The just, therefore, involves at least four terms; for the persons for whom it is in fact just are two, and the things in which it is manifested, the objects distributed, are two.
At this point it makes sense to insert a brief and somewhat formal explanation of social cooperation and agreement, where I presume the logic of propositional attitudes, and some logical facility in the reader, though I will explain the formulas I use:
Conscious social cooperation involves the following, with
"aCFa" = "person a tries to cause the fact that a has property F"
"aDFa" = "person a desires the fact that a has property F"
"aKFa" = "person a knows the fact that a has property F"
First, there is a simple definition of cooperation:
(1) CP(a,b,Fa,Gb) =def aCFa iff bCGb
Cooperation: a and b cooperate concerning Fa and Gb
which is to say that a tries to bring about Fa if and only if b tries to bring about Gb. Next, cooperation requires
(2) A(a,b,Fa,Gb) =def aDCP(a,b,p,q) & bDCP(a,b,Fa,Gb)
Agreement: Both desire the cooperation mentioned in (1)
which is to say a and b cooperate only if both a and b desire to cooperate, and for this again they need to both know the agreement
(3) KAC(a,b,Fa,Gb) =def aKA(a,b,Fa,Gb) & bKA(a,b,Fa,Gb) -
Knowledge of Agreement to Cooperate
which is to say that a and b both know that both desire to cooperate.
And now one can state when the successful social cooperation of a and b has taken place:
(4) KAC(a,b,Fa,Gb) & aCFa & bCGb & Fa & Gb -
Successful social cooperation of a and b about Fa and Gb
Successful social cooperation of a and b about Fa and Gb amounts for a and b to have knowledge of their agreement to cooperate and for both to have done their agreed parts successfully, which will bring to a whatever good was produced by Gb and will bring to b whatever good was produced by Fa.
Next, it is relevant to note the following supposed truth about agreements, as a sort of minimal condition for their fairness, and as explanation for the mutual willingness to cooperate:
(5) A(a,b,Fa,Gb) --> v(a,aCFa) <= v(a,bCGb) &
v(b,bCGb) <= v(b,aCFa)
which is to say that in a minimal fair agreement to cooperate, the persons involved like to exchange because each likes what the other can offer more than what he can offer to the other in exchange.
Note that (5) can be derived from a presumption like this about both a and b, here only formulated for a, and using "v(a,q)" = "the value of q for a":
(6) aDC(a,b,Fa,Gb) --> v(a,aCFa) <= v(a,bCGb)
which is to say that a desires to cooperate only if a believes a will get at least as good as he gives.
It is noteworthy that none of the above requires money or a market in any sense, for all that need be involved are the value-assessments of the parties involved, that should make for the agreement stated by (5).
And it is also noteworthy that there are, between humans, very many ordinary social transactions that conform to the above, and that amount the exchanges of kindnesses, mutual help, barter, friendliness, politeness etcetera, for most of the voluntary cooperation between humans conforms to the above, and needs nothing else, since if the conditions are met, both parties involved by their own values profit from the transaction (or at least don't loose by it).
However, it is true that the above may lead to situations that, at least after the fact of exchange, may be considered quite unfair - for example, as the Indians may have soon found after bartering away Manhattan for a handful of trinkets to the Dutch that they were deceived, and could have received much more than they got out of this nation of sly and dishonest traders.
One can take care of this difficulty in various ways, and I will here do it as follows:
(7) v($,Fa) sim v($,Gb) =def (xe$)xK(pr((Eye$)A(x,y,Fx,Gy))>1/2)
which is to say that I suppose acts and commodities in a society $ have a social value that is similar if and only if everybody in the society knows that they probably can find someone in the society to make a fair exchange of the goods or acts involved.
And now one can define
(8) FA(a,b,Fa,Gb) = def A(a,b,Fa,Gb) &
aK(v($,Fa) sim v($,Gb)) &
bK(v($,Fa) sim v($,Gb))
which is to say that a fair agreement amounts to getting a fair deal for a fair price, where the deal is fair because both parties agree they are willing to exchange, and the price is fair because both parties know they could get a similar deal elsewhere in the society, if they had tried.
Note though that it is still not necessary to use money, and that again there are many transactions between humans in a society that are fair agreements as defined, whether or not they involve money.
 (..) but this is the origin of quarrels and complaints - when either equals have and are awarded unequal shares, or unequals equal shares. Further, this is plain from the fact that awards should be 'according to merit'; for all men agree that what is just in distribution must be according to merit in some sense, though they do not all specify the same sort of merit, but democrats identify it with the status of freeman, supporters of oligarchy with wealth (or with noble birth), and supporters of aristocracy with excellence.
Hence what Aristotle is saying amounts to a kind of proportionality with regards to merit. Suppose a's merit in society or group $ is written as m($,a) and a's share in society or group $ of some good g is written as s($,a,g), and likewise for person b, then there is - something like - justice or equitability or fairness in the distribution of good things g if
m($,a) : m($,b) = s($,a,g) : s($,b,g)
It seems sensible to assume that this will tend to seem to be fair to a and b to the extent that they both agree on their relative merits.
