Definitions of Equality

Just definitions today, folks. We’ll need them later. You’d think a lot of these would be obvious. Experience shows they are not. This article is solely about equality of outcomes and not equality used in any other sense.

A philosophically contradictory, but the most desired, definition of equality is with respect to outcomes. Applied to individuals unconditionally, equality fails immediately for the obvious reason that no two individuals can be identical, and therefore they can never be equal. For example, and trivially, they cannot occupy the same space simultaneously.

Sensing, but not explicitly acknowledging this, equality mavens shift the burden from individuals to groups of individuals. Equality, or the lack of it, then becomes a statistical phenomenon, and logically possible, however unlikely.

Suppose we are interested in the equality of outcome for some specific measure, say, a score on a test. Equality of individuals is now logically possible. For example, all individuals can receive an identical score.

The difference is obvious: unconditional equality is impossible—not just unlikely, understand, but impossible. Conditional equality—where it is defined with respect to some finite, quantifiable, measure—is logically possible, even though, in practice, such an outcome is unlikely.

Again, equality mavens are not strictly concerned with this modification and still speak of equality with respect to groups. For test scores, it is, of course, logically possible for two groups to be identical. One way is as just mentioned: all individuals, regardless of group, receive an identical score, an outcome which guarantees all groups will attain equality. But this example is not what mavens usually seek (though they might secretly desire it).

If all we know about an individual is that he belongs to Group A, then we can use probability to quantify the chance that he receives a particular score on a measure of interest. That is, we can compute things like this statement: “Given that an individual is a member of Group A, then the chance he scores above 50 on the test is X”. We can do the same for members of Group B, Group C, and so forth.

One possible definition of statistical equality is the condition where, regardless of Group, each individual has the same X in sentences like this: “Given that an individual is of the Group of interest , then the chance he scores above Y on the test is X.” That is, equality means all groups are equal in X, for each possible value of Y. (Y can only take the [finite] values possible on the measure of interest.) This is predictive equality.

This does not, of course, imply that all individuals in all groups must receive identical scores; it is only a statement about our ability to predict individuals’ scores.

Statistical equality can also be the condition where, regardless of Group, each individual has the same X in sentences like this: “Given a previously observed set of scores and a member in the Group of interest, X-percent of individuals scored about Y.” Again, equality means all groups are equal in X, for each possible value of Y. This is observational equality.

It is of crucial importance to understand the differences between these two definitions: predictive and observational.

Suppose we had only two individuals, one from Group A and one from Group B. “A” scored 40 out of 100 (where higher is better), and “B” scored 60. In this case we do not have observational equality.

If we only had one member of each Group, they must all have the same score for observational equality to hold. And it should be obvious that if our observational data consists of small numbers of individuals, or large numbers of groups, or both, it is very likely that observational inequality will result.

If we liked, having seen the scores from our example, we could write a paper on the “observed discrepancy” and claim the two groups were not equal. And we would be on solid ground as long we resisted the temptation to extrapolate—which is the same as making a prediction—to other individuals in the two groups, whose test scores as not yet known.

This is because it might not be true that predictive inequality exists between the two groups. That is, our calculations might show that individuals from any group have the same probabilities for scores, given additional information about the historical performance of the groups on the measure of interest.

There are several other caveats. Approximate observational or predictive equality can exist, even when actual equality does not, and approximate equality might be “close enough” for equality seekers. A lot depends on what is meant by “approximate”, of course. As long as it unambiguously defined, and that definition is held to, then no problem exits.

Equality, particularly observational, but also predictive since it depends on observations, is variable: a sample of individuals now might show an inequality that does not obtain later, and vice versa.

Inequality, either observational or predictive, approximate or exact, might exist in one dimension, but equality might be present in others. Those who delight in discovering and reporting inequalities should be compelled to report all the dimensions for which inequalities were searched. For example, there are those who search for inequalities in measures of health, and only report the one (or those) which show the largest negative discrepancy with respect to their favorite group. The dimensions which favor their group, or are those which show equality, are frequently left unreported. This naturally exaggerates actual differences.


  1. Speed


    Please comment on the statement, “All men are created equal.”

