**By Steve Bolton**

…………Fuzzy set relations carry an added layer of complexity not seen in ordinary “crisp” sets, due to the need to derive new grades for membership in the resultset from the scores in the original sets. As I explained two weeks ago in this series of amateur self-tutorials, binary set relations like fuzzy intersections are a bit more complicated than reflexive set relations like fuzzy complements, since we have to perform fuzzy aggregation operations on multiple sets. The good news is that we can reuse most of the concepts introduced in the last article for fuzzy unions, which are handled in an almost identical way. With intersections, we’re basically requesting “all of the records that are members of a both sets,” but with unions, we’re asking SQL Server, “Give me any records that are members of either set.” When we use the fuzzy versions of either, that amounts to retrieving maximum values in the case of intersections and minimums in the case of unions. With the crisp sets SQL Server users work with every day, set relations are straightforward because no calculations of membership values are involved, but with fuzzy intersections there are myriad ways of defining these grades; fortunately, the mathematical functions for fuzzy unions are quite similar to those introduced in the last article. Last time around I provided sample code for four basic types of fuzzy intersections known as the standard intersection, drastic intersection, algebraic product and bounded difference; this week, I’ll demonstrate how to code the corresponding inverses, which are known as the Standard Union, Drastic Union, Algebraic Sum and Bounded Sum. Just as the concept of fuzzy intersections are best exemplified by a class of mathematical functions known as T-norms, so too are fuzzy unions typified by another class known as T-conorms. As we shall see, the authors and parameters of the most popular T-conorms are almost identical to those I coded for the last installment, based on my favorite mathematical reference on the topic, George J. Klir and Bo Yuan’s *Fuzzy Sets and Fuzzy Logic: Theory and Applications*. At the tail end of the last article I offered suggestions for coping with the odd data modeling issues that crop up in cases when it is desirable to persist parameter values for such advanced functions; I won’t rehash that here because the solutions are exactly the same for fuzzy unions. I’ll also omit discussion of the theorems and formulas that justify these functions, since it’s not really necessary to overload end users with that kind of information, for the same reason that it’s not necessary to give a dissertation on automotive engineering to get a driver’s license. Suffice it to say that T-conorms share many of the same mathematical properties as T-norms, except that they’re characterized by superidempotency instead of subidempotency, which you don’t really need to know unless you’re devising your own fuzzy aggregates. Instead, I’ll spend more time discussing the use cases for the different varieties of fuzzy set relations; suffice it to say for now that it boils down to looking for different shades of meaning in conjunctions like “or” and “and” in natural language, which often correspond to fuzzy unions and joins.

…………First, let me get the code for fuzzy unions out of the way. To readers of the last article, the structure of Figures 1 through 4 ought to look familiar. Once again, I’m using the output of two stored procedures I wrote for Outlier Detection with SQL Server, part 2.1: Z-Scores and Outlier Detection with SQL Server, part 2.2: Modified Z-Scores as my membership functions (you can of course test it on other membership functions that have nothing to do with Z-Scores), then storing the results in two table variables. As usual, the GroupRank and OutlierCandidate columns can be safely ignored (they’re only included in the INSERT EXEC because they were part of the original procedures), while the ReversedZScore column is used in conjunction with the @RescalingMax, @RescalingMin and @RescalingRange variables to normalize the results on the industry-standard fuzzy set range of 0 to 1. In fact, there aren’t any differences whatsoever in the code till we insert the results of the union in a third table variable, so that we can demonstrate several T-conorms with it later on; in practice, it may be possible to perform fuzzy unions without these table variables, depending on how the membership functions are being calculated. We don’t have to perform any mathematical operations on the records returned by ordinary crisp relations, but this is not true with their fuzzy counterparts. Using a regular INTERSECT operator in the sample code in the last tutorial would have been awkward to say the least, since it would have excluded unmatched values for the MembershipScores of both sets, which must be preserved to calculate the grade for membership in the fuzzy relation. The same problem arises when we try to implement a fuzzy union with the T-SQL UNION operator, so I had to resort to a workaround similar to the one used for fuzzy intersections, with a few cosmetic changes. I use a FULL JOIN here instead of an INNER JOIN and had to add IsNull statements and a CASE to substitute grades of 0 whenever the Value of one of the columns is NULL. The same two WHERE clauses I used after the join condition in the last tutorial are present here, but are buried in two subqueries; thanks to the null checks, they’re assigned membership grades of 0 on one of the two sets, whenever the WHERE clause removes them from one side but not the other.

