Untitled

How do bash's logical operators work (or why don't they work here)?
#!/bin/bash

$local_time=`date +%H%M`
if [ $local_time -ge 1430 ] && [ $local_time -le 2230 ] ||
   [ $local_time -ge 0300 ] && [ $local_time -le 0400 ]
then
   echo 'pass'
else
   echo 'skip'
fi

#!/bin/bash

local_time=$(date +%H%M)
if (( ( local_time > 1430  && local_time < 2230 ) || ( local_time > 0300 && local_time < 0400 ) )); then
  echo 'pass'
else
  echo 'skip'
fi

if [[ ( $local_time -ge 1430 && $local_time -le 2230 ) ||
      ( $local_time -ge 0300 && $local_time -le 0400 ) ]]
then
 ...