And it should also be mentioned here that equal merits are usually easier to agree to in cases where (i) a and b have both actively contributed to the production of the benefits that are shared out (ii) their merits depend mostly on their personal contributions to the production of these benefits and (iii) what a's and b's personal contributions have been is uncontested.
It becomes less obviously fair when these conditions are not met, and especially where a gets (disproportionally) more than b, because of some difference in caste, nobility, gender, ethnicity, religion etc. and b does not accept that these differences exist or that they are relevant for the distribution. (In rare cases, a may object as well, but as people are, on average, it is not common for people to protest that they get too much of what they desire.)
In any case, the clearly contentious cases of unequal distribution will be especially those where the one gets more than the other not because one made a greater contribution or as some clear personal merit or entitlement, but because the one belongs to a supposedly better kind of persons than the other, where the better kind does not involve personal acts or characteristics but some general characteristic that holds for many regardless of their actions, like race or gender or family.
However, it should be mentioned that during the greatest part of human history clearly unequal schemes of distribution of goods have been accepted because those who got less agreed or put up with the idea that those who got more were entitled to get more by reason of their birth in some group, or by reason of their group's defeating or enslaving those who got less.
Also, it should be noted that most men do think it fair if some that are special - such as: the ill, the weak, the old, the young - get special and favourable treatment, and are entitled to more than would be their fair share on account of their special condition. And this does not only hold for those with undeserved shortcomings, but often also for those with favorable characteristics, such as the beautiful, the brave, the witty.
And a final relevant consideration with respect to the stated proportionality is that it does seem to hold within many human groups; that it often gets the simpler form of "equal merits give equal rights or shares"; and that this is often argued negatively as in: nobody got special treatment or advantage, except perhaps those who are clearly special or disadvantaged.
 The just, then, is a species of the proportionate (proportion being not a property only of the kind of number which consists of abstract units, but of number in general). For proportion is equality of ratios (..)
This is a very important point to get Aristotle's sense right: He insisted that the just is "a species of the proportionate" and held, contrary to modern democratic notions, that not all people are equal.
It should be remarked at this point that modern democratic notions often involve a confusion between (i) the equality of all and (ii) the equality of all for the law, and that the former is a delusion and the latter is desirable, and also that the former is in fact rejected by all or nearly all if they personally benefit from an unequal distribution of something they desire, whereas the latter, even where it exists on paper, is difficult to enforce, since e.g. the poor, the ill or the stupid will often find it impossible to enforce their paper rights in practice.
 This, then, is what the just is - the proportional; the unjust is what violates the proportion. Hence one term becomes too great, the other too small, as indeed happens in practice; for the man who acts unjustly has too much, and the man who is unjustly treated too little, of what is good. In the case of evil the reverse is true; for the lesser evil is reckoned a good in comparison with the greater evil, since the lesser evil is rather to be chosen than the greater, and what is worthy of choice is good, and what is worthier of choice a greater good.
See first under .
Next, the main problem for a proportionally fair distribution will be whether the people who fall under such a distribution agree on the prior distinctions of merit that enter into the proportional sharing - for if you don't agree your neighbour is ten times as good, or contributed ten times as much as you did, you will not think it fair that he receives ten times as much as you do.
However, it should also be noted that in any more or less stable society the majority of its members mostly accept the distinctions of merits between groups, normally it seems because they have been born and educated in that society with those distinctions, which therefore appeared normal.
On corrective justice.
 The remaining one is the rectificatory, which arises in connexion with transactions both voluntary and involuntary.
Perhaps "compensatory" or "corrective" is better than "rectificatory".
 For it makes no difference whether a good man has defrauded a bad man or a bad man a good one, nor whether it is a good or a bad man that has committed adultery; the law looks only to the distinctive character of the injury, and treats the parties as equal, if one is in the wrong and the other is being wronged, and if one inflicted injury and the other has received it.
So here we see in what sense Aristotle means his "rectificatory" justice: Especially in those cases where the law, that is bound to treat equal cases equally, and is supposed not to consider individual merits, in so far at least as these are not involved in the case, and are judged apart from these merits.
 This is why, when people dispute, they take refuge in the judge; and to go to the judge is to go to justice; for the nature of the judge is to be a sort of animate justice; and they seek the judge as an intermediate, and in some states they call judges mediators, on the assumption that if they get what is intermediate they will get what is just.
Indeed, but it makes sense to add two reasons why people may go to court over disagreements they cannot easily settle otherwise: Because they assume that the judge will be impartial, and because they know what principles he will use to settle the case, for the former is necessary for a good judge, and the latter follows from the fact that minimally fair laws have been made public for all to know, and are supposed to be applied similarly to all cases they apply to at all, by a fair and impartial judge.