    … and …

    Where you said, “where higher is better” did you mean, “where larger is better?”

  2. Leonard Weinstein

    Where are you going with this?

  3. TomVonk

    Nobody is created equal , everybody is created different .
    So the reality is exactly the contrary of the first statement .
    The only case where equality makes at least half sense is equality of all members of a group with regards to some set of rules defined by the group .
    That is what one usually calls equality before the law .
    The law being only a set of man made rules regarding rewarded and punished behaviours , it makes sense to postulate that the rules apply to everybody .
    It doesn’t mean that it works (and it usually doesn’t) but there is at least no logical impossibility .
    For everything else given by the nature and distributed more or less normally (strength , intelligence , speed , creativity , voice , looks etc) inequality rules .
    An interesting case is economy (rich vs poor) .
    On one side people who have a specially developped quality (f.ex intelligence or voice) can use it to valorize it and , if they are free , become rich .
    On the other side people who have not a special quality may want to restrict the freedom so that there are rules interdicting to others to become rich .
    So in this particular case the natural inequalities may lead to the existence of rich and poor while man made rules may either enhance or reduce those economical differences .
    I said that this part was interesting because it is a case of nature vs culture opposition that made flow rivers of ink (and blood) since the man exists .

  4. Vinny Burgoo

    ‘Those who delight in discovering and reporting inequalities should be compelled to report all the dimensions for which inequalities were searched.’

    They should also be compelled to report all of the *groupings* for which inequalities were searched.

    When equality mavens can’t find sufficient evidence that individuals have suffered unequal treatment in one or more dimensions because of their membership of a particular grouping, they have taken to searching for other groupings that the retrovictims can join. This is because insufficient evidence for discrimination against individuals due to their membership of different groupings is summable.*

    For example, slight disrespect for a woman’s womanliness + slight disrespect for a homosexual’s homosexuality + slight disrespect for a minority-ethnic’s minority ethnicity = total disrespect amounting to ‘discrimination’ under the law. This is known as intersectional or multiple or [I’ve forgotten the term, but there are three in all and, as you’d expect of such mavens, the definitions are used both interchangeably and contradictorily] discrimination.

    *In UK law. And perhaps EU-wide law. Probably. Possibly. I threw out all my notes about this several months ago. My clear-out wasn’t my fault. Lawsuit in preparation.

  5. Bernie

    Within human society, what is valued may not be equality per se, but an absence of too much man-induced inequality. For example, parity in the NFL is regarded as a beneficial outcome pursued by the league. But too much parity becomes a nightmare as infamous tiebreaking rules need to be created in order to deal with ties. So one does not want equality per se, but rather a limit on the amount of inequality on a year over year basis. Dynasties are deemed bad for business as are perennial losers, viz., Detroit Lions. But these notions are strictly contextual – NFL parity is a goal within a largely man-controlled environment. Other domains of human activity may not be subject to human rules or correcting inequalities may prove to be counter-productive – Animal Farm is a pretty good description of such counter-productiveness.

    Any definition of equality will need to be operationalized and this, I suspect, will be a rather frustrating and largely semantic undertaking.

  6. Katie

    To Speed, I give the wisdom of Nathaniel Ames (1708-1764): “All men are created equal, but differ greatly in the sequel.” (Credit to Florence King, who enjoys this phrase greatly.)

  7. Katie

    Nathaniel was a doctor and almanac maker. His son, Fisher, is the one that Ms. King heaps her praise on. He was a contemporary of one T Jefferson. I’m sure the son picked up some witty remarks around the dinner table, and this is bound to be one of them. I am very much inclined to attribute the father as the originator, as even one B Franklin was something of a copycat.

  8. Doug M

    We hold these truths to be self evident, that all men are created equal — an interesting statement (axiom?) Mr. Jefferson. (What did his slaves think?) It defies the concept hereditary royalty. Divine right is bullshit. While some political philosophers of the Enlightenment had said such things, it was “revolutionary.”

    (Supreme executive power comes from a mandate from the masses not some farcical aquatic ceremony! Strange women lying in ponds is no basis for a system of government)

    Hobbs (ex Calvin) weighs in on the subject…
    Though there be found one man sometimes manifestly stronger in body or of quicker mind than another, the weakest has strength enough to kill the strongest, either by secret machination or by confederacy with others.

