TITLE: A connection between vector space operations and intuitionistic connectives?
QUESTION [10 upvotes]: As I was working my way through Steven Roman's Advanced Linear Algebra, I came across this in Chapter 11, exercise 1.

Let $U,W$ be subspaces of a metric vector space $V$. Show that
  (a) $U\subseteq W \implies W^\bot\subseteq U^\bot$
  (b) $U\subseteq U^{\bot\bot}$
  (c) $U^\bot = U^{\bot\bot\bot}$  

Additionally, $(U+W)^\bot = U^\bot \cap W^\bot$ and $U^\bot + W^\bot \subseteq (U\cap W)^\bot$. I'm having some trouble with the reverse inclusion (at least it's not as obvious as the other ones were).
All of these properties look strikingly similar to the behavior of connectives in intuitionistic logic, e.g. $p\rightarrow q \vdash \neg q\rightarrow \neg p$, $\vdash p \rightarrow \neg\neg p$, $\vdash \neg p \leftrightarrow \neg\neg\neg p$, $\vdash \neg(p \lor q) \leftrightarrow \neg p \land \neg q$, $\vdash \neg p \lor \neg q \rightarrow \neg(p\land q)$.
Is there a connection here? If so, what is it?

REPLY [6 votes]: Here's a perspective that unifies all the previous answers. Unsurprisingly, it is based in category theory.
The internal logic of a symmetric monoidally closed category is multiplicative intuitionistic linear logic (MILL). A monoidal category is one that has a suitable notion of tensor product, $\otimes$ (with unit $I$). It is symmetric if we have a natural isomorphism, $\sigma_{AB}:A\otimes B\cong B\otimes A$, such that $\sigma_{BA}\circ\sigma_{AB}=id_{A\otimes B}$. It is monoidally closed if we have $\text{hom}$ objects characterized by the tensor-hom adjunction. I'll write $A\multimap B$ for $\text{hom}(A,B)$.
An archetypal example of a symmetric monoidally closed category is $\mathbf{Vect}_k$, the category of $k$-vector spaces for a field $k$. Moving back to an arbitrary category, in the special case where $\otimes = \times$, the categorical product, we talk of a cartesian monoidal category. A cartesian monoidally closed category is called a cartesian closed category. In this case, the internal hom is the exponential object. For example, $\mathbf{Set}$ is a cartesian closed category with $X\multimap Y$ written $Y^X$, the set of functions from $X$ to $Y$. The internal logic of an arbitrary cartesian closed category is (a fragment of) intuitionistic propositional logic (or better minimal logic). $\mathbf{Vect}_k$ is not cartesian closed though.
The Galois connections that Stephen A. Meigs mentions generalize directly to adjunctions in symmetric monoidally closed categories. First, we have the tensor-hom adjunction itself: $(-)\otimes B \dashv B\multimap(-)$. This states that $\mathsf{Hom}(A\otimes B,C)\cong\mathsf{Hom}(A,B\multimap C)$ natural in $A$ and $C$ (and in fact also $B$, but this goes beyond what the adjunction immediately states). (Also, $\mathsf{Hom}(A,B)$ is the hom-set of arrows from $A$ to $B$, not to be confused with the internal hom, $\text{hom}$ which is an object of the category itself. As the name suggests, though, the internal hom "internalizes" many aspects of hom-sets.) The symmetry also gives us the following adjunction $((-)\multimap C)^{op}\dashv (-)\multimap C$ or $\mathsf{Hom}(A,B\multimap C)\cong\mathsf{Hom}(B,A\multimap C)$. As mentioned before, and as suggested by the notation, these adjunctions correspond to the logical rules for $\otimes$ and $\multimap$ in linear logic, which, in the cartesian case, correspond to $\land$ and $\to$ in (non-linear) propositional logic (minimal, intuitionistic, or classical).
Now MILL doesn't contain $(-)^\perp$. We can, however, do what they do in minimal logic which is to define the "negation" $\neg A$ (so $A^\perp$ for us) as merely $A\to R$ ($A\multimap R$) for some arbitrary but fixed proposition $R$. One natural choice is to choose $R=I$, the unit of the tensor product. (A more natural choice as far as negation is concerned is $R=0$ or $R=\bot$ but we have not assumed that structure.) All the rules you mention generalize in linear algebra from statements about $V^\perp$ to statements about $\text{hom}(V,W)$, i.e. we view $V^\perp$ as $\text{hom}(V,k)$ (where $k$ is the underlying field). $k$ is actually the unit for the tensor, i.e. $k=I$. This corresponds, for a general symmetric monoidally closed category, to defining $A^\perp\equiv A\multimap I$. You can verify almost all the laws you mention. One way to handle $\lor$ is to add categorical coproducts of which the direct sum of vector spaces, $\oplus$, is an example. (Note, how we handle disjunction is one of the places where intuitionistic and classical perspectives start to diverge significantly.) In this case, we have $(A\oplus B)\multimap C \cong (A\multimap C)\times (B\multimap C)$ as just a (somewhat disguised) special case of the very general and highly useful fact that right adjoints preserve limits, in this case taking products (in the opposite category, so coproducts) to products. We can make these statements about the subobjects $A\cap B$ and $A\cup B$ (given we have the appropriate structure to interpret these) which fits a bit better.
All of this is part of a larger pattern connecting logic, type theory, category theory, and computation.