Actually this is a good testing question because you are evaluating the client/server functionality between a device and a cloud service, and how the mobile client behaves when it changes from one state (eg logged on in cloud w/connection) to another state (logged on with no connection).
If the user is "offline" (e.g. changes to airplane mode) or has lost connection they wouldn't be able to receive any push notification anyway, so the logic/excuse from your developer seems a bit flawed. (Of course this assumes your app is smart enough not to allow the user to log on if there is no network connection available.)
Many apps that require a 'handshake' with the service will usually display some sort of message indicating that the device is unable to establish a network connection (no cell bars or no wifi) when attempting to perform some action.
But, if the log out functionality displayed this error message it would leave the client on the device in a funky state (user might think it is still logged in). So, most apps that I have seen will simply return the user to the log in page state of the app when the user presses log out on a mobile device.
In lieu of specs (or even with specs), in general you could look for consistency with how other mobile apps (how do other apps that are similar to yours) handle this situation.
Testing mobile apps for lost connections is an important aspect of mobile app testing. You should also consider possible scenarios such as temporarily lost connections (eg. 30 seconds, 5 minutes, 6 hours), switching to airplane mode (both cell and wifi off), switching on airplane mode with cell signal (eg CDMA, GSM, etc) off and wifi on, etc.