1. There are Christian philosophers who've done very detailed works on this question.
2. There is one school of thought that sin, intrinsically, by it's very nature, necessarily corrupts the intellect.
I happen to agree with this, and I don't think it's too hard to prove in numerous ways.
3. When asking the question about how much trust can we place in the intellect if it's corrupted... I don't think this is terribly complicated.
A. SIN:
If sin intrinsically corrupts the intellect, then the degree of corruption of the intellect should be somewhat commensurate with the degree of sin... or you could say something like this:
"The degree to which a person's intellect is faulty is probably commensurate to the degree to which the person disbelieves God and his Word... as more contradiction of TRUTH must necessarily create more logical error, and more corruption of the thought processes."
B. PURE LOGIC:
If we're talking about pure logic, well logic seems to be based on principles EXTERNAL to the human mind.
So much like using principles of scripture (which are also external to the human mind) the more we rely on logic, the more rational and sensible our thought processes will be.
The problem is that humans often fail to rely on logic, and instead rely on unchecked emotions emanating from our fallen and corrupt minds.
There is nothing wrong with emotions, but since our minds are fallen, we need to rely on scripture, and logic, to keep our emotions "in check", and to keep them from becoming wholly irrational. I think we'll find the Bible talks about keeping our thoughts and emotions in check... so this is very biblical.
C. NO LOGICAL ABILITY LEFT:
If you want to make the case our intellect is so corrupt we can't have any logical thoughts at all... well... this is provably false. We can take principles of logic, and use them to actually test our own reasoning. Furthermore, if we HAD NO LOGICAL ABILITIES LEFT... we wouldn't be able to walk across the room, or flip a light switch, or drink a glass of water. If we had NO LOGICAL ABILITIES INTACT we wouldn't be able to function in any way... and we certainly couldn't use language to talk about it.
4. So to sum things up:
A. We certainly have fallen and corrupted intellects.
B. Though our intellects are no longer perfect, they will sink further into corruption commensurate to the degree which we turn against God's perfect truth in scripture... as turning against truth forces us, necessarily, into more logical error.
C. Although our logic is often flawed because we fail to use logic, we can still demonstrably prove we all constantly use laws of logic in every action of everyday life... therefore, although our intellects are corrupted, we are not at all separated from logical ability.
D. Even in our fallen states, God has GIFTED US with special things to test and check the corruption of our minds, to stem the flow of wrong thoughts, and to help us think correctly
1. God has given us his PURE WORD, which we use to "prove all things", and which we use to CORRECT our own thinking.
2. God has given us principles of logic, which seem to infuse everything in creation... and which we can also use to check our own thinking.
3. God has given us a conscience, to steer us when our thinking goes off course in moral areas
4. God has given us the indwelling holy spirit, to "guide you into all truth"