⇒[~B⇒k~A]
(4) If A⇒kA, then k=1.
The main problem here would be that either the system would have to either
(a) be extremely fine-tuned to avoid two different inference paths to the same conclusion resulting in two different truth values (inconvenient)
(b) adopt something like V(B) = minimum of truth values of all possible inference paths starting from the axioms (unlikely to work)
(c) make this part of a temporal logic so that two different inference paths would occur at different times, and hence V(B, t0) need not be equal to V(B, t1) (that would end up being trivial),
(d) be inconsistent (unfortunate), or
(e) some variation that I have not thought of.

The reason I would like something like this is that, if one is to model human reasoning, one has the problem that humans put less confidence in conclusions when they are more abstract, i.e., when it takes more steps to arrive at the conclusions. (Perhaps I should be using confidence or preference values instead of truth values, but as far as I can see, these would be equivalent approaches.)
I am open to suggestions. Thanks.

Dale
Mentor
2020 Award
Sounds like you are thinking of fuzzy logic. I don’t know the details, just that the truth value of statements is not true (1) or false (0) but is any real number from 0 to 1.

Gold Member
Thanks, Dale, I understand that I am dealing in fuzzy (or "many-valued") logic, but there are many fuzzy logics, and I have not yet seen (but perhaps I'm not looking in the right places) a fuzzy logic which acts like the one I outlined. There are implications that are non-classical, such as the Łukasiewicz implication → such that V( A→B)= min {1-V(A)+V(B)), and others which are usually defined by truth tables. But in all these, implication remains transitive, and are not adequate to correspond to a lessening truth value according to the length of the reasoning chain.

Dale
Mentor
2020 Award
Sorry, I am out of my depth on that then. I don’t even know enough about fuzzy logic to recognize that it doesn’t have the properties you are looking for.

Gold Member
Ah, well, thanks for trying. Maybe someone who is more familiar with the area will weigh in.....

jedishrfu
Mentor
Fuzzy logic may act like a hysterisis loop too. It will stay at zero until some threshold value is crossed and then switch to a one. Similarly going from one to zero, it will stay at one until a different threshold value is reached before switching to zero.

I don’t have an explicit reference just from reading about it years ago. Wikipedia has an article that may be worth reading on multi valued logic which might help here.

Gold Member
Thanks, jedishrfu, I believe you are referring to the logics which can be used for analogs to nerve firings, no? This is an interesting logic, but remains a two-valued logic.
I know the Wiki article (and I have some books on multi-valued logics), but the type of logic which I am looking for is not described in any of these sources.

jedishrfu