Luke 10:4 (NASB)
Carry no money belt, no bag, no shoes; and greet no one on the way.
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On his second and larger campaign1, Jesus sent out a group of seventy-two disciples to "every city and place where he was about to go." (10:1) The disciples were being sent out as Christ's ambassadors.
While giving instructions, Jesus commanded:
Carry no purse, no bag, no sandals, and salute no one on the road. (10:4)
The prohibition of greeting anyone was to apply only during their journey. Why would Jesus prevent them from greeting anyone on the road?
"The harvest is plentiful, but the laborers are few" (10:2)
Just before Jesus told them what they were not to do, he impressed upon them the urgency and the magnitude of their task. They were in the Father's field. God was the "Lord of the harvest." The task was immense. The laborers were few. They were to remain focused on reaching their assigned towns or villages at the earliest, and to complete their tasks. Nothing should ever distract or slow them down.2 Social greetings in the Ancient Near East were not just about elaborate physical gestures or verbal pronouncements. It included exchange of pleasantries, news, and questions about the family, etc.3
"Go! Behold I send you out as lambs in the midst of wolves." (10:3)
The metaphor is clear. The mission was a delicate operation to be accomplished with utmost care as per Jesus' instructions. The majority of people out there were not friends or even neutral. They were adversaries who would look upon the disciples as pieces of meat on a plate. There probably was no need to create opportunities for a flare up of hostility or persecution through unnecessary interactions.
"Whatever house you should enter, first say, ‘Peace on this house.’ And if there be a son of peace there, your greeting shall rest on him; but if not, it shall return to you." (10: 5-6)
The disciples were commanded to pronounce a "greeting" of divine peace only over households among the target population. Among them, even among the wolves, there was a possibility of finding a son (or daughter) "of peace" — a potential point of entry for them and their message.
The disciples' greeting – while on God's mission as Christ's ambassadors – wasn't an ordinary "Hello." It was a channel of God's peace or blessings. Therefore, such greeting was reserved for people whom Jesus sent them to.
1 The first one is described in Luke 9:1-6.
2 Elisha's instruction to Gehazi is an example of such focus needed during an emergency. 2 Kings 4:29.
3 Windisch, TDNT 1:496–99. Cited by Darrell L. Bock, Luke, Baker Exegetical Commentary on the New Testament, Grand Rapids: Baker Academic, 1996.
Similarly, Elisha instructed his servant in 2 Kings 4:29
Elisha said to Gehazi, "Tuck your cloak into your belt, take my staff in your hand and run. Don't greet anyone you meet, and if anyone greets you, do not answer. Lay my staff on the boy's face."
Jesus sent his disciples out on a mission tour. He didn't want them to waste time and be sidetracked by friends or strangers. These are common tactics of Satan to delay or stall the mission. To avoid loitering and help them focus on the tasks at hand, Jesus commanded them not to perform these superficial activities.
There is also the story of the man of God from Judah in 1 Kings 13:
8 But the man of God answered the king, “Even if you were to give me half your possessions, I would not go with you, nor would I eat bread or drink water here. 9For I was commanded by the word of the Lord: ‘You must not eat bread or drink water or return by the way you came.’ ”
He obeyed, so far so good. But then later on, an old prophet
14 found him sitting under an oak tree and asked, “Are you the man of God who came from Judah?”
“I am,” he replied.
Big mistake! He socialized with him. Eventually,
24 as he went on his way, a lion met him on the road and killed him, and his body was left lying on the road, with both the donkey and the lion standing beside it.
To simplify the complications, it was best not even to greet anyone.
I don't want to "steal" an answer but would like to point out a comment by @www.gffg.info
Salute no man by the way - Salutations among the Orientals did not consist, as among us, of a slight bow or an extension of the hand, but was performed by many embraces and inclinations, and even prostrations of the body on the ground. All this required much "time;" and as the business on which the seventy were sent was urgent, they were required not to "delay" their journey by long and formal salutations of the persons whom they met.
In addition, Barnes goes on to give an example of what a greeting could entail:
"If two Arabs of equal rank meet each other, they extend to each other the right hand, and having clasped, they elevate them as if to kiss them. Each one then draws back his hand and kisses it instead of his friend's, and then places it upon his forehead. The parties then continue the salutation by kissing each other's beard. They gave thanks to God that they are once more permitted to see their friend - they pray to the Almighty in his behalf. Sometimes they repeat not less than ten times the ceremony of grasping hands and kissing."
The study note from the New World Translation for this verse provides this information:
greet anyone: Or “embrace anyone in greeting.” In certain situations, the Greek word a·spaʹzo·mai (“to greet”) may have involved more than saying “hello” or “good day.” It could have included the embraces and long conversation that may take place when friends meet. Jesus was not encouraging his disciples to be rude. Rather, he was emphasizing that his followers should avoid unnecessary distractions and make the most of their time. The prophet Elisha once gave similar instructions to his servant Gehazi. (2Ki 4:29) In both cases, the mission was urgent, so there was no time for delay.
It could have been a means of not jeopardising the allocation strategy.
The disciples were sent out in pairs in different directions to specific destinations. If people were greeted along the way the chance that two pairs of disciples would end up in the same town would be greatly increased.
Thus, the directive to not stop and converse with travellers along the way could have been a means to prevent corruption of the plan to allocate one pair of disciples to each town of Israel