Which Equality?

How equal are equal human beings?

Rudolf Kaehr Dr.^@

ThinkArt Lab Glasgow

Abstract

"All human beings are equal”. What does “equal” mean? What are the many definitions of “equal” and “equality"? From a Diamond perspective, concepts of sameness, from equality, similarity, bisimilarity to hetero-morphism and more are sketched in respect to their usage in ethical discourses, e.g Human Rights, of mono-, multi- and trans-cultural formations. What happens in such scenarios to the Golden Rule? The family of mankind? The brotherhood?

1.  Interdependency of context and composition

Life under the regulation of equality has stopped to be funny. To do the same, which can have strictly different meanings and significance, can end up in prison, deportation or execution if judged by identity-trained forces and institutions of our free society.

The following short study might be wrong in time. Things are still sub-human. The human rights not realized at all. On the other hand, what do we understand by equality if this term is defined only in a negative way, i.e. by exclusion of non-equality. Despite the wrong timing a conceptual effort to achieve a positive and constructive understanding of equality, anticipating futures to come, appears to be a reasonable entertainment.

From Lyrical Babe to Lyrical Terrorist; first female victim of Terrorism Act 2000

The Handbook³. Courtesy to the FBI⁴. But beware of what you are doing! There are 8 chapters to download?

This had been the Dark Ages in Britain at 2005. Evolved from untrained poets to certified scientists, today’s enemies of our society are demanding for much brighter advanced surveillance.

"It's extremely upsetting.” Hicham Yezza

The Home Office has said it does not comment on individual cases.⁵

Students today have not only to learn how to avoid plagiarism⁷but much more efficiently, how to self-censor their own thoughts and writings.

On the sixth day, without realising his freedom was imminent, he was told by a female police officer that the document he had looked at was deemed illegitimate for research purposes by the university, and if he ever looked at it again he could face further detention.⁶

When the same is different

If an action happens we have to ask in which context it happens. A diamond context is not pre-given as it is the case for Context Logic, it is co-created within the action which happens in the context the action happens. This is not a circular interdependency of action and context but hints to the complementarity of action and context.

That is, composition and context of composition are defined interdependently as acting complementarily to each other. There is no context without composition and there is no composition without context. In contrast to the concept of duality in formal systems, the concept of complementarity is not yet well understood.

Context Logic is introducing contexts secondarily to compositions, hence as special attributes of propositions and logical operations. (Goddard⁹)

Legal systems, which are not depending on extra-terrestrial sources, i.e. religion, metaphysics or similar, or on biologist speculations, like genetics or brain science, have to take the irreducible interdependency of composition and context of actions into account.

A very general and strong model of composition is realized with the mathematical Category Theory. Compositions of actions, regarded as mappings or morphisms, are defined by matching conditions of morphisms, commutativity and axioms of identity and associativity.

There is no conceptual space left by such a definition of composition for the context of categorical composition. That is, the conditions of the possibility of composition, its context, is not given with the composition but is pre-ordered to compositions as their axiomatic pre-conditions of compositions. Everything needed, like contexts, have to be introduced as a secondary step in the theory as an application of the theory on the base of the introduced definition of composition. But such a secondary concept of contexts has, as a second-level context, another systematic status than the primordial context of the pre-conditions of the axiomatics of compositions.

Context Logic, a hint

"A context is defined by a set of descriptions which give the time and place of utterance, the topic of conversation, the identifications made, and similar detailed information. Hence the context variables range over such sets.” (Goddard)

Based on classical sentential logic, with its negation, conjunction, disjunction and implication connectives, a new context symbol θ, ranging over the classical symbols, is introduced. If p is a sentence, then p(c) is a sentence in context c, or “p with respect to context c.”

To get a working formalism, a world assumption has to be accepted. Otherwise, the neat interchangeability of contexts in composed formulas wouldn’t work. A context-formula like .

The rules for the sentential context logic are requiring the interchangeability of the context of the parts with the context of the whole of the composition.
In this situation, the single world assumption has to be introduced. Otherwise, formulas with mixed context symbols,  cannot be treated in a direct intuitive and simple way. Hence the idea of context of this context logic is working on the base of a general standard context, only.

This hint should make it clear enough that the operation of composition, represented as a meta-variable, , for the logical connectives, is, as such, not touched at all. That is, the logical operators, connectives, are not involved in any kind of environments or contexts. They are dealing, context-free, with contextualized sentences. Only in this respect, they are connected with anything like contexts. And even this is, again, restricted to a general standard context as the general notion of all specific contexts.

Obviously, context logic is supposing a single world and a plurality of contextualized logics restricted to this single world, hence based on classical logic. The hierarchy is clearly established: first is logic, second context; both covered by a single contexture.

