On a certain occasion in ancient times there was to be a great wrestling match. The preparations were made and a crowd gathered. On one side stood Tiberius Claudius Marcianus; on the other side stood several challengers. As Tiberius stripped for action his opponents gasped at his powerful physique. They each cast a dubious glance at themselves and quickly asked to be excused. To put it more bluntly in our modern vernacular, they threw in the towel and quit!
The Greek word involved here was paraiteomai (par i TEH oh my). In addition to the meanings suggested above, it also meant to beg off. This definition serves well in the first of several uses of paraiteomal in the New Testament. In Luke 14:18 a great banquet was ready, but those who were invited began to make excuse. They began to beg off, refuse, and ask to be excused. In the more courageous context of Acts 25:11, Paul said if he had done anything for which he deserved to die, then he did not seek to escape death.
Sometimes the word also carries the connotation of aversion and repugnance. (This may be part of the reason the host of the banquet back in Luke 14 was so angry. His guests had not merely declined; they had spurned his invitation!) In this light we should look at the three times word is used in Hebrews 12. Verse 19 tells how the Israelites at Mt. Sinai begged off from listening to the voice of God. (See Exodus 20:18ff.) Verse 25 twice warns Christians against doing the same thing. When God speaks we must not ask to be excused! We must not refuse His commands or spurn His commission. Let's not beg off!