I can not separate Matthew 5:13 from 14-16 as the author of the list did:

“You are the salt of the earth; but if the salt loses its flavor, how shall it be seasoned? It is then good for nothing but to be thrown out and trampled underfoot by men.

Application/View 1

"Ye are the salt of the earth." These words (and those which follow to the end of verse 16) are frequently regarded as being spoken of God’s people at large, but this we think is a mistake. First, because such an interpretation is out of harmony with the immediate context. In our last chapter we called attention to our Lord’s changing of the pronoun in verse 11 from the "they" in verses 1-10 to the "ye." In verse 10 Christ enumerated the general principle that "blessed are they which are persecuted for righteousness sake," but in verse 11 He made particular application to His own ministers: persecution is the usual experience of God’s people, but it is the special portion of His servants. Clear confirmation of this distinction is found in verse 12, where the maligned ministers of Christ are bidden to rejoice because "so persecuted they the prophets which were before you"—not "the saints," but the official servants of God.

Thus, the "Ye are the salt of the earth" obviously has reference to those who now occupy the same position as did the "prophets" of old, namely those called of God to act as His mouthpiece and interpret His will. Additional proof is found in what immediately follows, where after further designating them the "light of the world" Christ added, "A city that is set on a hill cannot be hid a figure fitly pertinent to the officers of Christ, who are made a spectacle to the world. Finally, what is said in verse 15 plainly pertains to the ministers of God rather than to their hearers, for the candle on a candlestick again speaks of official dignity, and the giving "light to all that are in the house" is plainly the one man ministering to the many.

Matthew Henry begins his comments on these verses by pointing out, "Christ had lately called His disciples and told them they should be ‘fishers of men’ (4:19); here He tells them further what He designed them to be—the salt of the earth and light of the world: that they might be indeed what it was expected they should be." It is only in recent generations, when the spirit of socialism has invaded the religious realm, that this passage has been promiscuously applied to Christians. The two emblems which Christ here employed are very striking, and their order significant. He resembles His ministers to "salt" to humble them, for salt is cheap, common, and insignificant; to "light" to encourage them, for light is illuminating, conspicuous, elevated.

The passage we are now to ponder forms the second section of our Lord’s Sermon on the Mount. In it Christ touches upon the office of the apostles, and therein (according to their measure) that of all His ministers. It was a distinct division of His address, yet there is a manifest relation between it and the last one: only those whom the Lord pronounces "blessed," whose characters correspond to that which He portrayed in verses 1-11, are called by Christ to witness publicly for Him. The ministers of God must themselves first be seasoned by the Word: how could they fittingly apply salt to the consciences of others who had never felt the bite of it on their own? The design of these verses, then, is to stir up Christ’s servants to diligence and fidelity in declaring the will of God unto saint and sinner alike.

Thus, the first two sections of this Sermon are closely connected. The coherence of our present portion with the former stands thus: Christ had declared that there is a company on earth upon whom the Divine benediction rests. Anticipating the question, How do they attain to and maintain this felicity by such graces of the Spirit, which fits them for that estate? He answers, the preaching of God’s Word is the principal means to work in the heart those graces to which true happiness is promised. Because this is a high and holy privilege to bring men to this estate, Christ exhorted His ministers unto earnestness in their service by two weighty reasons, drawn from the properties of their work, and propounded by two similitudes.

"Ye are the salt of the earth" (verse 13). "Ye," that is those whom I have called to be apostles and set apart for the work of the ministry. Ye are "salt," not literally, yet by resemblance; yet not in regard of their persons, but of their lab ours. They are here likened to "salt": they were to season souls for God by making them savory in heart and life. From this emblem both ministers and people may learn their respective duties. Ministers are to dispense the Word, both Law and Gospel, in such a way as to express the qualities of salt. Now the properties of salt as applied to raw flesh or fresh meats are principally these: first, it will fret and bite, being of a hot and dry nature; second, it makes meat savory to our taste; third, it preserves meat from putrefaction by drawing out of it superfluous moisture.

Salt is an indispensable necessity of life. It is God’s great antiseptic in a sphere of decay. It is wrought into the very rocks and soil of earth so that the waters filtering through them become purified thereby. It is a necessary element of the blood, which is the life of our bodies. How well suited is it then as a figure of the Truth, by which means the soul is sanctified, for as salt arrests natural corruption, so the Word of God militates against moral corruption. This figure, then, furnishes clear direction to every minister of God as to his manner of preaching. Since the Word alone be the savory salt whereby souls are seasoned for the Lord, then it ought to be dispensed purely and sincerely. If salt be mixed with dust and rubbish it loses its pungency and efficacy, and if the Word be mingled with levity or exciting anecdotes its power is nullified.

This figure plainly warns the minister of his pressing need of fortitude. It is "salt" and not sugar candy he is to employ: something which the ungodly will be more inclined to spit out than swallow with a smile, something which is calculated to bring water to the eyes rather than laughter to the lips. The minister, then, must not expect faithful preaching to be acceptable and popular. It is contrary to nature for those whose consciences are pricked to be pleased with those who wound them. Christ’s servants must be prepared for their hearers to fret and set themselves against what searches out their corruptions. Such displeasure and opposition is a testimony that their ministry is "salt," that it has bitten into the depravity of their people. Instead of being discouraged and dismayed they are to perseveres endeavoring to season their congregation more and more with the pure salt of God’s Word.

