There's
always trust involved. You have to trust the DNS infrastructure, you have to trust your ISP, you have to trust the VPN provider. You don't have to trust them completely, but you have to trust them at least somewhat.
We take steps to reduce the amount of trust required, such as splitting that trust across many parties, so any one party hopefully can't betray us enough that it matters or that we don't notice, but there's still a lot of trust. For example, we use SSL certificates and certificate authorities that are known ahead of time to protect from problems on the network, but that requires you trust your OS and/or your browser, which is generally how you receive those certificate authorities. If I'm able to get my own CA on your system and trusted, and I can see your traffic, it doesn't matter whether you're using HTTPS connections.
A VPN provider might say they're not keeping logs, or that their servers are not beholden to a third party and traffic is not being analyzed, but ultimately all you have is their word on that. Ultimately, the only thing different between you connecting to the NSA and routing all your traffic (even if your traffic is mostly encrypted) through them so they can look at it and a VPN provider is that you trust the VPN provider when they say they aren't the NSA and they aren't looking at your traffic.