** Figure 1: Sample Code for a Fuzzy Union on Two Different Types of Z-Scores**DECLARE @RescalingMax decimal(38,6), @RescalingMin decimal(38,6), @RescalingRange decimal(38,6)

DECLARE @ZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

DECLARE @ModifiedZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint,

OutlierCandidate bit

)

INSERT INTO @ZScoreTable

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

— RESCALING

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

INSERT INTO @ModifiedZScoreTable

(PrimaryKey, Value, ZScore, GroupRank, OutlierCandidate)

EXEC Calculations.ModifiedZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@OrderByCode = 8,

@DecimalPrecision = ’38,32′

— RESCALING

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ModifiedZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ModifiedZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

— DERIVING THE Union

DECLARE @UnionTable table

(PrimaryKey sql_variant, Value decimal(38,6), MembershipScoreForSet1 decimal(38,6), MembershipScoreForSet2 decimal(38,6),

StandardUnion AS (CASE WHEN MembershipScoreForSet1 >= MembershipScoreForSet2 THEN MembershipScoreForSet1 ELSE MembershipScoreForSet2 END),

DrasticUnion AS (CASE WHEN MembershipScoreForSet1 = 0 THEN MembershipScoreForSet2 WHEN MembershipScoreForSet2 = 0 THEN MembershipScoreForSet1 ELSE 1 END),

AlgebraicSum AS (MembershipScoreForSet1 + MembershipScoreForSet2) – (MembershipScoreForSet1 * MembershipScoreForSet2),

BoundedSum AS (CASE WHEN MembershipScoreForSet1 * MembershipScoreForSet2 > 1 THEN 1 ELSE MembershipScoreForSet1 * MembershipScoreForSet2 END)

)

INSERT INTO @UnionTable

(PrimaryKey, Value, MembershipScoreForSet1, MembershipScoreForSet2)

SELECT CASE WHEN T1.PrimaryKey IS NULL THEN T2.PrimaryKey ELSE T1.PrimaryKey END,

CASE WHEN T1.Value IS NULL THEN T2.Value ELSE T1.Value END,

IsNull(T1.MembershipScore, 0), IsNull(T2.MembershipScore, 0)

FROM (SELECT PrimaryKey, Value, MembershipScore

FROM @ZScoreTable WHERE Value != 142) AS T1

FULL JOIN (SELECT PrimaryKey, Value, MembershipScore

FROM @ModifiedZScoreTable WHERE Value != 147) AS T2

ON T1.PrimaryKey = T2.PrimaryKey

WHERE T1.Value IS NOT NULL OR T2.Value IS NOT NULL

SELECT *

FROM @UnionTable

ORDER BY DrasticUnion ASC

__Figure 2: Sample Results from the Duchennes Table[ii]__

…………Basically, the four computed columns in Figure 1 implement the Standard Union, Drastic Union, Bounded Sum and Algebraic Sum by inverting the operations used for the corresponding fuzzy intersections.[iii] The only record which has a zero membership value for the first set also has a zero membership value for the second, hence it’s assigned a fuzzy union value of 0, so there a no Drastic Union values in between 0 and 1 in this case. A few weeks ago I had to order the fuzzy intersection results by the DrasticIntersection DESC in order to depict any difference in the return values of that more restrictive function, but this time I had to order by the DrasticUnion in the opposite direction to do the same. That is because the two operations are basically the inverse of each other, with the first being a maximum and the other a minimum.

…………The more complex T-conorms cited by Klir and Yuan and other fuzzy set experts also implement minimum bounds for membership in the resultset, just in a more specific manner than the four listed above.[iv] The code in Figure 3 is not much different from that of Figure 3 in the last article, except for the substitution of MIN operators for MAX, plugging in a reciprocal here and there and other such tweaks. Once again I used the VALUES trick posted by Jamie Thomson a few years back to perform row-by-row MAX and MIN operations, instead of using them to retrieve aggregates for the whole table.[v] The fuzzy unions are named after the same authors mentioned last week, with those by József Dombi, M.J. Frank, Horst Hamacher, Yandong Yu, Didier Dubois, Henry Prade. Ronald R.Yager, Michio Sugeno and Siegfried Weber segregated into the top query. For the sake of convenience and readability, I put the four T-conorms developed by Berthold Schweizer and Abe Sklar in the second query. A couple of differences include the @OneDividedByLambdaParameter and complicated CASE statement for the SchweizerAndSklar3 fuzzy union, which are designed to strain out some opportunities for Divide By Zero and Invalid Floating Point Operation errors. The WHERE clause in both SELECTs are designed to do the same through brute force, instead of cluttering the code, which is only for demonstration purposes. I suggest more strongly than ever that my calculations be rechecked if accuracy is important for your use cases, since there are few step-by-step examples with sample data in the literature for me to validate these on.

