Truncated CI Wavefunction In a calculation of for example a Li-Atom in a DZ basis we have 4 MO.

For the alpha string (containing two electrons), that means we have for a FCI calculation the determinants: 
|12>, |13>, |14>, |23>, |24>, |34> (electrons in MO 1 and 2, 1 and 3 and so on).

If I wanted to do a truncated CI calculation (CIS for example) only singly excited wavefunctions would contribute. 

My (maybe extremely dumb) question now is: Are the determinants |23> and |24> singly excited or doubly excited? Is it the excitation 
$$a^\dagger_3a_1|12&gt; = |32&gt; = -|23&gt;$$
 or is it 
$$a^\dagger_2a_1 a^\dagger_3a_2|12&gt; = |23&gt;$$
 Or is it both? And the coupling (1,-1) so to say determines whether this in used as a single or doubly excited determinant?

EDIT: From the "Molecular Electronic Structure Theory" book by Helgaker, Jorgensen and Olsen, from the chapter 11.8.5 (Direct CI algorithms for RAS Calculations) I read "In CISD theory, there are three such classes (string classes): class 0, with no electrons in virtual space; class 1, with one electron in virtual space; and class 2, with two electrons in virtual space".

This for me reads as if |23> would count as a class 1 string and not a class 2.
In configuration interaction (and in coupled cluster) we typically think of excited determinant being caused by single particle excitations of the form $\hat{t}^a_i = \hat{a}^\dagger_a\hat{a}_i$ where $i$ labels a virtual orbital and $a$ labels an occupied orbital, causing an excitation of the Hartree Fock reference determinant $\Psi_0$.
The resulting CI expansion would go:
$$
\Phi_\text{CI} = \Psi_0 + \sum_{i\in occ\\ a \in vir} \hat{t}^a_i \Psi_0 + \sum_{i,j\in occ\\ a,b \in vir} \hat{t}^{ab}_{ij} \Psi_0 + \sum_{i,j,k\in occ\\ a,b,c \in vir} \hat{t}^{abc}_{ijk} \Psi_0 + \: ...
$$
Hence, in the calculation on Li mentioned above, we would not consider the excitation $\hat{t}^2_1 = \hat{a}^\dagger_2\hat{a}_1$ in the first place as it would annihilate the Hartree Fock determinant $|12&gt;$ containing a particle in orbital 2.
By this logic the determinant $|23&gt;$ is a singly excited determinant caused by $\hat{t}^3_1= \hat{a}^\dagger_3\hat{a}_1$, or class 1 as described by Helgaker.
Note:
As determinants $|23&gt;$ and $|32&gt;$ only differ by a multiplicative constant (-1) only one needs to be considered and the resulting coefficient/cluster amplitude will take care of the relative sign.