Cowbee's reply is pretty solid. I'll approach from a different angle.

The real learning lies in examining the details of your three points.

Point 1. Is it true that most modern (Western) philosophy depends on the belief that logic will allow us to get to the truth in some capacity?

Yes. But what do you understand Modern Western philosophy to be? Modern Philosophy is a specific academic term meaning philosophy from Decartes in the 1600s to Kant in the late 1700s and ending by the very early 1800s.

What you're describing is also true of Ancient Greek Philosophy and Medieval Philosophy. But you don't seem concerned about those because they have been surpassed. Well Modern Philosophy was surpasses about 220 years ago. So why are you asking this question in this way? Why do you care about a philosophical tradition that's over 200 years old?

Perhaps you used Modern in the colloquial sense, to mean "present day philosophy" or contemporary philosophy. Most present day philosophy does not hold the belief that logic is the basis of truth.

After Kant essentially triggered the end of the Modern period of philosophy, logic stopped being the sole basis for knowledge in philosophy. The next 100 years saw the emergence of Hegel, Marx, Kierkegaard, and Nietzsche, all of whom undermined logic as the basis of truth. Then a century after Kant we get phenomenology, linguistics, and postmodernism, all of which fundamentally undermine logic entirely. That's more than a century of contemporary philosophy at this point.

So, an important question, is your first premise actually a misunderstanding you have about philosophy, and if so, despite philosophy having undermined logic for a century, how did you come to believe in this first premise?

I ask this not because you're position is somehow deficient or laughable but because I experienced much the same coming to awareness that what I thought philosophy had arrived at and what philosophy had actually arrived at were very different, and the process of both learning where philosophy is and learning why I was so wrong about it were incredibly enlightening.

Point 2. You put "quantum mechanics" in opposition to "human logic". What is human logic and why qualify logic with the word "human"? Philosophy from the Ancient Greeks through Modernity saw logic as universal, not specific to humans. The idea that there could be a logic that is distinctly human comes out of philosophy after the Modern Philosophical period has ended. So perhaps postmodernism has had an impact on you that you're unaware of.

If logic is distinctly human, or if the logic we know is distinctly human, what is its origin? Is it a tool we created? Is it innate in our bodies? Is it an accidental emergence that has no meaning at all?

If logic can be human, are there other logics that are not in opposition with quantum mechanics? Are those logics a valid basis for truth? Could we adopt non-human logic to arrive at truth?

Point 3. What are "correct conclusions"? What does it mean for a conclusion to be correct? Is it a description of the language being used or is it a description of a mental state that language can only approximate?

Perhaps there's no such thing as correct conclusions at all and there never was. Perhaps the concept of correctness is actually the thing that is human. In that case quantum mechanics is the least of our worries as the whole of existence might stand in opposition to the very concept of correctness.

And yet, correctness as a concept is here, so it must exist in some sense, mustn't it?

Wrapping Up

I think that stepping back from one's own questioning, going back to the beginning, and observing one's own words is a powerful path of learning. We back into our quest for knowledge a certain set of presuppositions that we have inherited without realizing it. Part of the process is questing forward, and part of the process is questing backwards. Forward we gather new experiences and insight, backward we dismantle the presuppositions that bind and constrain our movement. We can then quest forward again with entirely new degrees of freedom.

Great question! Thanks for bringing it to the community for us to participate!