How personal was Paul's relationship to Jesus?
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There is evidence that Paul did in fact see the Lord. He saw him in glory on the road to Damascus - See Acts 9:4; 22:7; 26:14 – as one example although it could be argued that he only saw the glory and not the Lord himself. In Galatians, Paul says that he was given the gospel by revelation from Christ.
Galatians 1:11,12 But I certify you, brethren, that the gospel which was preached of me is not after man. For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.
This passage indicates that he received the Gospel directly from the Lord. Perhaps he was taught face to face as the disciples were for 40 days following the Lord's resurrection and prior to his ascension. See Acts 1:1-11
The following passage, 1 Corinthians 15:8 clinches it. He says directly that he has seen the Lord - And last of all he was seen of me also, as of one born out of due time. Here is the entire passage in context:
1 Corinthians 15:3-10 For I delivered unto you first of all that which I also received, how that Christ died for our sins according to the scriptures; 4 And that he was buried, and that he rose again the third day according to the scriptures: 5 And that he was seen of Cephas, then of the twelve: 6 After that, he was seen of above five hundred brethren at once; of whom the greater part remain unto this present, but some are fallen asleep. 7 After that, he was seen of James; then of all the apostles. 8 And last of all he was seen of me also, as of one born out of due time. 9 For I am the least of the apostles, that am not meet to be called an apostle, because I persecuted the church of God. 10 But by the grace of God I am what I am: and his grace which was bestowed upon me was not in vain; but I laboured more abundantly than they all: yet not I, but the grace of God which was with me.
I hope you have found this helpful.