I have long been interested in notions of social justice and equality, but somehow it all felt a bit loosey-goosey and amorphic. To be honest, I feel this way about the entire composition of government, politics, and jurisprudence, and other power structures, but those are topics for some other days. Also, I won’t endeavour to speak to the artificial income-market construct, so for the purposes of this post, I’ll take this as given, as anachronistic and quaint as it might otherwise be. Nor will I discuss whether the system itself, apart from the equality question, is optimal or even makes any sense from a broader vantage, or whether competition has a role in an otherwise coöperative society.
Along with empty virtue notions as freedom, liberty, and justice, (topics for another day) equality is a post-Enlightenment Age catchphrase. As with its counterparts, it sounds nice; it has a nice ring to it, but it is just as specious. These are words invoked to raise emotions, but as with a pointillist’s painting, if you attempt to scrutinise them too closely, they become unintelligible.
Aside from maths, equality is an issue for sociology as well as political philosophy. In maths, the concept is tautological.
“In sociology, the study of the causes and consequences of inequality in its various forms – class, race, gender, power, status, knowledge, wealth, income – is one of the most pervasive themes of the discipline. In philosophy, theories of justice and rights are centrally concerned with the problem of justifying and criticising different kinds of inequality.” 
Sociologist, Bryan Turner, identified four flavours of equality:
Equality of Condition
Equality of Opportunity
Equality of Outcome
Political Philosophy is more concerned with accepting the sociological definition as given and discussing it in the negative sense of inequality.
In maths, we have likely been made accustomed with equalities since grade school. In fact, that’s what makes the other notions so compelling. It all appears to be so tidy and scientific.
1 = 1
2 = 2
1 + 1 = 2
Amazing, right? Two values balanced on either side of an equals sign. The problem is that these equations—these equalities—are logical tautologies. They are equal because this is in fact how they are defined, defined in the same manner as we define a red lorry as red. Equal in this context is not useful for us, save as a familiar reference.
“We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.”
Aside from an unsubstantiable claim that these so-called “truths” are somehow “self-evident”, this weakest form of equality is a claim “that all men (read: people) are created equal” at birth because “their Creator” (read: God, an obvious metaphysical nod) “endowed” them with this aspect.
Translating this into common parlance, this states that we are all equal in the eyes of God [sic]. In essence, to commence a racing metaphor, this means that we all get to participate in the race, and that sounds good, right?
The problem with ontological equality is that it outright ignores existing inequalities, so whilst you may be equal in God’s eyes, that’s where it ends. Essentially, you get an empty promise. If you’ve got something to say about, say a prayer; God’s got operators standing by. At least you get to play the game.
Equality of Condition
Where ontological equality leaves off, equality of condition steps up. Beyond the metaphysical promise, it claims that all people get to start at the same position, that we each get to start at the same starting line. That sounds fair, right?
The problem with equality of condition is that whilst you may get a place at the starting line, you still face any systematic or structural adversity in play, whether that be discrimination, wealth disparities, access to quality education or other public services, and so on, at least you get to play the start at the same place.
Equality of Opportunity
My Libertarian associates seem to love this one. In fact, in their world, the only conversation is about equality of opportunity (also known as formal equality of opportunity) versus opportunity of outcome. It’s a cage match to the death, and opportunity is their champion.
This flavour of equality doesn’t claim that a person has the right to start in the same place—only that the rules will be the same.
The problem with equality of opportunity is that it makes a specious claim. Besides ignoring the condition and situation and any past infractions—letting bygones be bygones (especially when they have given you the advantage that you wish to retain).
A parallel would be to allow a steroid-pumping athlete to compete by rationalising that, well, the race has started, so let’s just keep playing. Afterall, we’re playing by the same rules, so that makes it fair. Forgive and forget, right?.
The privileged live in better neighbourhoods, have access to better schools, can afford tutors and summer programmes. Many live in more stable family environments growing up, and they have access to networking benefits. This is further reinforced in university, and, like compound interest, the earlier one starts, the greater the effects of compounding.
Again, equality of opportunity might sound good on the surface, but, yet again, it disintegrates on scrutiny. Its main purpose is a feel-good head fake to keep one’s eyes off the prize.
Substantive Equality of Opportunity
A subset of equality of opportunity is substantive equality or fair equality of opportunity. Under this model, additional remediation is asserted to the disadvantaged person. This might be a familiar concept to golfers, who have handicaps. The goal is to—whilst also enforcing a similar rule set—accomodate those with some head start advantage. In the everyday context, it could be providing additional funding or resources to underprivileged children.
The problem with substantive equality of opportunity is that the deficiencies are multifaceted, the system itself is too complex to account for all material dimensions and measures, and most assessments are normative in nature.
Equality of Outcome
Equality of outcome is particularly pernicious. It claims that in the end, everybody wins, and everyone gets the same prizes.
This is a potential result of Communism, that is if the definition is taken to the absurd. This is a common criticism by some when every participant receives a participation prize—a manifestation of the notion that everybody is special.
The problem with equality of outcome is that, among other things, not everybody wants the same thing, so this logic basically boils down to I want what I want if what you have is what I want as long as everyone else who also wants what I want has it, too. Of course, we could reduce this down from actual equality—apples for apples and oranges for oranges—into value equality, where everyone has access to some comparably equivalent value (whatever that might mean, especially insomuch as different people assign different values to the same goods and services).
Kurt Vonnegut depicted this in his short story, Harrison Bergeron (PDF), where, not being able to raise certain persons, people were instead reduced to the lowest common denominator, so rather than elevate the cognitive ability of low IQ people, the solution was to diminish the capacity of higher IQ people, so as to produce the same—albeit lower—results.
In the end, ontological equality is nothing more than vapour; equality of condition fails to account for material differences among people and their situations; equality of opportunity also fails to account for disparities of condition; and equality of outcome is an unrealistic pipe dream that would be too complicated and complex to implement.
In order to further communication, if that is indeed even the purpose (rather than obfuscation, which I feel may be the prime motive), we need to use different concepts, to find new terminology.
 Equality: Sociological & Philosophical Perspectives, Brighouse & Wright, 2009 (https://www.ssc.wisc.edu/~wright/Sociology%20915%20&%20Philosophy%20955%202009%20syllabus.pdf, retrieved 5-9-2017)
Also note, that I reserve the right to make inline edits to this post in an attempt to extend, clarify, and otherwise elucidate this topic on equality.