It was established earlier on that the rule of reflexivity for an equivalence relation could be derived from the rules of symmetry and transitivity. (Often it is simply stated as one of the three properties of an equivalence relation, even though it is entailed by the other two). Let’s now rewrite this derivation using the language of truth-values.

We have, already, these two statements:

Rule of symmetry: Id(a, b) ⊆ Id(b, a), which entails that Id(a, b) ≡ Id(b, a)

Rule of transitivity: Id(a, b) ∩ Id(b, c) ⊆ Id(a, c)

If we apply these to the “path” from a to b and back again, we get:

Id(a, b) ∩ Id(b, a) ⊆ Id(a, a)

but because Id(a, b) ≡ Id(b, a), Id(a, b) ∩ Id(b, a) is just the same as Id(a, b), so we can simplify the above to the pair of statements:

Id(a, b) ⊆ Id(a, a)

Id(b, a) ⊆ Id(b, b)

which, taken together, give us:

Rule of Reflexivity: Id(a, b) ⊆ Id(a, a) ∩ Id(b, b)

Now let us consider the case of the completely isolated element which is not even connected to itself, e.g. where the truth-value of a ∼ a is 0. In this case, we can say that Id(a, a) = 0, and therefore Id(a, b) ⊆ 0. This re-establishes the rule that an element unconnected to itself cannot be connected to any other element: we can now say that if the truth-value of a ∼ a is 0, then so is the truth-value of any a ∼ b.

The inexistent, then, is any element that is minimally connected to any other element, including itself; and it is its minimal degree of connection to itself that prescribes its minimal degree of connection to any other element.

What about elements for which the truth-value of a ∼ a is neither 0 (minimal) nor 1 (maximal)? These elements have an intermediate degree of existence, which can be measured as the value of Id(a, a). This is just what Badiou says: for every element in the “support set” A, that element exists to precisely the extent that it is connected to itself, or to precisely the extent that it is true that a ∼ a. Moreover, no element can be more strongly connected to any other element than it is to itself: an element’s degree of existence prescribes the maximum degree to which it can be connected to any other element.

We now have the basic vocabulary of Badiou’s theory of the object; the concepts of the “phenomenal component”, “atom” and “real atom” are all built up from the foundations I’ve laid down here. At bottom, these foundations are very simple: they define a relation among elements of set governed by the rules of symmetry, reflexivity and transitivity, in which degrees of “relatedness” between pairs of elements can be logically compared and combined. What, then, is this theory for? I would suggest that it does the following:

i) It formalises some basic intuitions about the internal composition of objects: that their elements may be related to each other, and can be partitioned into discrete collections based on how they are related. Objects are not simply disparate collections of unrelated pieces: they are often internally articulated, with distinct components.

ii) It introduces a notion of logical compatibility among relationships, which shows how one set of relationships may constrain others, bringing about a kind of emergent integrity. Objects are not chaotic or random in their internal organisation: one aspect of an object’s organisation reinforces or limits another.

iii) It gives a clear sense to the concept of inexistence: an element of an object inexists relative to that object if it does not participate in any relationship with any element of that object, including itself. The inexistent is, so to speak, absolutely withdrawn: its presentation establishes that not every aspect of an object is visible or - to take it from another angle - logically consequential.

iv) The inexistent in turn provides the zero-degree of the concept of existential intensity, which enables us to speak of elements of an object having a greater or lesser degree of existence relative to the object in which they appear. An object is not an “all or nothing” proposition, immediately transparent in every respect: parts of it may veil other parts - or, alternatively, bring them into relief.

The theme of the “real atom” and the materialist postulate is more difficult (and there is worse to follow), but equally philosophically suggestive; in a (possibly much) later post, I hope to show why. Certainly some readers will wonder whether all the maths is worth it; I can only say that in my own experience so far, perseverence brings not only clarity but also the sublime feeling of having picked up and learned to use a new tool of thought - it’s not wholly unlike getting your hands on a synthesizer for the first time…