Disable button onclick but enable another button

You should write functions:

function disablefirstbutton() {
    document.getElementById("firstbutton").disabled = true;
    document.getElementById("secondbutton").disabled = false;
}

function disablesecondbutton() {
    document.getElementById("secondbutton").disabled = true;
    document.getElementById("firstbutton").disabled = false;
}

<button id="firstbutton" onclick="disablefirstbutton()">
<button id="secondbutton" onclick="disablesecondbutton()">

If you don't mind using jQuery, here it is!

$(function() {
    $('button').click(function() {
        var classname = $(this).attr('class');

        if(classname == 'positive')
        { 
            $('button.positive').attr('disabled', 'disabled');
            $('button.negative').attr('disabled', false);
        }
        else
        {
            $('button.negative').attr('disabled', 'disabled');
            $('button.positive').attr('disabled', false);
        }
    });
});

Check this code, it's in your flavour and working:

Code Snippet -

<button
    onclick="this.disabled=true;document.getElementById('down7913').disabled=false;"
    type="submit"
    class="positive"
    name="up7913"
    id="up7913"
  >
    First
  </button>

  <button
    onclick="this.disabled=true;document.getElementById('up7913').disabled=false;"
    type="submit"
    class="negative"
    name="down7913"
    id="down7913"
  >
    Second
  </button>