Fuzzy set theory

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Corbin Simpson

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Sep 24, 2020, 1:57:32 PM9/24/20
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coi

la gleki asks in private communication whether I've figured out fuzzy set theory semantics.

# The boring parts

What are truth values? We'll need to presume them in order to get started. The truth values are a partially ordered collection with at least two members, often called "true" and "false" or "top" and "bottom" or "1" and "0". (A partial order is like a less-than-or-equal relation, but might not relate all elements.) We require that "false" is less-than-or-equal to "true".

To do more with truth values, we also have conjunction and disjunction, with the requirement that "false" absorbs conjunction and is a unit for disjunction, and "true" absorbs disjunction and is a unit for conjunction. In demonstrative terms, "false and X" should be "false", "false or X" should be "X", "true or X" should be "true", and "true and X" should be "X". We also want a negation which sends "true" to "false", "false" to "true", is an involution, and obeys the De Morgan laws. By being involutive, I mean that negated negation is just identity, and the De Morgan laws are simple classical rules relating negation and conjunction/disjunction.

The Booleans are the typical truth values, with the typical multiplication and addition for conjunction and disjunction, and the typical (only possible) negation. Today, though, we'll consider the real numbers instead, with 0 as false and 1 as true. We have our choice of conjunction and disjunction (reader's exercise: Look up "t-norm") but we will choose two-argument minimum and maximum. Since this is a t-norm, we satisfy De Morgan's laws. The real numbers are big enough to do measure theory and thus probability theory; we will want to take integrals later.

To be aggressively concrete, here's some Python code.

    zero = 0.0
    one = 1.0
    conjunction = lambda x, y: min(x, y)
    disjunction = lambda x, y: max(x, y)
    negation = lambda x: 1.0 - x

What are sets? Classically: A set is a collection of elements. A fuzzy set is a collection of elements with associated truth values for each element; these associated values form a membership function on the set, sending each element to its truth value. We can think of membership in a fuzzy set as "graded" or "by degrees"; the question is not whether an element is a member of a fuzzy set, but by how much an element is a member.

In the traditional constructive sense, a set is a procedure for enumerating elements. A fuzzy set enumerates a truth value alongside each element, indicating the degree to which the element belongs to the set. When disjunction is used to tally up the truth values per element, then we recover the normal constructive notions of cardinality and membership. Note that the question of membership is still as hard as before, but now there's an additional question for each grade: Is the given element a member with at least a truth value above the chosen grade? That is, constructively, "is it ever at least half-true?" is as hard as "is it ever true?"

What is negation? The complement of a fuzzy set has all of the normal problems of asking what is not among its elements. Fuzzy set theory is always defined relative to a universe of discourse. For Lojban, this has amusing consequences. Suppose {mlatu} designates a fuzzy set (discarding species for now). Then {na mlatu} would also designate a fuzzy set. Since I am not a cat (with high probability) my grade in {na mlatu} must be quite truthy, but {na} is an involution and so I am a member of {na na mlatu} == {mlatu} with low but not-quite-zero probability.

We might be able to fix this with first-order many-sorted logic, which enforces type constraints on elements. This isn't really visible classically, but constructively we could imagine that enumerating {mlatu} will either include me with tiny/zero probability if we're using full second-order logic, or never include me when using first-order many-sorted logic. In the latter, there is a "sort of sets of cats" and I simply am not of the correct type to be considered as a cat. (cf my earlier mail "Difficult logical questions")

Maybe this isn't a problem, though. After all, actually being completely certain about something is well-known to be a problem which impedes Bayesian reasoning, Cromwell's Rule. It is far safer to default to things being almost certainly, but not quite false, when considering possibilities.

What about relations? Everything immediately generalizes cleanly, but let's spell it out. A fuzzy (binary) relation has, for each pair of related elements, an associated truth value. Think of a relation as enumerating rows of tuples; to make it fuzzy, we are appending a truth value onto each tuple. Returning to {mlatu}, the fuzzy relation between mlatu1 and mlatu2 indicates the degree to which each given cat is of each species.

