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What do you call a logic that is a gradient between
a gradient between two extremes
and a single point.

So, for simplicity, let’s say an upside-down triangle (▼)…

In my case, specifically, the top corners of that triangle are “False” and “True”, and the bottom corner is “Don’t know”.
So the vertical axis is how sure we are, and the horizontal axis is normal one-dimensional fuzzy logic.

The best I could come up with it “fuzzy ternary logic”. But that’s no good, since the two dimensions are separate things. While “dual fuzzy logic” implies a cube with two corners at the bottom too.

So I thought there’s probably a professor out there who spend years on deep-diving into this and it is probably a whole sub-field of logic. :)

The reason I’m asking, is because this seems to represent the logic of scientific research best, yet I haven’t ever seen a name for it. (Mostly because most of the time, vertical axis is unfortunately ignored in science communication.)

(As you can probably tell, I’m not a professional philosopher by any stretch. So be kind. :)

Evi1M4chine
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  • This is similar to [Many-valued two-dimensional logic](https://en.wikipedia.org/wiki/Vector_logic#Many-valued_two-dimensional_logic), the set of truth values there is a triangle. – Conifold Apr 22 '23 at 21:08
  • You might like to know about 'four cornered argumentation' in Buddhist Mahayana thought: https://en.wikipedia.org/wiki/Catu%E1%B9%A3ko%E1%B9%ADi – CriglCragl Apr 22 '23 at 21:24
  • @CriglCragl: Ah, very interesting! This simply adds the possibility of something being *both* true and false at the same time. …Though reality, as far as I perceived it, seems not to have this unless you count quantum superposition, and has relativity instead, which solves things nicely by parametrizing it with the context of the observer. (Something I suspect is also what’s actually going on for what we now know as superposition.) … But anyway, thank you for this! – Evi1M4chine Apr 23 '23 at 08:11
  • **I don’t know which answer to to accept, as they are *both* good, and I don’t know how they relate to each other.** – Evi1M4chine Apr 23 '23 at 08:53
  • "This statement is false." Where do you place that in your system? A little more on applying the catuskoti: https://aeon.co/essays/the-logic-of-buddhist-philosophy-goes-beyond-simple-truth The aim of Buddhist logic is quite different, Nagarjuna brought the catuskoti to prominence, and he used it to conclude: "The victorious ones have said That emptiness is the relinquishing of all views. For whomever emptiness is a view, That one has accomplished nothing." – CriglCragl Apr 23 '23 at 11:42
  • Is one of the "gradient between" redundant or intentional? – Barmar Apr 23 '23 at 17:01
  • @Barmar: It is very much intentional. – Evi1M4chine Apr 24 '23 at 12:59
  • @CriglCragl: My question was not about making statements about statements, but about input data (like measurements, sensory input, experiencing things). But if one would experience someone making a statement, and judge that as false, then it would go to the top left at the corner of “False”. … Although in the “real world” (as far as I experienced it), there is of course no such thing as a “100% false”, as that would require infinite measurements unless the universe has a finite number of temporal states. :) – Evi1M4chine Apr 24 '23 at 13:04
  • I don't think you can separate sensing & logic in that way, because so much depends on expectation vs what is recieved, which is the foundation of true/false. Discussed here: 'Why is a measured true value “TRUE”?' https://philosophy.stackexchange.com/questions/81655/why-is-a-measured-true-value-true/81664#81664 There is such a thing as "100% false", in systems where that is by a definition, & the criterion for the definition applies, eg electronic logic gates (nb the excluded middle is also defined). – CriglCragl Apr 24 '23 at 13:28

2 Answers2

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the vertical axis is how sure we are, and the horizontal axis is normal one-dimensional fuzzy logic.

