Here I will point to only two examples of logical errors that seem incontrovertible.  There are others, but these are sufficient to reveal the limitations of his inspiration.
 
The Cretans
The first is his self referential statement in the letter to Titus.  Titus had been left in Crete to attend to missionary tasks, and Paul later wrote to him to more clearly define his duties.  Early in this letter, referring to the Cretans, he came forth with this astonishing statement:
One of themselves, a prophet of their own said, “Cretans are always liars, evil beasts, lazy gluttons.” This testimony is true (Titus 1:12).
But, if Cretans are always liars, this statement by one of them must be false.  Paul says it is true.  Paul has surely fallen into a trap of his own making.  If the illogic is not obvious to you, perhaps it will help if I present it as follows, keeping in mind that the Cretan whom Paul was quoting is Epimenides.
 1. Paul writes that Epimenides says that Cretans are always liars.
 2. Epimenides was a Cretan (a prophet of their own).
 3. Therefore Epimenides always lies.
 4. Paul says that what Epimenides said is true.
This is a pure conundrum.  It is nonsense.  Paul says that Epimenides' testimony is true, the very testimony that says that Epimenides is a liar!  Surely this kind of puny logic must have a merely human origin, the mind of Paul.
 
Who Must Die?
The second error in logic is just as obvious, and much more serious, as displayed in Romans 7:1-3:
Do you not know, brethren – I am speaking to those who know the law – that the law is binding on a person only during his life?  Thus a married woman is bound by law to her husband as long as he lives, but if her husband dies she is discharged from the law concerning the husband.  Accordingly, she will be called an adulteress if she lives with another man while her husband is alive.  But if her husband dies she is free from that law, and if she marries another man she is not an adulteress.
To assist you with the perception of error in this logic, again I will break it down as follows:
 1. The law is binding on a person only during his (or her) life.
 2. Therefore, the law binds a married woman to her husband as long as she
     lives.
 3. Therefore, when she dies, she is free from the law concerning her husband.
 4. But Paul sets her free when the husband dies, not when she dies!
By the premise, "the law is binding on a person only during his life," the married woman is yet bound to her husband, even though he is dead, and she will continue to be bound to him until she dies!  It is, of course, true that the woman is bound to her husband only as long as he lives, but this is by a separate provision in the law.  This fact is undoubtedly what tripped Paul here – this together with the gender distinction that made it difficult to see the woman as bound by the law, apart from her husband, in any case.

Paul next proceeds to compound the error by applying this principle to the spiritual experience of the disciples:
Likewise, my brethren, you have died to the law through the body of Christ, so that you may belong to another, to him who has been raised from the dead in order that we may bear fruit for God (Romans 7:4).
He seems to intend that the analogy be applied in this way:
1. You were married to the law.
2. Because Christ died, you are freed from the law.
3. Being free from the law, you may now marry Christ (may belong to another).
But according to the premise stated, you are the one who must die to be free from the law, not someone else, such as Christ.  But we see how Paul must have intended to apply the principle: you are associated with Jesus in his death, therefore you have also died in some spiritual sense and through that death freed from the law that bound you to the law.  But the fact is that Jesus is the only one who has died so as to conform to the principle, for you are yet walking around in your flesh.

To compound the faulty logic, if the lesson drawn from it is to conform to the marriage analogy that is given, then it is your husband, the law, that must have died to free you to marry Christ – but Paul's faulty analogy kills off both you and Christ (everybody but the law!) and allows the law to live.  These failures in logic justify us in concluding, at the very least, that when he wrote these lines he was not writing under the inspiration of the Holy Spirit, but was simply recording whatever came to the mind of the man.  He was not particularly intelligent or he would have spotted the fallacies in his arguments, even without the aid of the Spirit.
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