Expressional or equational programming

Functional languages do have statements.

   a : Int -> Int

   a x = x + 2

both `:` and `=` are statements, not expressions.

This is sometimes true, the `=` equivalent in Clojure is an expression, though:

   user=> (def a (def b 2))
   user=> a
   #'b

Lisps generally only have expressions (every form returns a value.

Lisp runtime implicitly runs each top-level expression as statement. When you run (def x y) it's not only true that this expression returns 'x, it's also true that it alters the static state of the program by adding 'x to the scope.

To make a purely statementless language, I think you need to make everything happen in a single monadic computation. E.g. in pseudo-Haskell-ish:

    main : Scope -> StaticIO Scope
    main scope0 = do
      scope1, x' <- def scope0 "x" Int
      scope2, xv <- dec scope1 "x" 3

But the main part is still statement.

At least in all of the discussions I have seen, the difference between statements and expressions is whether they return a value, not whether they have side-effects. So (def x b) is an expression, because it returns a value. {} in C is a statement, even though it has no side effects.

Now, there are some statements in CL as far as I remember - I believe (declaim x integer) is a kind of statement, at least in the sense above.

But you can write a CL program that doesn't use a single statement.

The return value of some forms is implementation dependent, but all forms return a value in CL:

    CL-USER(1): (defparameter *foo* (declaim (special x)))    
    *FOO*
    CL-USER(2): *foo*
    (X)
    CL-USER(3): (defparameter *bar* (declaim (type integer x)))
    *BAR*
    CL-USER(4): *bar*    
    (X)

In other cases, e.g. `(values)` which is an expression that returns no values, the value is nil

They are usually called declarations rather than statements, I think.

How do any of those qualify as statements? Statements represent actions in imperative programming languages. Declarations and definitions are not actions.