I

The principle of equality is a fundamental principle of morality and justice.  The principle says that similar cases must be treated similarly; if two individuals are the same in all morally relevant respects, then they must be treated the same.  The principle of equality likewise says that if there is a distinction in the way individuals are treated, this distinction must rest on a morally relevant difference between the individuals (this is just the contrapositive of the principle).  The principle of equality is a purely formal requirement of morality, a minimal requirement that any moral action, arrangement, or system must meet.  The principle of equality itself says nothing about what are morally relevant respects or differences.  What one person thinks is morally relevant may not be what another thinks is morally relevant.  But that is a different matter.

Open any text book on moral theory or applied ethics and you will find the principle of equality prominently featured.  For example, Ronald Munson in his widely used medical ethics text says: "Theories of justice differ significantly, but at the core of all theories is the basic principle that 'Similar cases ought to be treated in similar ways.'"  Munson calls this "the formal principle of justice."[1]  Other versions of the principle of equality are more epistemic in nature but still embody the same idea:  "Whenever we treat items differently, we must be able to give a reason."  If we can give no morally relevant reason for difference of treatment, then we are acting arbitrarily.  Any system that systematically embodies such arbitrary modes of treatment cannot be a moral system.  Ian Hacking puts this nicely in his review of Amartya Sen's book Inequality Reexamined.  Hacking says:

 I do not think that the case for equality derives its justification from a consensus about equal concern for others. The fact that people want equality in some respect‑‑even those who would curtail even a minimal system of welfare or health care‑‑is, I think, grounded not in concern for all, but in pure reason.....  If a proposed system of social arrangements treats people differently in a certain respect, it must give reasons for doing so.  Every difference, every case of privileged or inferior status, demands a reason...  Of course one person may give a reason for a difference, such as wealth or race, that another finds odious or irrelevant.  I am concerned only with Sen's question of why all plausible (even if repugnant) social theories invoke equality of something as a starting point before they try to show why inequality may be justified.  If you go back to first principles, it seems, you must go back to equality of something.  Otherwise, you would have a difference without a reason.....  I see equality as a logical demand, an instance of what Leibniz called the need for a sufficient reason.[2]

Although the principle of equality is purely formal, it is often used in moral argument and has certainly been a key part of the arguments against racial, gender, and other sorts of discrimination.  Recently it has been used by people supporting animal rights.  Gary Francione, a Rutgers law professor and founder of the Rutgers Animal Rights Law Center, argues on the basis of the principle of equality that "...animals deserve the same basic rights as humans in terms of physical safety and protection from exploitation."  "Once people and animals are identified as relatively similar, it is against the principle of justice to treat them differently."[3]

II

The reason it is impossible to be moral, is that the principle of equality, given certain undisputed empirical facts, leads to contradictions.  It is a principle that cannot be satisfied.  Since this fundamental principle of morality cannot be satisfied, it is impossible to be moral. 

Among the empirical facts that force the principle of equality to contradiction is the nontransitivity of similarity (NTS).  NTS occurs when A is indistinguishable in respect R from B, and B is indistinguishable in respect R from C, but A is distinguishable in respect R from C.  That NTS occurs is an undisputed empirical fact.  Indeed, NTS occurs whenever R is an observational property.  In his book on discriminibility and vagueness Timothy Williamson notes the phenomenon of NTS. 

We know independently that [indiscriminibility relations of direct observation] are not [transitive].  Given a series of red paints, each very slightly darker than the one before, I may be able to discriminate between the first red and the last without being able to discriminate between any red and its immediate successor (all by the naked eye).[4]

Although NTS clearly applies to direct sensory observation, it just as clearly applies elsewhere as well.  Indeed, any vague predicate will allow a series of items in which no item is distinguishable as to applicability of the predicate from its neighbors, but the endpoints are distinguishable.  Max Black in his classic article on vagueness offers the following example of such a series:

One can imagine an exhibition in some unlikely museum of applied logic of a series of "chairs" differing in quality by least noticeable amounts.  At one end of a long line, containing perhaps thousands of exhibits, might be a Chippendale chair: at the other, a small nondescript lump of wood.[5]

In such a series each item may be discriminible overall from its neighbors, but no item is discriminible in terms of chairness from its neighbors.  So each item in the series is just as much of a chair as its neighbors, and yet the item at one end, the Chippendale, is a chair, unquestionably and absolutely, and the item at the other end is not a chair at all.