This would be called "probabilistic fuzzy logic."

causative
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  • I've seen a couple of other examples of this sort of 2-dimensional logic. One, I think by Boolos, involved a combination of probability and a sort of confirmation logic as proposed by Popper. – David Gudeman Apr 22 '23 at 19:08
  • Interesting! So to see if I understood this correctly: The “probabilistic” represents what I called the “vertical axis”, yes? – Evi1M4chine Apr 23 '23 at 08:12
  • Hmm… The “both the concepts of probability of truth and degree of truth in a unique framework” from the linked paper might answer that. But just to be sure, I’d appreciate getting a sanity check before making that assumption. :) – Evi1M4chine Apr 23 '23 at 08:23
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    @Evi1M4chine Yes, probability is a way to represent how sure you are about something, which is your vertical axis. – causative Apr 23 '23 at 09:11
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(The three-valued logic of Łukasiewicz represents the corners of your triangle. He began with a three-valued modal logic; it was later generalized to n-valued as well as infinitely-many-valued variants, both propositional and first order.

Łukasiewicz logic was motivated by Aristotle's suggestion that bivalent logic was not applicable to future contingents, e.g. the statement "There will be a sea battle tomorrow". In other words, statements about the future were neither true nor false, but an intermediate value could be assigned to them, to represent their possibility of becoming true in the future.

-from the Wikipedia article Łukasiewicz logic

The third or middle truth value can be interpreted as "Unknown", but alternate interpretations such as "Possibly and possibly not", "Neither proven nor disproven" and "Contingent: Neither necessary nor impossible" are also possible.

Fuzzy logic was developed independently and assigns real numbers between 0 and 1 to the truth values. Łukasiewicz logic, (or at least a modest extension of it which defines a "strict Łukasiewicz conditional") and fuzzy logic are both examples of a deMorgan algebra (a generalization of Boolean algebra).

As far as I know, the connections between Łukasiewicz logic, especially the infinite valued version, and fuzzy logic have not been fully explored.

Confutus
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  • Hmm… I cannot interpret this “intermediate value”… Would that be the “We don’t know yet?”. – Evi1M4chine Apr 23 '23 at 08:16
  • Is it true that this is not a fuzzy logic though? Or is it not a part of what defines this logic, whether it is fuzzy or not? – Evi1M4chine Apr 23 '23 at 08:19
  • Ignoring the fuzziness question, are there differences between @causative’s “probabilistic fuzzy logic” and this? (Ignore these comments by me if they should better be a new question and aren’t just for clarification. :) – Evi1M4chine Apr 23 '23 at 08:20
  • Just read the wiki article a second time, and it says ”t-norm fuzzy logic”. But it still looks like it is a logic of discrete values. I guess it is fuzzy and I have to read up some more. :) So thank you too. This is very useful! – Evi1M4chine Apr 23 '23 at 08:26
  • The intermediate value has several possible interpretations. "Don't know" usually works. "Possibly, and possibly not" is another good one. "Neither proven nor disproven" is another. In some contexts, "contingent: neither necessary nor impossible" – Confutus Apr 23 '23 at 08:28
  • To be precise, since *Łukasiewicz logic* seems to be generalized to any number of possible values nowadays, one could call my case “three-valued Łukasiewicz logic”…? – Evi1M4chine Apr 23 '23 at 08:29
  • Yes, all those interpretations fit. It’s broader than just “Don’t know”. I would dare to even use “equal superposition of all possible states” in quantum physics, or “perfect white noise” in music production. :) – Evi1M4chine Apr 23 '23 at 08:33
  • Łukasiewicz 3-valued logic represents the three boundary cases, while fuzzy logic has as many truth values as the real numbers between 0 and 1 inclusive. However, they do have a similar algebraic structure. – Confutus Apr 23 '23 at 08:44
  • Well, now I don’t know anymore. :)) … I guess the algebraic structure of what I meant would be `[ (truth ⋅ confidence) | truth <- zeroToOne, confidence <- zeroToOne] where zeroToOne = [0..1] :: [Float]` , written in Haskell programming language style, where `⋅` would be an operator that limits `truth` as much, as `confidence` is close to `0`. – Evi1M4chine Apr 23 '23 at 08:48
  • Both are deMorgan algebras (a generalization of Boolean algebra). Well, I use a slight extension of Łukasiewicz 3-valued logic by defining and using a strict Łukasiewicz conditional, so I get somewhat unorthodox results. – Confutus Apr 23 '23 at 08:58