    Parity in sports — while the league trumpets the virtue of parity, they lie. Sports fans (and the leagues that sell the product) want a rivalry between superior teams, who show-down year after year. Same goes for individual sports. Federer – Nadal, Agasi – Sampras, Connors – McEnroe, who cares what else goes on in tennis. Heck, fans even seem to like Tiger clearly superior to the rest of the golfing world.

  9. Briggs


    To clarify: I am speaking in this note solely about equality of outcomes and not about equality in any other sense. I did not make this clear enough; I’ll fix it above.


    “All men are created equal” was Jefferson’s. I do not agree, and there is no evidence it is true—nor desirable. With regard to outcomes or any physical or biological measure. See Katie’s comments for a humorous take. It is, however, a pleasant thought to many.

    Leonard Weinstein,

    We’ll soon see.

    Tom Vonk,

    Amen, brother. I hope to find time to discuss the definitions of equality of treatment, which is a more nobler goal, but one which I think is also impossible to always attain.

    Vinny Burgo,

    Quite right: the flexibility of group membership should certainly be stressed. Sorry to hear about your lawsuit.


    You’re right as far as the NFL etc. go, but think of the schools in which winning at sports is discouraged, or where everybody gets a ribbon in a contest, and so on. A truly equal society would be like one gray, unmoving mass—sort of like China was after the Cultural Revolution, after all the blood had dried and was swept away.

    Doug M,

    On the other hand, if some lady is giving me a magical sword, I’m taking it.

  10. Ari

    It always amazed me how vehemently people argue in favor of “equality.” This is especially pernicious amongst my fellow libs, who seem to believe that equality (that is, the notion that all people are somehow the same) is itself a necessary starting point for a just society.

    What is particularly troubling is the blind adherence to the idea that any belief in any sort of inequality amongst individuals or groups means that one must necessarily therefore be bigoted toward that group. I remember being treated rather harshly in class for having the temerity to defend Larry Summers’s statement that we should study differences in men and women before making broad conclusions.

    I was called “sexist” and “ignorant.” When I pointed out that men and women are NECESSARILY different (though not necessarily superior or inferior), I could almost feel knives stabbing me. I blurted out, “if

    I didn’t see what was so controversial about that statement. Men and women ARE different: this is just a fact. Of course the response was, “But women shouldn’t be limited by society!” and “Who says women can’t do the same jobs?” I had no idea what any of that had to do with my statement, but I just shut up after that.

    A few classmates sheepishly agreed with me later in private, but I’ll be damned if I wasn’t the evil misogynistic bastard from hell after that.

  11. 49erDweet

    Ari, it’s the mindless – actually senseless resorting to “if you think this then you must think that” assumptions by friends and associates who believe they know everything that starts an intelligent and self-aware person on the pathway to independent adulthood. You may still consider yourself are a liberal. That you think for yourself is evidence your liberality is evolving. Welcome to the real world, however you turn out.

  12. JH

    I would say that the words “same/different” and “equal/unequal” are interpreted identically when comparing two quantities mathematically. But, I prefer saying that we are born different. In my view, the statement that we are born unequal has a negative connotation and involves the evaluation of a person’s value and a sense of inferiority. I don’t think anyone is born inferior. Our children are born equally dear to us, but with different qualities and levels of abilities.

    There appears to be a confusion between equality and sameness in our society. To me, gender difference and inequality don’t represent exactly the same thing. Group (social) differences can lead to group (social) inequalities. For example, gender differences have resulted in gender inequalities in education and society in Afghanistan.

  13. J. Peden

    Here’s what I learned back in prepostmodern Philosophy about trying to impose “equality” and the like, so it’s naturally longwinded:

    “Unconditional” equality is impossible unless you change the definition of “equality”; for the simple reason that if you tried to make equality “unconditional” you would be trying to impose equality a priori onto reality – whereas the word “equality” instead gets its meaning only because of the possibility of “unequality”, i.e., more or less, better or worse, etc.. So you still have to find out who or what is “equal” according to what variable you are looking at. Just as Briggs said, if I’m receiving it correctly.

