Try the following commands from a command line:
sudo add-apt-repository ppa:johanvdw/saga-gis
sudo apt-get update
sudo apt-get install saga
The problem may be that synaptic has not updated it sources because the signing key was not imported. This should be the most easy way to fix that error.
[update]
It seems the wxwidgets package for raring was uploaded but not build for some reason. This is now fixed. You should be able to install saga on raring.
This annoying issue occurs because libgdal was forked between libgdal1 and libgdal1h last year. Ubuntu GIS stable uses libgdal or libgdal1, Ubuntugis unstable uses libgdal1h.
GDAL is a translator library for raster geospatial data formats.
As a library, it presents a single abstract data model to the
calling application for all supported formats. The related OGR
library (which lives within the GDAL source tree) provides
a similar capability for simple features vector data.
You will have to start from fresh.
Make sure you uninstall all software remnants of Ubuntugis Stable / Unstable. Uninstalling libgdal, libgdal1, libgdal1h will remove all software that depends on it. Then check your repos, make sure Ubuntugis Stable & Unstable isn't there.
Then add these repos for QGIS 2.2, Saga 2.1 and Postgres 9.3.3 & PostGIS 2.1.1.
http://www.qgis.org/en/site/forusers/alldownloads.html#ubuntu
Johan Van de Wauw's saga-gis repository
https://wiki.postgresql.org/wiki/Apt
I usually install QGIS first then SAGA, Grass, PostgreSQL 9.3.3, PostGIS 2.1.1, postgresql-contrib-9.3 and finally pgAdmin3.
There are a few small issues. This will install Grass 6.4.3-2, which might not work from Processing in QGIS 2.2. Grass 6.4.3-3, provided by Ubuntugis Unstable, is supposed to fix the problem (but did not work for me on Ubuntu 13.10).
Also, only Ubuntu GIS Unstable provides pgRouting as far as I know (with PostgreSQL 9.1 & PostGIS 2.1.0). If you want pgRouting you will have to compile it from source, it only takes few minutes. If Configure complain that Postgresql-9.1 is not installed, install postgresql-server-dev-9.1 and try again.
Best Answer
So I tried some more things and came up with a solution. I will go over what I did step by step and hopefully this will work for others.
First thing I did was uninstall qgis as completly as I know how
Then I removed the Ubuntugis-unstable PPA
I went to the qgis website and added the qgis Dufor deb and deb-src to my software & updates application
deb http://qgis.org/debian raring main
deb-src http://qgis.org/debian raring main
Next I added the keyserver info by typeing the following commands into the terminal
Once I had everything added to my list of repositories I used the update command
From here I ran 4 commands to install qgis and saga. I had to do each command separately, whenever I tried to do them all in one go I would get error messages
Once all of this had been completed I was able to run qgis and saga at the same time and sextante was able to find and execute saga modules.