To recap: so far, we have an image of an “object” as something with an internally differentiated structure, within which elements appear as a) connected to each other and b) grouped into discrete parts based on their connections. Mathematically, this structure is like an equivalence relation, except that it also supports the presentation of “inexistent” elements that are not connected to anything, even themselves. Rather than simply listing all of the connected pairs of elements, we take all of the possible pairs of elements in the structure and assign a value to each saying whether the elements in the pair are connected or not.

The notation we use for this structure is (A, Id), where A is the set {a, b, c…} of elements in the structure (which Badiou calls the “support set” of the object), and Id is the function which assigns to each pair of elements a value representing how connected they are. This value is taken from a special set, Ω, which contains all of the “truth values” for Id. In the simplest case, the only two available values are 0 (not connected) and 1 (totally connected), and so Ω is the set {0, 1}. I will shortly be considering the more complicated case, where Ω provides a range of truth-values, which are arranged in a kind of lattice known as a Heyting algebra. First of all, however, an adjustment in terminology is needed.

In our original example, two elements could be either connected or not connected. If we see the function Id(a, b) as assigning a truth-value to the statement a ∼ b (”a is connected to b”), then this suggests another way of putting it: in our original example, the statement that two elements were connected could be either completely true (1) or completely false (0).

The rule of symmetry, for example, can then be restated as: Id(b, a), or the truth-value of b ∼ a, is equal to Id(a, b), or the truth-value of a ∼ b. In fact, we can weaken this slightly to the following:

Rule of symmetry: Id(b, a) ⊆ Id(a, b)

In other words, the truth-value of b ∼ a is less than or equal to the truth-value of a ∼ b: b is never any more connected to a than a is to b. (Note that we use the symbol ⊆ to mean “less than or equal to” when comparing truth-values, and that 0 ⊆ 1).

(It turns out that this way of stating the rule of symmetry actually entails that Id(a, b) ≡ Id(b, a), but we get this “strong” statement for free as a consequence of the “weaker” statement Id(a, b) ⊆ Id(b, a). I leave the proof of this as an exercise for the reader).

What is the advantage of this restatement in terms of truth-values? It means that we can uphold the rule of symmetry even when a ∼ b is neither completely true nor completely false. Let’s take as our range of truth-values the following “diamond”:

“True” is 1 and “False” is 0; we’ll use the symbols ⌊ and ⌋ for “right deviation” and “left deviation” respectively. Here, then:

0 ⊆ ⌊

0 ⊆ ⌋

⌊ ⊆ 1

⌋ ⊆ 1

and because ⊆ is transitive, obviously 0 ⊆ 1 as well. Note however, that neither ⌊ ⊆ ⌋ nor ⌋ ⊆ ⌊ (this statement in itself probably identifies me as a right-deviationist).

Now if two elements of an object, a and b, are neither completely connected nor completely disconnected, then the strength of their association can be measured by one of these “intermediate” truth values, ⌊ and ⌋. We can also say, using our revised rule of symmetry, that if the truth-value of a ∼ b is ⌊, then so is the truth-value of b ∼ a. But what we now have to do is restate the rules of transitivity and reflexivity in terms of truth-values so that we can say for example what the value of Id(a, b) might be if:

Id(a, c) = ⌊

Id(b, c) = ⌋

This will be the subject of the next post.