mysqld can't start without any info mysqld can't start without any info, that is the only information I get is 'Starting MySQL database server: mysqld . . . . . . . . . . . . . . failed!'.

I edited the my.cnf file to enable logging on /var/log/mysql/mysql.log but it apparently doesn't want to log anything.

So I don't know anything. I know there are many similar questions, but in most cases people are able to tell something more from the log file(s).

BTW, it is happening after abnormal system shutdown.

Update 1: in my.cnf file I can see a line like this:
Error logging goes to syslog due to /etc/mysql/conf.d/mysqld_safe_syslog.cnf.

When I open the file I can see this:
[mysqld_safe]
syslog

What does it mean? How can I see error log?

Update 2: when I do 
mysqlcheck -u root -p --all-databases

I get "Got error: 2002: Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock' (2) when trying to connect"

Update 3: OK, now I know what syslog is:) It says: Can't start server : Bind on unix socket: No space left on device
So now I have a final question - on which device?
the answer is there MySQL can not write to the directory you are pointing inside my.cnf, and it dose not have the right permissions. to fix this problem make sure the user MySQL has owner privileges on the directory that my.cnf is pointing for example lets say the base_dir is located under /MySQL , data_dir is located under /MySQL/data and the socket is located under /tmp in this example you need to make sure /MySQL and /MySQL/data are owned by MySQL user, for /tmp since its a default directory for Linux user MySQL should have read,write and execute privileges. also you need to make sure the partition that contain the data_dir is not full "contain enough free space"