 Therefore the just is intermediate between a sort of gain and a sort of loss, viz. those which are involuntary; it consists in having an equal amount before and after the transaction.
See under  for a fairly precise, full and general theory, that explains fairly well why people cooperate, and what is involved in this.
On corrective justice and reciprocity.
 Some think that reciprocity is without qualification just, as the Pythagoreans said; for they defined justice without qualification as reciprocity. Now 'reciprocity' fits neither distributive nor rectificatory justice (..)
And it does not because "an eye for an eye" and similar rules do not make reference to the merits of the parties involved, and hence also not to the fact that such supposed equalities may be quite unequal if the parties involved are not equal.
See under  for proportional justice.
 But in associations for exchange this sort of justice does hold men together - reciprocity in accordance with a proportion and not on the basis of precisely equal return. For it is by proportionate requital that the city holds together. Men seek to return either evil for evil - and if they cannot do so, think their position mere slavery - or good for good - and if they cannot do so there is no exchange, but it is by exchange that they hold together.
Yes, this seems to me to be correct - but it should be noted, also with reference to  and , that this does require a considerable agreement on the relative merits of the people in the society who receive shares in proportion to their supposed merits.
 Now proportionate return is secured by cross-conjunction. Let A be a builder, B a shoemaker, C a house, D a shoe. The builder, then, must get from the shoemaker the latter's work, and must himself give him in return his own. If, then, first there is proportionate equality of goods, and then reciprocal action takes place, the result we mention will be effected. If not, the bargain is not equal, and does not hold; for there is nothing to prevent the work of the one being better than that of the other; they must therefore be equated.
I think I have explained this tolerably well under , and I insist once more that this explanation goes beyond money and transferable commodities, and applies to many more things. Also, there is a sharpening of it under .
It may be also interesting to remark that the present consideration of Aristotle, and similar ones in his Politics, seem to have inspired both Ricardo and Marx in their theories about the economical value of commodities.
And since I am here writing about the economical value of commodities, let me also remark that there seem to be two basic approaches to that: One that insists that the value of a commodity is somehow proportional to the amount, kind and quality of labor expended on it to make it, and another that insists that the value of a commodity is somehow proportional to the demand for it.
Aristotle addresses a related problem in the next point.
 For it is not two doctors that associate for exchange, but a doctor and a farmer, or in general people who are different and unequal; but these must be equated. This is why all things that are exchanged must be somehow comparable. It is for this end that money has been introduced, and it becomes in a sense an intermediate; for it measures all things, and therefore the excess and the defect - how many shoes are equal to a house or to a given amount of food. The number of shoes exchanged for a house (or for a given amount of food) must therefore correspond to the ratio of builder to shoemaker. For if this be not so, there will be no exchange and no intercourse.
As I have explained under  and , it seems to me that people exchange many things under wider conditions than may occur in a market, or than may required be for profit, or than can be measured (reasonably) in terms of money.
But my main reason to remark this is very probably one Aristotle would have agreed to, namely that there are more goods than economical goods, and more things of value than can be measured by money.
 And this proportion will not be effected unless the goods are somehow equal. All goods must therefore be measured by some one thing, as we said before. Now this unit is in truth demand, which holds all things together (for if men did not need one another's goods at all, or did not need them equally, there would be either no exchange or not the same exchange); but money has become by convention a sort of representative of demand; and this is why it has the name 'money' (nomisma) - because it exists not by nature but by law (nomos) and it is in our power to change it and make it useless.
It is quite interesting, also economically and politically, that Aristotle says that it is "demand, which holds all things together", for in this he agrees with most modern economists, but not with Marx and his followers, who held, rather, that the proper "unit" to measure commodities by is not money, or demand, but labour: The time that it takes average skilled labour to produce it.
 There will, then, be reciprocity when the terms have been equated so that as farmer is to shoemaker, the amount of the shoemaker's work is to that of the farmer's work for which it exchanges. But we must not bring them into a figure of proportion when they have already exchanged (otherwise one extreme will have both excesses), but when they still have their own goods.
First note that here, unlike in the previous note, Aristotle does formulate something like a labour theory of value, when he writes "the amount of the shoemaker's work is to that of the farmer's work for which it exchanges".
Secondly, this involves something like the following, with "f" for "farmer", "w(f)" for "work of farmer" and "v(w(f),t)" for "value of work of farmer done in time t", and likewise for the shoemaker:
(1) v(f,t) : v(s,t) = k IFF v(w(f),t) : v(w(s),t) = k
which is considerably more demanding than the theory I stated under  or  since it assumes far more is settled than I assumed there.
Third, Aristotle does make a very justified remark when he says that what matters for an exchange to be made is what the participants think of it before it is made, and not after it, and this holds for my theory under  as much as for his proportional exchanging, using something like (1).
And for those with some interest in economics, it should be mentioned that we have here something like the original basis of labour theories of value, such as were also used and developed by Ricardo, Marx and Sraffa.