In a case of development or decay there need be no possibility of discrimination of an item from its neighbors.  An adult human person develops from a fertilized egg.  If we consider the series broken up into one second segments (or smaller if you like), then there is no detectable difference between each item and its neighbor, but there is a huge difference between a fertilized egg and an adult human person. 

It does not matter whether Black's series of items is actual or only possible.  For the purposes of moral reasoning it is irrelevant whether there actually are such series, although I believe there certainly are many besides the developing fetus.  I am confident that using actual examples we could construct a series of items in which no item is distinguishable from its neighbor in terms of morally relevant characteristics, and in which the first item is a normal adult human person and the last is a rock. As long as it is possible, even counterfactually, to construct a series of items involving morally relevant features to which NTS applies, and I claim that it is always at least counterfactually possible, the argument I am about to give in the next section goes through. 

III

The principle of equality does not, of course, imply its converse--"Items that differ in morally relevant respects must be treated differently."  Nevertheless a related principle which I call "the principle of differential treatment"‑‑"It is morally required or at least allowed to treat sufficiently dissimilar cases differently"--is clearly required by morality.  Morality must recognize that sometimes differences of treatment are justified, and that such differences of treatment can only be justified by differences in the cases.  The principle of differential treatment does not hold that all differences are sufficient to justify difference of treatment, just that some are.  Consider some morally relevant characteristic R.  Presumably items will differ in R, even to the limiting case where some items have R and others lack it. If a difference in having or lacking R does not ever require or at least allow difference in treatment, then in what sense is R morally relevant?   So minimally I would say that satisfying the principle of differential treatment is required for a characteristic to be morally relevant.[6]

The principle of equality, the principle of differential treatment, and the fact of NTS with respect to morally relevant characteristics are sufficient to generate contradictions.  They form an inconsistent triad.  Consider the following simple model: It is morally required to treat items indistinguishable with respect to morally relevant characteristic R similarly.  It is morally required (or at least allowed) to treat items distinguishable with respect to R differently.  Now we have the situation that A is indistinguishable from B in respect R, B is indistinguishable from C, but A is distinguishable from C.  Thus it is morally required to treat A and C similarly, because A and B must be treated similarly and B and C must be treated similarly, but it is also morally required (or allowed) to treat A and C differently.  This situation will arise wherever A and C are required (or allowed) to be treated differently and NTS applies to respect R, but this is just the situation with all morally relevant characteristics.[7]  Thus it is impossible to be moral.

For purposes of illustration I have assumed a simple three item model.  In practice, the number of items is usually vast, but that makes no essential difference to the logic of the situation.  The contradiction will emerge in any situation in which B (the first item in the series) is sufficiently distinguishable in respect R from E (the last item in the series) to be treated differently, but no item is sufficiently distinguishable in respect R from its neighbors to be treated differently.  Each item is morally required by the principle of equality to be treated similarly to its neighbors, so eventually by a series of steps we will arrive at the conclusion that B must be treated similarly to E.  The argument will proceed as follows: "Item 2 is indistinguishable from B, so must be treated similarly, item 3 is indistinguishable from item 2, so must be treated similarly,..., E is indistinguishable from E‑1, so must be treated similarly."  On the other hand, since B and E are sufficiently distinguishable, by the principle of differential treatment, difference of treatment of B and E is required (or at least allowed).  Here we have the contradiction: We are required to treat B and E similarly, and we are not required to treat them similarly. 

This contradiction at the heart of morality may not seem to make it impossible to be moral, just very difficult, for after all I can still be moral in situations where I treat neighbors in the series the same.  It may seem that the contradiction only arises when I consider what to do with respect to end points.  But we still get a contradiction even when we consider immediate neighbors‑‑say, items n and n+1.  Since items n and n+1 are indistinguishable with respect to R, we are required to treat them similarly.  Let us suppose that we are required to treat B and E differently, because they are sufficiently distinguishable in respect R (the argument works as well with "allowed to treat them differently" but is less dramatic).  Since B+1 is indistinguishable from B with respect to R, and E‑1 is indistinguishable from E with respect to R, we are required to treat B+1 differently from E‑1 (because whatever applies to E applies to E‑1, and whatever applies to B applies to B+1).  Now just continue to march until you reach item n from B, and item n+1 from E. You will by this argument be required to treat item n differently from item n+1.