** Figure 3: Code for Several Popular Types of T-Conorm Fuzzy Unions**SELECT PrimaryKey, Value, MembershipScoreForSet1, MembershipScoreForSet2,

1 – ((1 – MembershipScoreForSet1) * (1 – MembershipScoreForSet2)) / (SELECT MAX(Value) FROM (VALUES (1 – MembershipScoreForSet1),(1 – MembershipScoreForSet2),(@AlphaParameter)) AS T1(Value)) AS DuboisPradeTConorm, — basically the same as the intersection except we add some 1 minuses — I adapted this trick from http://sqlblog.com/blogs/jamie_thomson/archive/2012/01/20/use-values-clause-to-get-the-maximum-value-from-some-columns-sql-server-t-sql.aspx

Power(1 + Power(Power(CAST(1 / MembershipScoreForSet1 AS float) –1, @LambdaParameter) + Power(CAST(1 / MembershipScoreForSet2 AS float) –1, @LambdaParameter), CAST(1 / @LambdaParameter AS float)), –1) AS DombiTConorm,

1 – Log(1 + (((CAST(Power(@SParameter, 1 – MembershipScoreForSet1) AS float) –1) * (Cast(Power(@SParameter, 1 – MembershipScoreForSet2) AS float) –1)) / (@SParameter –1)), @SParameter) AS Frank1979TConorm, — basically the same as the intersection except we add some 1 minuses

MembershipScoreForSet1 + MembershipScoreForSet2 + (@RParameter –2) * (MembershipScoreForSet1 * MembershipScoreForSet2) / (@RParameter + (@RParameter –1) * (MembershipScoreForSet1 * MembershipScoreForSet2)) AS HamacherTConorm,

(SELECT MIN(Value) FROM (VALUES (1), (Power(Power(MembershipScoreForSet1, @OmegaParameter) + Power(MembershipScoreForSet2, @OmegaParameter), 1 / CAST(@OmegaParameter AS float)))) AS T1(Value)) AS YagerTConorm,

(SELECT MIN(Value) FROM (VALUES (1), ((MembershipScoreForSet1 + MembershipScoreForSet2 + (@LambdaParameter * MembershipScoreForSet1 * MembershipScoreForSet2)))) AS T1(Value)) AS YuTConorm,

(SELECT MIN(Value) FROM (VALUES (1), ((MembershipScoreForSet1 + MembershipScoreForSet2 – (@LambdaParameter / @OneDividedByLambdaParameter) * (MembershipScoreForSet1 * MembershipScoreForSet2)))) AS T1(Value)) AS WeberTNorm

FROM @UnionTable

WHERE MembershipScoreForSet1 != 0 AND MembershipScoreForSet2

!= 0

SELECT PrimaryKey, Value, MembershipScoreForSet1, MembershipScoreForSet2,

1 – Power((SELECT MAX(Value) FROM (VALUES (0), (Power(MembershipScoreForSet1, @PParameter) + Power(MembershipScoreForSet2, @PParameter) – 1)) AS T1(Value)), 1 / CAST(@PParameter AS float)) AS SchweizerSklarTConorm1, — just the inverse of the same intersection

Power(Power(MembershipScoreForSet1, @PParameter) + Power(MembershipScoreForSet2, @PParameter) – (Power(MembershipScoreForSet1, @PParameter) * Power(MembershipScoreForSet2, @PParameter)), 1 / CAST(@PParameter AS float)) AS SchwiezerAndSklar2,

CASE WHEN 1 – MembershipScoreForSet1 <= 0 AND 1 – MembershipScoreForSet2 >= 0 THEN 1 – EXP(-1 * Power(Power(1, @PParameter) + Power(Abs(Log(1 – MembershipScoreForSet2)), @PParameter), 1 / CAST(@PParameter AS float)))

