ORA-48108 & ORA-01034
You can not use ALTER SYSTEM when the instance is not started, but you can not start the instance because of an incorrect parameter value. Create a PFILE (if you do not already have one), edit that, then start the instance.

$ sqlplus / as sysdba
SQL&gt; create pfile from spfile;
SQL&gt; exit
$ vi $ORACLE_HOME/dbs/init${ORACLE_SID}.ora


Set the value of diagnostic_dest as required, then save the file.

$ sqlplus / as sysdba
SQL&gt; create spfile from pfile;
SQL&gt; startup


Note: the directory must exist.

Answer to comment:


  For UNIX or Linux, the platform-specific default location (directory)
  for the SPFILE and text initialization parameter file is:
  
  $ORACLE_HOME/dbs
  
  Oracle Database locates your initialization parameter file by
  examining filenames in the following order:
  
  
  spfile$ORACLE_SID.ora 
  spfile.ora 
  init$ORACLE_SID.ora
  


Or, for a test, you can startup the instance by manually specifying the pfile, for example:

SQL&gt; startup pfile=/u01/app/oracle/product/11.2.0/dbhome_1/dbs/initORCL.ora