Google Sheets IF statement with OR logic error?

The formula I created for a Google Sheet is not working. Seems to be a logic error but I'm not sure what I'm doing wrong here.

This formula is being entered into F3:

=IF(OR(C3="",D3="",E3=""), "", TODAY())

I thought this would do it. If C3 is blank OR D3 is blank OR E3 is blank, then leave the cell blank, else show today's date. It's treating it like an AND statement and only putting today's date if all three cells (C3, D3, E3) have something in them.