The hearer also is to receive instruction from this figure. Hereby each one may see what he is in himself by nature: depraved and corrupt, as unsavory flesh and stinking carrion in the nostrils of God, or else what need of salt? How this should humble and cause us to lay aside all pride and self-righteousness. Again, every one must learn hereby to suffer the word of reproof, whereby his secret sins are discovered and denounced. When our conscience is searched we must be willing for salt to be rubbed into it, for mortification precedes salvation. The hearer must give all diligence to be seasoned with this heavenly salt so that the thoughts of his heart, the words of his mouth, and the actions of his life may be acceptable to God (Col. 4:6). If we sit under the ministry of the Word (oral or written) and be not seasoned thereby our case is doubly evil (Judges 9:45).

"But if the salt have lost his savor, wherewith shall it be salted? It is thenceforth good for nothing but to be cast out, and to be trodden underfoot of men" (v. 13). This was brought in by Christ to move His servants unto fidelity and diligence in their ministry by the danger attending the opposite. Infidelity in the ministry is like unsavory salt: ineffectual, worthless, despicable, subject to a fearful curse. This is the great danger of the pulpit: to become men-pleasers, to yield unto the demand for smooth speaking, to tickle the ears of their auditors with novelties. Such preachers become unsavory salt, unprofitable in their ministry, failing to season souls so that they are acceptable to God. Salt is useless when it loses its virtue and acrimony. Ministers become such when through lack of prayer and continuous study they fail to increase in spiritual knowledge, or when adopting false doctrine they preach error, or when they cease to denounce sin, or when they fail to practice what they preach.

The greatness of the danger attending ministers who become unfaithful and unprofitable is here pointed out by Christ in His words "wherewith shall it [i.e. the salt—cf. Mark 9:50] be salted." Those who depart from fidelity are very seldom, and then only with great difficulty, recovered and restored. Read what is recorded of the false prophets in the Old Testament and of false apostles in the New and where is there an instance that any repented? The same solemn principle is exemplified in the case of almost all those preachers who have forsaken Protestantism and gone over to Rome. How diligently, then, do ministers need to take to heart that injunction, "Meditate upon these things; give thyself wholly to them; that thy profiting may appear to all. Take heed unto thyself, and unto the doctrine; continue in them: for in doing this thou shalt both save thyself and them that hear thee" (1 Tim. 4:15, 16); and again, "But thou, O man of God, flee these things [cf. 5:10]; and follow after righteousness, godliness, faith, love, patience, meekness" (1 Tim. 6:11).

The unprofitableness of unfaithful ministers is expressed in the words "it is thenceforth good for nothing": just as unsavory salt is become worthless to season meat, so unfaithful ministers are valueless to God and man. The curse resting upon such is, "it is cast out and trodden under foot of men," that is, such preachers are condemned both by the Lord and by their fellow men. "Therefore have I also made you contemptible and base before all the people, according as ye have not kept My ways, but have been partial in the law" (Mal. 2:9), such was the fate pronounced upon the renegade priests of old. No doubt Christ was here making an oblique reference to the scribes and Pharisees of His day, affirming their unprofitableness and announcing the impending doom of Judaism. Solemn beyond words is this verse, and prayerfully should it be laid to heart by all Christian ministers.

Lost its savor (mwranthi).
The verb is from mwrov (dull, sluggish, stupid, foolish) and means to play the fool, to become foolish, of salt become tasteless, insipid (Mark 9:50). It is common in Syria and Palestine to see salt scattered in piles on the ground because it has lost its flavor, "hae tint its tang" (Braid Scots), the most worthless thing imaginable. Jesus may have used here a current proverb.
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What metaphor does Jesus use in Matthew 5:13?

SALT (ISBE article) (Easton) (Torrey's Topic on Salt)

 

WHAT METAPHOR DOES JESUS USE IN Matthew 5:13?

SALT

 

WHAT IS THE SIGNIFICANCE OF SALT IN OT (Lev 2:13, Ezra 6:9-10; Ezek 43:23-24 Nu 18:19 2 Chr 13:5)?
Salt highly valued – For priests, sacrifices & covenants ~ symbolized permanence

Salt was a part of a soldier’s pay = salt and salary are related “He’s not worth his salt”

WHAT DO WE LEARN ABOUT THE EFFECT OF SALT IN 2KI 19-22?
Elijah to heaven, Elisha's Miracle of salt purifying testified = God's messenger.

Salt ~ purification

WHAT DOES JESUS SAY IN MT 5:13 -- IS THIS A COMMAND?
It is a statement of fact = You ARE the salt of the world

WHO IS THE SALT OF THE WORLD?

WHAT MAKES THEM SALTY IN CONTEXT?
Kingdom citizens = 7 fold character of Mt 5:3-9 (poor in spirit, etc)

WHAT WAS SALT USED FOR IN JESUS' DAY?
Preservative = prevents decay, restrains corruption where it is present
Seasoning - adds flavor (see Colossians 4:5 below)

HOW WOULD THIS APPLY – WHAT DOES THE METAPHOR MEAN?
Salty Christians preserve their surroundings