(Category theory time! Let R be the real numbers and Rel the category of sets and relations; the Goguen category Rel(R) has fuzzy relations as objects and is monoidal closed (so that its arrows are also fuzzy relations); under any choice of t-norm, we can proceed as normal. Everything is obviously correct; just pipe the truth values through each diagram and use disjunction to make everything commute.)

Finally, set union and intersection are given by disjunction and conjunction respectively on their truth values. Simple and sweet.

# The bikeshed parts

Lojban appears to already have {jei} set up to handle fuzziness. Simply allow {du'u} to generalize to {jei} and we're done, right?

Well... The first objection that I know of is that {jei} is not much used, and so we have to have the typical discussion of whether or not formalism and clarification are appropriate things to do. Maybe a different abstractor should be used; maybe {du'u} should have a du'u3 for truth values or probabilities; etc.

A more interesting objection is that {du'u} and {jei} can't code for truth values without somehow reducing all propositions down to truth values, e.g. {lo du'u mi mlatu} is obviously just "false", while {lo jei mi mlatu} is some small non-zero epsilon. However, this is reductive and not how truth values actually work. Any rich-enough logic (say, that can express Peano arithmetic) suffers Tarski's Undefinability and will have propositions which simply do not have consistent truth values assigned. In other words, the membership function can be partial. (cf "Difficult logical questions" and constructivism)

A deeper objection to the mere existing syntax comes from the idea of "Boolean blindness", the sort of obliviousness to Booleans caused by overexposure to classical logic. {li pa} and {li no} are poor ways to say "true" and "false" in the symbolic sense. I don't have suggestions here, just a highlight of the problem.

Another problem that I don't have suggestions for handling is the inability to talk about fuzzy selbri. There ought to be a BAI so that {lo broda BAI li PA} performs some sort of graded selection and only considers those broda1 which grade at least PA in membership/relation. But not only is there no such BAI, there are not even any reasonable candidates among the gismu for {fi'o} nonce tagging.

Which gismu? {pacna} specifically predicts information not yet known to pacna1; it's an agent making a prediction using a model. {cunso} describes random variables with cunso3 being a probability distribution function (PDF) rather than a fuzzy grade; it is possible to build random variables from fuzzy sets but they are not the same thing. Both {pacna} and {cunso} seem to require the passage of time, which is also a problem. {cu'o} and {lakne} are possible candidates but also have the same problems with time.

In the framing set up so far, la gleki's central question for me was whether we can have a selbri with a structure like "x1 is an element of fuzzy set x2 with probability/grade x3".

We've covered some issues around this sort of selbri: Negation causes impredicative quantification over all sorts of elements, and we can't just obtain the selbri by adding a BAI to {cmima}. Another issue: Should this selbri itself be "crisp"; that is, non-fuzzy? It's known that fuzzy logic can't tell whether sets are crisp, so it might not matter. We can't even tell if the set of truth values is crisp; there are usually uncomputable real numbers, unless we're constructive and we use an axiom to force all real numbers to be computable ala Turing.

That said, I'll start the bikeshed by proposing {raucmi}, "c1 is an element of fuzzy set c2 with membership grade g1 (g2=unity, g3=pa jei c1 cmima c2)" which neatly does the minimum required.

Finally, what if somebody doesn't want fuzzy logic? Then they can go back to Boolean logic. Pick a threshold near 1, like 0.99. Then send all truth values above 0.99 to "true", all values below 1 - 0.99 to "false", and everything in the middle to "inconclusive". This is a standard three-valued logic. When the threshold is 0.5, then two-valued Boolean logic is the result. But we should stress that this is always an arbitrary choice rather than a core feature of fuzzy logic; there's no way to force somebody to not interpret {du'u} as {jei}.

I hope that this clarifies fuzzy set theory a bit. As you can hopefully see, most of the issues of set theory are inherited unchanged, and we get many new difficult questions about probability theory. Indeed, the next chapter would be to explore measures on fuzzy sets.

mi'e la korvo .i di'ai

gleki.is...@gmail.com

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Sep 26, 2020, 4:51:15 AM9/26/20
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Em quinta-feira, 24 de setembro de 2020 às 20:57:32 UTC+3, Corbin Simpson escreveu:

la gleki asks in private communication whether I've figured out fuzzy set theory semantics.
...
In the framing set up so far, la gleki's central question for me was whether we can have a selbri with a structure like "x1 is an element of fuzzy set x2 with probability/grade x3".