    This is a simple feature of how words work to have meaning. Therefore, it likewise makes no sense to worry about “everything being a dream”, thinking you might be only dreaming all the time, but which contradicts the actual use of the word “dream” to also apply to “non-dream” states such that they both get their meaning only by contrasting with each other as possible experiencial features of consciousness. But that is something which your suppostion has just contradicted by saying essentially that non-dream states can’t exist, or the equivalent, that “you can’t ever know they exist” or really that you wouldn’t know one if you saw one, because everything could equally be a “dream”.

    So now you are actually back to square one, right from where you started in trying to analyze reality in reference to “dreams”, right before you changed the definition of the word “dream” to one which really makes no sense. This is a well known propaganda trick: take a word that has meaning, change it so that it actually doesn’t have the same meaning, but act like it does have the same meaning. “Liberal” has strayed from “Classical Liberal”, for example, to be almost the exact opposite of what a Classical Liberal does to arrive at decisions: groupthink vs individual rational thought including doing real Science.

    You could change the definition of “dreaming” to try to say the same thing you were worried about to begin with. So you’d wake up each morning and dutifully say, “It all could be a dream”, but then you’d have to still go ahead with life and make up a new word for the differing states of consciousness you still routinely experience. Eventually you wouldn’t even care about “dreams” as newly defined, unless perhaps Big Brother made you repeat the mantra to keep you well managed, or to get you to “betray” yourself as to actually having a brain capable of thinking for itself by questioning or denying that “perception is reality”.

    You can likewise deconstruct “relativism”, subjectivism, solipsism, determinism, nihilism, da-daism, “we can’t ever know reality” ism and every other damn thing a lot of people try to simply impose upon reality and thinking, because all they are really doing is arguing about word definitions, then probably also constructing a fantasy world consisting of new word definitions and some other sayings and mantras, which are constructed so as to in effect be no more than familiar noises or appearances, etc., mostly unrelated to reality.

    For another example, take when the sophisticates say, “perception is reality”, resorting necessarily also to full blown subjectivism as an extreme case of “reality” being only each person’s “perceptions”. Just tell them to shut up, which follows from what they think their own “claim” means, because all responses follow: my perception is as good as yours, and, btw, you can’t even know what mine is, or what my words even mean – because they are all defined by my subjective “perceptions” which are totally unique by definition since only I can have them and therefore also to the extent that no one can then really be sure what they are when it comes to anyone else’s. There can’t even be any standards to judge them by.

    Continuing with what to tell the “perception is reality” sophisticates: and so I don’t know what your words mean either, and therefore we’re not actually communicating anything to begin with, though according to you, you can’t even know what I’m saying right now, either, so watch me “leave”, but I might instead be trying to “stomp down the grass” or whatever your perception tells you it is. Again, according to the postulate “perception is reality”, word meanings only consist of one’s subjective perceptions, too, so language between people is impossible – which is obviously false.

    But from “perception is reality” everything gets crazy, so anything goes. Just look at “Climate Science”.

    Or tell them that all they are doing when they say “perception is reality” is giving you the definition of “delusionalism” or that they might really be indicating that they are intent upon deluding other people.

    Almost the same thing goes for “equality of wealth” as unconditional or imposed. No one really knows what “equality of wealth” is, unless it’s defined as, say, having “identical” things like houses, refrigerators, pets or “income”. Just think what fun it would be to have to continually be taking “wealth” from someone who decided to put some “income” into adding a garage, while someone else was partying it away into nonexistence. Or when you would try to impose “equality” of “health” by taking some away from anyone having “too much”.

    The Ants seem to have it figured out equality of wealth about as well as it can be: “Power to the Ants!” And to the Commies. Slavery is Utopia, dontcha know. Slavery is pretty equal, except for those who have selected themselves to “manage” the equality.

  14. Briggs


    Maybe all of your children are equally dear to you; mine are to me, too: but this is not true in general. Saying we are born unequal does have negative connotations, but that is only because we have an ordering of preference for abilities, skills, beauty, etc. etc. Attempting to soften it by saying “born different” is not particularly useful.

