Many-valued inference possible: extant?

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In summary, the conversation discusses the possibility of a type of implication relation in many-valued logic that weakens the truth value. The speaker suggests defining a new rule for implication and mentions some potential issues with this approach. They also mention the need to model human reasoning and the lack of an existing fuzzy logic that fits their criteria. Another speaker suggests fuzzy logic with hysterisis loops as a potential solution.
  • #1


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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.
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  • #2
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.
  • #3
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.
  • #4
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.
  • #5
Ah, well, thanks for trying. Maybe someone who is more familiar with the area will weigh in...
  • #6
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.
  • #7
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.
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1. What is many-valued inference?

Many-valued inference is a type of logical reasoning that allows for more than two truth values, typically in the form of three or more truth values such as "true," "false," and "unknown." It is used to handle complex or uncertain information in a logical and systematic way.

2. How is many-valued inference possible?

Many-valued inference is possible because it expands upon traditional binary logic by allowing for more than two truth values. This is achieved through the use of different logical systems, such as fuzzy logic or three-valued logic, which can handle uncertain or contradictory information.

3. What is the difference between many-valued inference and binary logic?

The main difference between many-valued inference and binary logic is the number of truth values that each can handle. Binary logic only has two truth values (true and false), while many-valued inference allows for three or more truth values, making it more flexible and applicable to a wider range of situations.

4. What are some applications of many-valued inference?

Many-valued inference has a wide range of applications, including artificial intelligence, decision making, and natural language processing. It is particularly useful in situations where there is uncertainty or incomplete information, such as in medical diagnosis or financial forecasting.

5. Is many-valued inference widely accepted in the scientific community?

Many-valued inference has been studied and used in various fields since the early 20th century and has gained acceptance in the scientific community. It is recognized as a valuable tool for handling complex or uncertain information and has been applied in numerous real-world scenarios with successful results.

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