somewhat. My question was about a predicate "blalaseta = x1 is a member of set x2 in property x3"  but it should probably be extended to "x1 is a member of x2 (set of elements of the same type as x1) in property x3 (ka, property of each element of x2) to the degree x4 (li, belongs [-1:1])".
Then x4=li ni'u pa would be equal to x3=li pa but with x3 having the predicate of the property inverted with {na}. Or alternatively, x4 could be limited to [0:1]. Additionally, x5=epistemology could be added or it all could be split into something like 

A boi S boi P boi J boi E blalaseta => J jei Aboi Sboi Pboi ckini* kei be E 
(obviously ckini needs a bit of pretuning)


  

gleki.is...@gmail.com

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Sep 26, 2020, 4:55:24 AM9/26/20
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Em quinta-feira, 24 de setembro de 2020 às 20:57:32 UTC+3, Corbin Simpson escreveu:

Another problem that I don't have suggestions for handling is the inability to talk about fuzzy selbri. There ought to be a BAI so that {lo broda BAI li PA} performs some sort of graded selection and only considers those broda1 which grade at least PA in membership/relation. But not only is there no such BAI, there are not even any reasonable candidates among the gismu for {fi'o} nonce tagging.


That's not something geeky. It was asked before for translating "I barely speak Lojban" because it modifies not the arguments (neither Lojban nor me are fuzzy). 

My attempt for such a brivla
"enki = x1 is bound to x2 by relationship x3 (relationship with two {ce'u} for x1 and x2); Used to more precisely express relationship (whether fuzzy or not) between terbricmi (predicate slots) and arguments (sumti) filling those slots. See also ckiniva'ejeisela'u." can be considered a failed one (indeed, how is it different from ckini?) 

Jacob Thomas Errington

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Sep 26, 2020, 10:03:47 AM9/26/20
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On Sat, Sep 26, 2020 at 01:55:24AM -0700, gleki.is...@gmail.com wrote:
> That's not something geeky. It was asked before for translating "I barely
> speak Lojban" because it modifies not the arguments (neither Lojban nor me
> are fuzzy).

What's wrong with {.i mi so'uva'e lo ka se jbobau} ?

Also, I reckon this also a translation problem. Perhaps {.i mi tolcre
lo ka se jbobau} or {.i mi tolcre lo ka tavla fo lo lojbo} more
accurately captures what one means to say in "I barely speak Lojban."

> My attempt for such a brivla
> "enki = x1 is bound to x2 by relationship x3 (relationship with two {ce'u}
> for x1 and x2); Used to more precisely express relationship (whether fuzzy
> or not) between terbricmi (predicate slots) and arguments (sumti) filling
> those slots. See also ckini

How is this any different from {ckini} ?

.i mi'e la tsani mu'o

--
Jacob Thomas Errington
W: https://jerrington.me/
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gleki.is...@gmail.com

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Sep 26, 2020, 4:04:47 PM9/26/20
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Em sábado, 26 de setembro de 2020 às 17:03:47 UTC+3, la tsani escreveu:
On Sat, Sep 26, 2020 at 01:55:24AM -0700, gleki.is...@gmail.com wrote:
> That's not something geeky. It was asked before for translating "I barely
> speak Lojban" because it modifies not the arguments (neither Lojban nor me
> are fu

What's wrong with {.i mi so'uva'e lo ka se jbobau} ? 

Also, I reckon this also a translation problem. Perhaps {.i mi tolcre
lo ka se jbobau} or {.i mi tolcre lo ka tavla fo lo lojbo} more
accurately captures what one means to say in "I barely speak Lojban."

> My attempt for such a brivla
> "enki = x1 is bound to x2 by relationship x3 (relationship with two {ce'u}
> for x1 and x2); Used to more precisely express relationship (whether fuzzy
> or not) between terbricmi (predicate slots) and arguments (sumti) filling
> those slots. See also ckini

How is this any different from {ckini} ?


Well the initial goal of enka was applying fuzzy relation to two arguments of the selbri, not to make the predicate fuzzy as a whole 

bangu2 is fuzzily related to bangu1 while bangu1 has a strong relationship with bangu3 
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