How to get only present and past posts with post_date

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You need to pass a strtotime() compatible string or an array instead of 'today'.

You have several options…

  1. You could change 'today' to 'now'. This will use the current time not midnight last night.
  2. You could change 'today' to 'today + 1 day'. This will be the end of today. Not exactly what you are asking for but worth mentioning.
  3. You could generate it with PHP like date('d-m-Y H:i:s'). This is unnecessary extra work since it will need to be parsed just like 'now'.

Of all of these options, 1 is the simplest and most comprehendable.

Documentation

https://developer.wordpress.org/reference/classes/wp_query/#date-parameters

https://www.php.net/strtotime