<aside> <img src="/icons/arrow-right-basic_gray.svg" alt="/icons/arrow-right-basic_gray.svg" width="40px" /> Thoughts are connections, and connections compose.
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We now define relation composition in binary logic, and see that it is dual to how relations transform.
Let $\mathrm{A,B,C}$ be sets, and let $R:\mathrm{A\,|\, B}$ and $S:\mathrm{B\,|\,C}$ be 0/1-relations.
Example. $\mathrm{A}$ is animals, $\mathrm{B}$ is plants, and $\mathrm{C}$ is molecules. $R(\mathrm{a,b})$ is “animal $\mathrm{a}$ Eats plant $\mathrm{b}$” $S(\mathrm{b,c})$ is “plant $\mathrm{b}$ Contains molecule $\mathrm{c}$”
The composite relation $R\circ S: \mathrm{A\,|\, C}$ relates $\mathrm{a}$ to $\mathrm{c}$ when
$$ \text{there is some } \mathrm{b:B}\;\;\text{ so that }\\ R \text{ relates } \mathrm{a} \text{ to } \mathrm{b} \text{ and } S \text{ relates } \mathrm{b} \text{ to } \mathrm{c}. $$
Formally, we can write this as follows.
$$ (R\circ S)(\mathrm{a,c}) = \mathsf{\Sigma} \mathrm{b:B}.\; R(\mathrm{a,b}) \times S(\mathrm{b,c}) $$
So the composite relation is a sum of products: an $R\circ S$-connection from $\mathrm{a}$ to $\mathrm{c}$ is given by an $R$-connection and ($\times$) an $S$-connection, along some ($\mathsf{\Sigma}$) $\mathrm{b:B}$.
If you’ve ever done linear algebra, this is matrix multiplication: for each pair $\mathrm{a:A}$ and $\mathrm{c:C}$,
| $\mathbf{R}$ | $\mathrm{a}$ | $\cdots$ |
|---|---|---|
| $\mathrm{b}_1$ | $R(\mathrm{a,b_1})$ | $\cdots$ |
| $\mathrm{b_2}$ | $R(\mathrm{a,b_2})$ | $\cdots$ |
| $\cdots$ | $\cdots$ | $\cdots$ |
| $\mathbf{S}$ | $\mathrm{b}_1$ | $\mathrm{b_2}$ | $\cdots$ |
|---|---|---|---|
| $\mathrm{c}$ | $S(\mathrm{b_1,c})$ | $S(\mathrm{b_2,c})$ | $\cdots$ |
| $\cdots$ | $\cdots$ | $\cdots$ |