WHEN 1 – MembershipScoreForSet1 >= 0 AND 1 – MembershipScoreForSet2 <= 0 THEN 1 – EXP(-1 * Power(Power(1, @PParameter) + Power(Abs(Log(1)), @PParameter), 1 / CAST(@PParameter AS float)))

ELSE 1 – EXP(–1 * Power(Power(Abs(Log(1 – MembershipScoreForSet1)), @PParameter) + Power(Abs(Log(1 – MembershipScoreForSet2)), @PParameter), 1 / CAST(@PParameter AS float))) END AS SchwiezerAndSklar3,

((1 – MembershipScoreForSet1) * (1 – MembershipScoreForSet2)) / Power(Power((1 – MembershipScoreForSet1), @PParameter) + Power((1 – MembershipScoreForSet2), @PParameter) – (Power(1 – MembershipScoreForSet1, @PParameter) * Power(1 – MembershipScoreForSet2, @PParameter)), 1 / CAST(@PParameter AS float)) AS SchwiezerAndSklar4 –basically the same as the intersection except we add some 1 minuses,

FROM @UnionTable

WHERE MembershipScoreForSet1 != 0 AND MembershipScoreForSet2

!= 0

** Figure 4: T-Conorm Results** (click to enlarge)

…………So what’s the point of going to all of this trouble to devise fuzzy unions and intersections? In many use cases this level of computation wouldn’t be of much benefit, but for others we can leverage the different shapes of the function returns depicted in Figure 4 to model different types of uncertainty; in fact, we could combine them into new T-norms and T-conorms more suited to our purposes if necessary.[vi] Theoreticians prefer them not only because they mesh well with fuzzy intersections and unions well, but because they’re well-studied. Apparently, there are at least five main research papers[vii] that detail which use cases match the particular T-norms and T-conorms I’ve introduced in the last couple of articles, but unfortunately, I was unable to get my hands on any of them because of my limited budget.

…………I did manage to find some more recent papers on tangential matters like linguistic connectives, such as AND and OR, which were at least helpful in outlining the main principles for selecting the right ones.[viii] Such discussions usually lead in two directions, one of which is using AND/OR logical operations to aggregate opinions and factor in costs in Decision Theory. Klir and Yuan include a well-written section on how to perform such related tasks as weighting goals, actions and constraints; aggregating multiperson decisions and criteria over multiple stages; selecting fuzzy or crisp responses and using fuzzy input states; and implementing it all with state transition matrices, a topic I touched on briefly in A Rickety Stairway to SQL Server Data Mining, Algorithm 8: Sequence Clustering.[ix]

…………The other direction is towards constructing fuzzy logic arguments from the same building blocks, like NOT, AND and OR. Some surveys have found surprisingly subtle shades of meaning in ordinary linguistic terms like AND and OR, which factor into the selection of the right T-norms and T-conorms. Once I can get my hands on those sources I may be tack another article onto the back end of this series, but for now I will have to leave readers dangling. One other suggestion I might offer for matching use cases is to use the Empirical Distribution Functions (EDFs) I introduced in my tutorial series Goodness-of-Fit Testing with SQL Server, perhaps along with scoring systems like the Kolmogorov-Smirnov Test. In crude terms, these are akin to measuring how closely two histograms match each other, so plugging in the T-norms and T-conorms alongside an ideal curve ought to give us a good starting point. Linear regression is a related technique in the same vein. Some alternatives mentioned elsewhere by Klir and Yuan include using maximum likelihood estimation (MLE) methods, curve fitting and neural nets to derive appropriate parameter values for the T-norms and T-conforms.

**The Bridge to Fuzzy Logic**

As is often the case in the fuzzy set field, simplicity quickly gives way to complexity; just as there are many more ways of defining fuzzy complements, intersections and unions than their crisp counterparts, so too are there many types of fuzzy logic. I’ll omit any in-depth discussion of alternatives like Lukasiewicz logic, since they’re beyond the range of ordinary SQL Server use cases. I’ll limit my comments to pointing out that the translation of fuzzy complements, intersections and unions into fuzzy NOT, AND and OR statements is your ticket across that bridge, should you need to cross it. Also keep in mind that there may be trolls beneath that bridge. Some of the steps are well-traveled, like certain multi-valued logics, which are really a matter of common sense; even a child can grasp the concept of inserting answer like “maybe” between a Boolean range of simple “yes” and “no” answers. Other innovations in the field of fuzzy logic smell of intellectual shock value, which may help a scholar to publish rather than perish; watch out for some of the tell-tale signs, like the justification of contradiction as a form of knowledge or misuse of subjective evidence. In such cases, dreaming up a sly mathematical justification for madness may get a person tenure, but does nothing to advance the cause of human knowledge. A solid mathematical scaffolding is available for the less glitzy forms of fuzzy logic in case you need to climb it, but be aware that it requires some mental gymnastics to navigate.