From a logical point of view the introduction of a context, local or global, is circular. A context is defined by attributes of a logic that is defined by the attributes of this logic that is defined by the context of this logic.
Today, such an approach to contexts is covert, more or less, by Modal Logic.

As far as I know, there is no other theory than the Diamond Category Theory, that is introducing composition as an irreducible and basic complementarity of context and composition. The context of composition and the composition of the context are in an interdependency interplay. Such an interplay is escaping the annoyance of logical circularity.

If we give up the single world hypothesis and are accepting a multitude of interacting worlds, represented as contextures, situations of sameness (equality), hence, have to be studied inside contextures, as intra-contextural morphisms, between contextures of a polycontextural constellation, as trans-contextural mappings, and between acceptional and rejectional configurations of contextures, as diamondal interactions.

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2.  Laws in mono-, poly- and transcultural formations

2.1.  Mono-cultural formation

2.1.1.   A message of equality

The Golden¹⁰Rule of Ethics is well placed in a mono-cultural setting. It even could be taken as a definition of mono-culturality along its rule of reciprocity.
In technical terms, reciprocity could be regarded as an isomorphism between two entities, relations or mappings.
But first, we learn from the tale of equality as it is given to us by Him.
Let's this story being told by one who knows and writes it well, John K. Williams¹¹from the Acton Institute:

Before exploring this question, a prior question must be addressed. What is meant by the noun “equality,” the adjective “equal,” and other cognates?

"Simply, the word “equality” and its cognates indicate a relationship between some quality or feature, two or more entities, or states of affairs. In respect of this quality or feature, the realities being compared are identical. Two pieces of wood might be equal in length.
Suppose someone were to assert that “All human beings are equal.” Such a person is claiming that in terms of some quality or property or characteristic, all people are identical and thus interchangeable. But what quality, what property, what characteristic?

I submit that it is impossible to specify any single physical, intellectual, or emotional characteristic that all human beings possess to the same degree. What strikes one about human beings is, surely, the uniqueness of each, not the sameness of all. [...]

"It does not help matters if, instead of claiming that all human beings are equal, one insists instead that all human beings ought  to be treated equally.

Actually, devotees of “equality” can never be satisfied. People are objectively speaking “unequal”: There is no quality, property, or characteristic-physical, intellectual, or emotional-that all human beings possess to the same degree. [..]

"Indeed, maybe the expression, “all human beings,” itself enshrines that elusive “something.” I suggest the following. Simply by virtue of their shared humanity, all human beings are actually or potentially capable of formulating their own vision of the “good life” and are striving to make that vision a reality. All, actually or potentially, can initiate self-directed, purposive behavior, the object of which is the creation of a “good life.” All, that is to say, bear the imago Dei.[...]

Hence, all enjoy equal human rights. The God-like capacity of an individual to be, albeit within limits, self-directing and self-determining cannot morally be trespassed upon by any other human being, however wise or however powerful. [...]

"This is the “equality” that matters, because this vision of human equality mirrors the equality that we human beings enjoy in the sight of God. He does not perceive us as identical clones. He knows each of us in his or her uniqueness.[...]

Is there any chance to define equality, or more generally, equivalence, in a mono-cultural environment without involving, as a sine qua non of conceptual consistency, God? Here the Christian God, obviously.
What happens to the poor guys, which are not equal and don’t even believe in God?

2.1.2.   A mathematical metaphor

Also in math, two entities are never equal, they are equal only up to isomorphism. And like in religion, the unique concept of the abstractness of isomorphism is guaranteeing diversity a unifying umbrella.

Take John Baez’ example!

Stripped off of the categorical arrows, the we get the arithmetic formula:

Who is the lucky guy who has all these m’s and n’s in his pocket? Actual or potential?

Even with a very generous concept of togetherness as equivalence, the role of negation and the whole underlying logic of the argumentation and construction, is not offering enough negational complexity to deal with non-isomorphic situations properly.

There is still a clear logical symmetry, classical or intuitionist (constructivist), between the possible states of a statement. Morphisms are equivalent or they are not equivalent. That’s it. Nothing more. This is obvious, if we consider the axiom of identity in category theory. Diamonds are based on both, identity and difference, at once. Hence their objects are not self-identical units, i.e objects, but differential bi-objects.
Obviously, they are not covered by classical logic alone.
The classical mathematical metaphor for equality and equivalence is basically ruled by a single logical negation.
As we will see, multi-cultural formations are necessarily multi-negational systems.

2.2.  Poly-cultural formation

2.2.1.  From polysemy to polycontexturality