2.1 Formal Equality
When two persons have equal status in at least one normatively relevant respect, they must be treated equally with regard to this respect.This is the generally accepted formal equality principle that Aristotle formulated in reference to Plato: “treat like cases as like” (Aristotle, Nicomachean Ethics, V.3. 1131a10-b15; Politics, III.9.1280 a8-15, III. 12. 1282b18-23). Of course the crucial question is which respects are normatively relevant and which are not. Some authors see this formal principle of equality as a specific application of a rule of rationality: it is irrational, because inconsistent, to treat equal cases unequally without sufficient reasons (Berlin 1955-56). But most authors instead stress that what is here at stake is a moral principle of justice, basically corresponding with acknowledgment of the impartial and universalizable nature of moral judgments. Namely, the postulate of formal equality demands more than consistency with one's subjective preferences. What is more important is possible justification vis-à-vis others of the equal or unequal treatment in question — and this on the sole basis of a situation's objective features.
2.2 Proportional Equality
According to Aristotle, there are two kinds of equality, numerical and proportional (Aristotle, Nicomachean Ethics, 1130b-1132b; cf. Plato, Laws, VI.757b-c). A form of treatment of others or as a result of it a distribution is equal numerically when it treats all persons as indistinguishable, thus treating them identically or granting them the same quantity of a good per capita. That is not always just. In contrast, a form of treatment of others or distribution is proportional or relatively equal when it treats all relevant persons in relation to their due. Just numerical equality is a special case of proportional equality. Numerical equality is only just under special circumstances, viz. when persons are equal in the relevant respects so that the relevant proportions are equal. Proportional equality further specifies formal equality; it is the more precise and detailed, hence actually the more comprehensive formulation of formal equality. It indicates what produces an adequate equality.
Proportional equality in the treatment and distribution of goods to persons involves at least the following concepts or variables: Two or more persons (P1, P2) and two or more allocations of goods to persons (G) and X and Y as the quantity in which individuals have the relevant normative quality E. This can be represented as an equation with fractions or as a ratio. If P1 has E in the amount of X and if P2 has E in the amount Y, then P1 is due G in the amount of X′ and P2 is due G in the amount of Y′, so that the ratio X/Y = X′/Y′ is valid. (N.B. For the formula to be usable, the potentially great variety of factors involved have to be both quantifiable in principle and commensurable, i.e., capable of synthesis into an aggregate value.)
When factors speak for unequal treatment or distribution, because the persons are unequal in relevant respects, the treatment or distribution proportional to these factors is just. Unequal claims to treatment or distribution must be considered proportionally: that is the prerequisite for persons being considered equally.
This principle can also be incorporated into hierarchical, inegalitarian theories. It indicates that equal output is demanded with equal input. Aristocrats, perfectionists, and meritocrats all believe that persons should be assessed according to their differing deserts, understood by them in the broad sense of fulfillment of some relevant criterion. And they believe that reward and punishment, benefits and burdens, should be proportional to such deserts. Since this definition leaves open who is due what, there can be great inequality when it comes to presumed fundamental (natural) rights, deserts, and worth — and such inequality is apparent in both Plato and Aristotle.
Aristotle's idea of justice as proportional equality contains a fundamental insight. The idea offers a framework for a rational argument between egalitarian and non-egalitarian ideas of justice, its focal point being the question of the basis for an adequate equality (Hinsch 2003). Both sides accept justice as proportional equality. Aristotle's analysis makes clear that the argument involves the features deciding whether two persons are to be considered equal or unequal in a distributive context.
On the formal level of pure conceptual explication, justice and equality are linked through these principles of formal and proportional justice. Justice cannot be explained without these equality principles; the equality principles only receive their normative significance in their role as principles of justice.
Formal and proportional equality is simply a conceptual schema. It needs to be made precise — i.e., its open variables need to be filled out. The formal postulate remains quite empty as long as it remains unclear when or through what features two or more persons or cases should be considered equal. All debates over the proper conception of justice, i.e., over who is due what, can be understood as controversies over the question of which cases are equal and which unequal (Aristotle, Politics, 1282b 22). For this reason equality theorists are correct in stressing that the claim that persons are owed equality becomes informative only when one is told — what kind of equality they are owed (Nagel 1979; Rae 1981; Sen 1992, p. 13). Actually, every normative theory implies a certain notion of equality. In order to outline their position, egalitarians must thus take account of a specific (egalitarian) conception of equality. To do so, they need to identify substantive principles of equality, discussed below.
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