…………At first glance, even the most reliable brands of fuzzy logic can lead to apparent violations of Aristotle’s three laws of logic, like the Law of Contradiction, the Law of the Excluded Middle and the Law of Identity.[x] How can a thing be yet not be at the same time, or exist in a ghost-like state between being and not-being, or be something other than what it is? To keep from having your mind blown, just remember from Implementing Fuzzy Sets in SQL Server, Part 2: Measuring Imprecision with Fuzzy Complements how records have can have partial membership in both a fuzzy set and its complement. That’s an easily digestible example of how a thing “can be” and “not be” at the same time; the false discrepancy arises from applying ordinary crisp, Boolean-valued logic to a set that was defined on a fuzzy range of values. In the same way, many of the apparent violations of Aristotle’s principles[xi] only arise from similar misapplications of fuzzy to crisp sets and vice-versa; in fact, this is usually the hidden culprit when we encounter the kind of manic misuse of fuzziness I addressed in Implementing Fuzzy Sets in SQL Server, Part 1: Membership Functions and the Fuzzy Taxonomy. It would be a mistake to think that Aristotelian logic is incapable of incorporating fuzzy set theory – as many authors in the field seem to suggest – given that Aristotle himself mentioned in his *Metaphysics* that ambiguity of definitions only leads to the *appearance *of contradictions.[xii]

**Putting Fuzzy Logic to Use**

Once we get over this bump in the road, it is possible to construct towers of logical propositions from fuzzy sets, using tried-and-true methods like generalized modus ponens, generalized modus tollens, generalized hypothetical syllogisms, the intersection/product syllogism, approximate reasoning, multi-valued and interval-valued reasoning, as well as making inferences from “quantified propositions.” [xiii] This includes the Kleene-Dienes, Reichenbach-Lukasiewicz and Goguen brands of fuzzy implications, which are a fuzzified version of regular syllogistic match that leads into the topic of fuzzy expert systems.[xiv] As Klir and Yuan point out, “To select an appropriate implication for approximate reasoning under each particular situation is a difficult problem. Although some theoretically supported guidelines are now available for some situations, we are still far from a general solution to this problem.”[xv] Fuzzy logic techniques can even enable us to make exotic apples-and-oranges comparisons of a kind rarely seen in ordinary speech, like Taner Bilgic and I.B. Turksen’s examples of “John is taller than he is clever,” “Inventory is higher than it is low,” “Coffee is at least as unhealthy as it is tasty,” and “Her last novel is more political than it is confessional.”[xvi] These are all examples of sentences with two disparate and imprecisely defined objects, each of which has membership functions measured on the same standard fuzzy scale of 0 to 1. The question becomes even more complicated when we use sentences with two objects and two subjects, like “I am taller then you are skinny,” which in fuzzy set terms means “I belong to the set of tall people more than you belong to the set of skinny people.”[xvii]

…………Fuzzy logic certainly has many legitimate uses, but I’ll omit any further discussion of the topic because of the obvious complexity involved. To that we can tack on the lack of direct application to use cases SQL Server users are liable to encounter anytime soon. Data mining and decision support are definitely within the realm of legitimate SQL Server use cases, but other types of “soft computing” like fuzzy expert systems represent advanced cases that are far beyond the scope of this series. For those with a need to know, Klir and Yuan include an entire chapter on how to assemble fuzzy expert systems out of individual components like knowledge bases, knowledge acquisition modules, explanatory interfaces, “blackboard interfaces” and inference engines, which are composed of chains of fuzzy IF-THEN clauses.[xviii] Of course, scores of books have been written on fuzzy expert systems since then, most of which seem to be just as heavy on arcane fuzzy math. Two articles from now, I’ll explain how a much simpler fuzzy math construct known as a fuzzy number can be easily implemented in SQL Server to address certain use cases that we encounter every day, particularly in modeling fuzzy linguistic statements like “about half” and “almost all” seen in Behavior-Driven Development (BDD) and user stories. First I’ll dispense with another topic that DBAs and data miners encounter on a daily basis, alongside set relations like unions, intersections: the all-pervasive LEFT JOIN operator. In the next installment, I’ll take a stab at explaining its striking absence in the fuzzy set li

p. 77, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[ii] These figures were derived from the Duchennes muscular dystrophy dataset made publicly available by Vanderbilt University’s Department of Biostatistics, which I have been using for practice data for the last couple of tutorial series.

