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. :)

  • This is similar to Many-valued two-dimensional logic, the set of truth values there is a triangle.
    – Conifold
    Commented Apr 22, 2023 at 21:08
  • You might like to know about 'four cornered argumentation' in Buddhist Mahayana thought: en.wikipedia.org/wiki/Catu%E1%B9%A3ko%E1%B9%ADi
    – CriglCragl
    Commented Apr 22, 2023 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!
    – anon
    Commented Apr 23, 2023 at 8:11
  • 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.
    – anon
    Commented Apr 23, 2023 at 8:53
  • "This statement is false." Where do you place that in your system? A little more on applying the catuskoti: aeon.co/essays/… 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
    Commented Apr 23, 2023 at 11:42

2 Answers 2


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."

  • 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. Commented Apr 22, 2023 at 19:08
  • Interesting! So to see if I understood this correctly: The “probabilistic” represents what I called the “vertical axis”, yes?
    – anon
    Commented Apr 23, 2023 at 8:12
  • 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. :)
    – anon
    Commented Apr 23, 2023 at 8:23
  • 1
    @Evi1M4chine Yes, probability is a way to represent how sure you are about something, which is your vertical axis.
    – causative
    Commented Apr 23, 2023 at 9:11

(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.

  • Hmm… I cannot interpret this “intermediate value”… Would that be the “We don’t know yet?”.
    – anon
    Commented Apr 23, 2023 at 8:16
  • 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?
    – anon
    Commented Apr 23, 2023 at 8:19
  • 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. :)
    – anon
    Commented Apr 23, 2023 at 8:20
  • 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!
    – anon
    Commented Apr 23, 2023 at 8:26
  • 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
    Commented Apr 23, 2023 at 8:28

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