    J. Peden,


  15. TomVonk

    Actually it is useful (and fun) to look at this question from the point of view of the standard set theory .
    Having a set S and elements of the set , what is equality ?
    Equality is a relation of equivalence (If A=B then B=A , if A=B and B=C then A=C , A=A) .
    It is impossible (and vain) to want to say more .
    However as we have defined a relation of equivalence we are legitimate to ask questions about equivalence classes .
    For example “How many elements are in the equivalence class of A ?”
    To answer this question it is necessary to specify the relation of equivalence by assigning properties to the elements of the set .
    The largest property is appartenance .
    Then I may define the equality by “A=B if and only if A and B are in S” .
    Follows that the equivalence class of A is the whole set .
    If S is mankind then this definition of equality gives “human are equal if they are human .”
    There is a quite clear circularity and this kind of equality definition , even if legitimate from the logical point of view , yields a very uninteresting and poor equivalence relation with only one trivial equivalence class .
    Everybody is equal .
    On the other extreme we can define a set of N measurable properties Pi of elements of the set and define equality by “A=B if and only if Pi(A) = Pi(B) for all i” .
    There exists a finite N for which the equivalence class of every A has only 1 element , A .
    If S is mankind , then this definition of equality gives “2 persons are equal if they have the same height , weight , IQ and last name .”
    This definition where N=4 is probably already sufficient that the equivalence classes for the 6.5 billions people reduce in the crushing majority to 1 element .
    Nobody is equal .
    So the last question , still from the purely logical point of view in the frame of the set theory , would be whether there can be an order (preference) for the definitions .
    Well this one is a meta question because it demands for properties of equality that go beyond the fundamental necessary (and sufficient) property of being a relation of equivalence .
    Clearly we can distinguish the equality definitions by the number of equivalence classes that they induce in the set . And/or by the distribution of the elements among the equivalence classes .
    We could say that rich equality definitions are those who induce interesting (interesting = non trivial like 0+0=0) structures in the set .
    We could justify that we “prefer” rich definitions because they give us something to study and lead to discoveries of new things .
    But somebody could just wave his hand and say that on the contrary it is the trivial sets that should be preferred because they are simple and there is nothing to look for .
    In the end I suspect that it is because of this wide field of subjectivity in the “preferred” definitions that politicians like the notion of “equality” so much .

  16. Briggs


    I love it!

  17. Rich

    J. Peden.
    Well. Am I a butterfly or not?

  18. Bernie

    The set approach makes sense to me, but who said the argument for equality has to be supported by logic? Like the rules of any game, they are essentially arbitrary and may include inconsistencies.

  19. Yooper Paul

    Re: All Men are created Equal.

    He meant this from God’s perspective, not Man’s.

  20. JH

    Interesting comments!

    Mr. Briggs,
    I really didn’t mull over the utility of softening the statement–“Well this one is a meta question” because “subjectivity is truth” might just hold here. ^_^ Perhaps it’s not useful to the world; I am honestly delighted that you’ve appraised my opinion. Let me just say that I have always considered all my friends are different from me, but not unequal.

  21. DAV

    The very act of testing in itself violates the idea of equality. The point of testing is discrimination, is it not? A test by definition has to partition into at least two categories and some categories are often more desirable than others.

  22. Ari


    Well, I’m liberal in the way that the South Park guys are “conservative” (not really.) I’m a liberal out of convenience, simply because I don’t identify as “conservative.” If I had to peg myself as anything, I’m closest to a libertarian, but I’m too pragmatic to be that, either.

    I’m just not a good bi-partisan American.

    On another note, it’s interesting to note that a society where all institutional inequalities were removed may (will?) end up being even more unequal in outcome. In fact, because I believe that inherent inequalities should be heeded in policy and society I cannot fully commit myself to being truly “libertarian.” Hobbes rang too true for me to ignore the fact that some kind of authority must be in place to mitigate inequalities and allow for some kind of just society.

