Yes, the GOP and this president are the party of falsehood. This president, and the modern GOP, have fundamental principles that are founded on lies. Also, I am not distracted when right wingers try to excuse their party by whataboutism -- saying that "all politicians lie". That is false. All humans at times are deceptive, that is true. Obama wanted a healthcare bill that covered more poor people. He was honest about that. The GOP wants tax giveaways for the rich, and they cloak that in a "healthcare bill". When your fundamental principles are lies, you are not a trustworthy person nor someone who helps the country.
Your point about liars robbing you of trust is good, but I'd say it's even worse than that. It's not just about believing any particular statement. The president is intentionally telling lies to confuse us, to make democracy difficult, to make it hard to know what to believe so people have to rely on him.
This is what Putin and Chavez did too. Several writers have discussed this use of lies:
http://www.nybooks.com/daily/2017/05/13/the-autocrats-language/
https://www.nytimes.com/2016/12/04/business/rutenberg-lessons-in-free-speech-from-pussy-riot.html
http://www.vanityfair.com/news/2017/03/is-trumps-chaos-a-move-from-the-kremlins-playbook
IS TRUMPS CHAOS TORNADO A MOVE FROM THE KREMLINS PLAYBOOK?
Its terrifying to think that the Trump administration is simply winging it, in a swirl of lies, contradictions, and Twitter rants. A scarier possibility is that there is, in fact, a plan, taken straight from Putin 101.