[iii] p. 78, Klir and Yuan.

[iv] *IBID.*, p. 82.

[v] Thomson, Jamie, 2012, “Use VALUES Clause to Get the Maximum Value from Some Columns,” published Jan. 20, 2012 at the __SQLBlog__ web address http://sqlblog.com/blogs/jamie_thomson/archive/2012/01/20/use-values-clause-to-get-the-maximum-value-from-some-columns-sql-server-t-sql.aspx

[vi] pp. 75, 83, Klir and Yuan.

[vii] *IBID*., p. 95. These are the five sources, for anyone interested in implementing these T-norms and T-conforms:

- Thole, U.; Zimmerman, Hans J. and Zysno, P., 1979, “On the Suitability of Minimum and Product Operators for the Intersection of Fuzzy Sets,” pp.167-180 in
__Fuzzy Sets and Systems__, April 1979. Vol. 2, No. 2. - Yager, Ronald R., 1979, “A Measurement-Informational Discussion of Fuzzy Union and Intersection,” pp. 189-200 in
__International Journal of Man-Machine Studies__, March 1979. Vol. 11, No. 2. - Yager, Ronald R., 1982, “Some Procedures for Selecting Fuzzy Set-Theoretic Operations” pp. 235-242 in
__International Journal of General Systems__, 1982. Vol. 8. - Zimmerman, Hans J. 1978, “Results of Empirical Studies in Fuzzy Set Theory,” pp. 303-312 in
__Applied General Systems Research, NATO Conference Series__. Vol. 5. - Zimmerman, Hans J. and Zysno, P., 1980, “Latent Connectives in Human Decision Making, “pp. 37-51 in
__Fuzzy Sets and Systems__, July 1980, Vol. 4, No. 1.

[viii] Among them were Alsina, C.; Trillas E. and Valverde, L. , 1983, “On Some Logical Connectives for Fuzzy Sets Theory,” pp. 15-26 in __Journal of Mathematical Analysis and Applications__. Vol. 93; Dubois, Didier and Prade, Henri, 1985, “A Review of Fuzzy Set Aggregation Connectives,” pp. 85-121 in __Information Sciences__, July-August, 1985. Vol. 36, Nos. 1-2.

[ix] pp. 391-405, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J. On this particular page, they’re extending the meaning of the term even further, to complex network topologies.

[x] Dawkins, Jamie, 2010, “Aristotle’s Three Laws of Thought,” published July 26, 2010 at The Year to the Present blog address https://jamescarterdawkins.wordpress.com/2010/07/26/aristotles-three-laws-of-thought/

[xi] For a handy table of these, see p. 8, Klir and Yuan.

[xii] I’ve read very little Aristotle first-hand, so I’ll have to rely on second-hand quotes from him, such as this one: “It is impossible, then, that ‘being a man’ should mean precisely ‘not being a man’, if ‘man’ not only signifies something about one subject but also has one significance. … And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call ‘man’, and others were to call ‘not-man’; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact. (Metaphysics4.4, W.D. Ross (trans.), GBWW 8, 525–526).” Dawkins, Jamie, 2010, “Aristotle’s Three Laws of Thought,” published July 26, 2010 at __The Year to the Present__ blog address https://jamescarterdawkins.wordpress.com/2010/07/26/aristotles-three-laws-of-thought/

[xiii] pp. 235, 239, Klir and Yuan.

[xiv] pp. 304, 306-307, Klir and Yuan.

[xv] p. 312, Klir and Yuan.

[xvi] p. 16, Bilgic, Taner and Turksen, I.B. August 1994, “Measurement–Theoretic Justification of Connectives in Fuzzy Set Theory,” pp. 289–308 in __Fuzzy Sets and Systems__, January 1995. Vol. 76, No. 3.

[xvii] *IBID*., p. 17.

[xviii] pp. 302-324, Klir and Yuan.