    And I’ll tell ya, there are times when I think we’d be better off with a sovereign than the wailing masses…

  23. TomVonk

    “The set approach makes sense to me, but who said the argument for equality has to be supported by logic? Like the rules of any game, they are essentially arbitrary and may include inconsistencies.”
    It is not only the argument (I prefer definition to argument) for equality that has to be supported by logic .
    It is any argument as long as it is supposed to communicate a statement with well defined values for true and false .
    The problem of inconsistency is that it is strongly pathological and destroys ANY argument .
    That’s why Hilbert was so keen to find a proof that arithmetics is not inconsistent and Gödel brought the proof that it indeed is not but that this privilege is paid by incompleteness .
    Inconsistent definitions/axioms/rules have the property that both A and non A can be “proven” true .
    Such definitions lead to a mad (in the clinical sense of the word) universe .
    Among many madnesses in this universe , equality would not be a relation of equivalence so about any statement like “Fishes are equal to witches because photons cackle” would be a valid and true statement .
    Nobody would survive long in a universe governed by mad rules .
    There is something in the evolution that made our brains such that we don’t admit inconsistencies in matters where survival is at stake .
    However for some obscure reason the brains can tolerate it in poetry .
    Probably because they know that it is not a serious matter 🙂

  24. TomVonk

    Btw to eliminate temptations for word games .
    Different is just an antonyme of equal .
    So non equal (unequal) and different are really synonymes .
    If somebody thinks that unequal and different are not synonymes then he must have a very unusual definition of equality .

  25. Bernie

    I agree that inconsistency is a dangerous and slippery slope for the likelihood of extended rational discourse. However, with all due respect to Hilbert, the reality of human discourse, as ethnographers have demonstrated, is that much of it is ambiguous, imprecise, inconsistent and illogical. Moreover, politicians and marketers are masters of creating slogans that when analyzed are difficult if not impossible to understand.

  26. TomVonk

    However, with all due respect to Hilbert, the reality of human discourse, as ethnographers have demonstrated, is that much of it is ambiguous, imprecise, inconsistent and illogical.
    Well while any rational disourse is founded on (standard) logics there are indeed cases in natural languages when a discourse is ambiguous (e.g not well defined) or even inconsistent (e.g mad) .
    However to the contrary of what you write you , it is not “much” , it is a small part .
    And even in this small part one has to distinguish ambiguity which can be healed by defining better and inconsistency which is deadly .
    I cannot find an example of a set of political statements that would be REALLY really based on inconsistent axioms .
    If such an example existed , anybody could immediately deduce from such axioms that there exist statements that are both true and wrong and find them .
    And a politician caught in saying that both A and non A are true would be shown as being both mad and ridiculous .
    While madness is not necessarily an obstacle to political carriers , ridiculousness definitely is .
    I believe that what you take for inconsistency is actually only an error in deriving valid statements from consistent axioms but not really a statement based on inconsistent axioms .

  27. Bernie

    I also doubt that politicians will be effective if they claim both A and non A are true – but primarily if A and non A are simple statements such that the inconsistency is readily recognized as reflected by your nomenclature. However, more typically IMHO many politicians make political promises that include, when carefully analyzed, a large or small component of A and non A. The current Medical Reform proposals provide numerous examples of A and non A, e.g., more benefits and lower costs! Most of the inconsistencies and ambiguities stem from the inherent complexity or disguised form of the statements about A and non A. The ability to argue A and non A is the hallmark of a confidence trickster – above average returns with no risk, are key components of Ponzi schemes. Bernie Madoff clearly demonstrated that A and non A can be maintained for an extended period of time among smart, rational and normally skeptical investors.

    I think that you are more optimistic about the presence of logic and rationality in human endeavors than I am.

  28. J. Peden

    Rich says:
    6 January 2010 at 8:38 am
    J. Peden.
    Well. Am I a butterfly or not?

    I only know that if you want to be one, you must sing this song every morning:

    Oh, I’m such a happy buttefly
    Just a hap hap happy flutterfly
    Tra la la la la, Tra la la la la la la la

    The music of my happy wings
    Gives life and love to everything
    Tra la la la la, Tra la la la la la la la

    I learned this from an ancient Cartoon, but it did seem to work there – probably “Betty Boop”.

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