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I thougt about propositional logic and boolean algebras and how propositional logic is (at least from one point of view) not really about $\land,\lor,\neg,...$ but about boolean operators, i.e. n-ary functions which map $0$-$1$-sequences of length $n$ into $\{0,1\}$, and how to use a finite subset of those (base) to encode all of them.

With boolean algebras it's a little bit similar, one way to think of them is as a mathematical structure associated with a set $X\supseteq\{0,1\}$ which enlarges each boolean operator $f$ to an $X$-valued operator ${\cal X}(f)$ such that this operation is compatible with function composition, i.e. $${\cal X}(f)({\cal X}(g_1),\dots,{\cal X}(g_n))={\cal X}(f(g_1,\dots,g_n)).$$ But any boolean algebra is also a lattice (via $\land$ and $\lor$) and any lattice structure can be described as partial order. I just wonder if it's also possible to describe the partial order on a boolean algebra in terms which do not prioritize the operations $\land$ and $\lor$.

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A poset is the underlying poset of a Heyting algebra iff as a category, it has finite products, finite coproducts, and is cartesian closed. Every Boolean algebra is a Heyting algebra, so these are necessary conditions for a poset to be the underlying poset of a Boolean algebra. Heyting algebras are to intuitionistic logic as Boolean algebras are to propositional logic; see, for example, this blog post.

The remaining question is which Heyting algebras are Boolean algebras. In every Heyting algebra we can define a negation operator

$$\neg x = (x \Rightarrow \bot)$$

where $\Rightarrow$ denotes implication / exponential and $\bot$ denotes false / the initial object. There is a natural map $x \to \neg \neg x$, and a Heyting algebra is a Boolean algebra iff this natural map is always an isomorphism, or in other words, iff double negation holds.

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  • $\begingroup$ Thanks so much for the answer and the link to your blog post, this was exactly the question I set out to answer (how to invent intuitionistic logic), the thing with boolean algebra was just something I came across. So that is really helpful :) $\endgroup$ – fweth Dec 4 '15 at 22:08

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