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show/hide this revision's text 3 clarify relation to boolean domain

I think a lot about logical universes of discourse, spaces whose abstract types are built up from the boolean domain $\mathbb{B} = \lbrace 0, 1 \rbrace$. I like to picture them as venn diagrams, which I imagine as having an object layer of abstract type $\mathbb{B}^k$ and a paint layer of abstract type $\mathbb{B}^k \to \mathbb{B}$. But the medium of venn diagrams is already overheated at $k = 4$, so it necessary to devise any number of artful dodges.

One trick that takes us a long way is factoring a high dimensional universe of abstract type $(\mathbb{B}^k, \mathbb{B}^k \to \mathbb{B})$ into a bundle of universes, a base universe of type $(\mathbb{B}^{k_1}, \mathbb{B}^{k_1} \to \mathbb{B})$ with a universe of type $(\mathbb{B}^{k_2}, \mathbb{B}^{k_2} \to \mathbb{B})$ attached to each point of the base, where $k$ is partitioned into $k_1$ and $k_2$,

Obviously, a lot depends on a happy choice of the partition $k = k_1 + k_2$, and there is probably no such thing as the one right answer, though many are likely to be more revealing than others.

Here's one place where I tried out several different genres of representation for these logical bundles.

Differential Logic and Dynamic Systems

Unfortunately, the graphics got mangled in a change of computer platforms some years back and I had to replace a few of them with ascii art until I can get back to redrawing them.
show/hide this revision's text 2 clarify pronoun antecedent

I think a lot about logical universes of discourse, spaces built up from the boolean domain $\mathbb{B} = \lbrace 0, 1 \rbrace$. I like to picture them as venn diagrams, which I imagine as having an object layer of abstract type $\mathbb{B}^k$ and a paint layer of abstract type $\mathbb{B}^k \to \mathbb{B}$. But the medium of venn diagrams is already overheated at $k = 4$, so it necessary to devise any number of artful dodges.

One trick that takes us a long way is factoring a high dimensional universe of abstract type $(\mathbb{B}^k, \mathbb{B}^k \to \mathbb{B})$ into a bundle of universes, a base universe of type $(\mathbb{B}^{k_1}, \mathbb{B}^{k_1} \to \mathbb{B})$ with a universe of type $(\mathbb{B}^{k_2}, \mathbb{B}^{k_2} \to \mathbb{B})$ attached to each point of itthe base, where $k$ is partitioned into $k_1$ and $k_2$,

Obviously, a lot depends on a happy choice of the partition $k = k_1 + k_2$, and there is probably no such thing as the one right answer, though many are likely to be more revealing than others.

Here's one place where I tried out several different genres of representation for these logical bundles.

Differential Logic and Dynamic Systems

Unfortunately, the graphics got mangled in a change of computer platforms some years back and I had to replace a few of them with ascii art until I can get back to redrawing them.
show/hide this revision's text 1 [made Community Wiki]

I think a lot about logical universes of discourse, spaces built up from the boolean domain $\mathbb{B} = \lbrace 0, 1 \rbrace$. I like to picture them as venn diagrams, which I imagine as having an object layer of abstract type $\mathbb{B}^k$ and a paint layer of abstract type $\mathbb{B}^k \to \mathbb{B}$. But the medium of venn diagrams is already overheated at $k = 4$, so it necessary to devise any number of artful dodges.

One trick that takes us a long way is factoring a high dimensional universe of abstract type $(\mathbb{B}^k, \mathbb{B}^k \to \mathbb{B})$ into a bundle of universes, a base universe of type $(\mathbb{B}^{k_1}, \mathbb{B}^{k_1} \to \mathbb{B})$ with a universe of type $(\mathbb{B}^{k_2}, \mathbb{B}^{k_2} \to \mathbb{B})$ attached to each point of it, where $k$ is partitioned into $k_1$ and $k_2$,

Obviously, a lot depends on a happy choice of the partition $k = k_1 + k_2$, and there is probably no such thing as the one right answer, though many are likely to be more revealing than others.

Here's one place where I tried out several different genres of representation for these logical bundles.

Differential Logic and Dynamic Systems

Unfortunately, the graphics got mangled in a change of computer platforms some years back and I had to replace a few of them with ascii art until I can get back to redrawing them.