Concise Bible Dictionary:

A term frequently occurring in scripture expressing an attribute of God which maintains what is consistent with His own character, and necessarily judges what is opposed to it—sin. In man also it is the opposite of lawlessness or sin (1 John 3:4-74Whosoever committeth sin transgresseth also the law: for sin is the transgression of the law. 5And ye know that he was manifested to take away our sins; and in him is no sin. 6Whosoever abideth in him sinneth not: whosoever sinneth hath not seen him, neither known him. 7Little children, let no man deceive you: he that doeth righteousness is righteous, even as he is righteous. (1 John 3:4‑7)); but it is plainly declared of man that, apart from a work of grace in him, “there is none righteous, no, not one” (Psalm 14:1-31<<To the chief Musician, A Psalm of David.>> The fool hath said in his heart, There is no God. They are corrupt, they have done abominable works, there is none that doeth good. 2The Lord looked down from heaven upon the children of men, to see if there were any that did understand, and seek God. 3They are all gone aside, they are all together become filthy: there is none that doeth good, no, not one. (Psalm 14:1‑3); Rom. 3:1010As it is written, There is none righteous, no, not one: (Romans 3:10)). But God has, independently of man, revealed His righteousness in the complete judgment and setting aside of sin, and of the state with which, in man, sin was connected. This was effected by the Son of God becoming man and taking on the cross, vicariously, the place of man as under the curse of the law, and in His being made sin and glorifying God in bearing the judgment of sin. Hence grace is established on the foundation of righteousness. The righteousness of God, declared and expressed in the saints in Christ, is thus the divinely given answer to Christ having been made sin. On the other hand, the lake of fire is an eternal expression of God’s righteous judgment. At the present moment God’s righteousness is revealed in the gospel and apprehended by faith.
This is an entirely different principle from that on which the Jew went, namely, that of seeking to establish their own righteousness, and not submitting to the righteousness of God (Rom. 10:33For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God. (Romans 10:3)). Their father Abraham believed God, and it was counted unto him for righteousness; and the faith of the believer is counted to him for righteousness, apart from works (Rom. 4:3,53For what saith the scripture? Abraham believed God, and it was counted unto him for righteousness. (Romans 4:3)
5But to him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness. (Romans 4:5)
Christ Jesus is made unto us righteousness from God (1 Cor. 1:3030But of him are ye in Christ Jesus, who of God is made unto us wisdom, and righteousness, and sanctification, and redemption: (1 Corinthians 1:30)). He is the end of the law for righteousness to all those who believe.
Besides the above, there is the practical righteousness which characterizes every Christian. By knowing God’s righteousness he becomes the servant of righteousness. The bride of the Lamb is represented as “arrayed in fine linen, clean and white:” which is “the righteousnesses of the saints” (Rev. 19:88And to her was granted that she should be arrayed in fine linen, clean and white: for the fine linen is the righteousness of saints. (Revelation 19:8)).
The doctrine of the imputed righteousness of Christ, though largely acknowledged in Christendom, is not found in scripture. The explanation generally given of the doctrine is that Christ having perfectly kept the law, His obedience has formed a legal righteousness that is imputed to the believer as if the latter had himself kept the law. One passage of scripture proves this view to be incorrect: “If righteousness come by the law, then Christ is dead in vain” (Gal. 2:2121I do not frustrate the grace of God: for if righteousness come by the law, then Christ is dead in vain. (Galatians 2:21)). The force of the doctrine is to maintain the validity of the law in application to believers; and it stands in the way of their apprehending their death to the law by the body of Christ, so as to be married to Christ raised up from the dead, to bring forth fruit to God (Rom. 7:44Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God. (Romans 7:4)).

From Anstey’s Doctrinal Definitions:

This has to do with how God is able to save sinners without compromising what He is in Himself. “The righteousness of God” is about God’s acting in love to save sinners, but at the same time, not compromising what He is as a holy and just God (Rom. 3:2121But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets; (Romans 3:21)).
Man’s sin has seemingly put God in a dilemma. Since “God is love” (1 John 4:99In this was manifested the love of God toward us, because that God sent his only begotten Son into the world, that we might live through him. (1 John 4:9)), His very nature calls for the blessing of man, for He loves all men (John 3:1616For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. (John 3:16)). But at the same time, “God is light (1 John 1:55This then is the message which we have heard of him, and declare unto you, that God is light, and in him is no darkness at all. (1 John 1:5)), and thus His holy nature justly demands that man should be judged for his sins (Heb. 2:22For if the word spoken by angels was stedfast, and every transgression and disobedience received a just recompence of reward; (Hebrews 2:2)). If God acted according to His heart of love and brought men into blessing without dealing with their sins, He would cease to be holy and just. On the other hand, if God acted according to His holy nature and judged men according to the claims of divine justice, all men would be justly sent to hell, and none would be saved—and the love of God would remain unknown. How then can God save men and still remain just? The gospel announces this. It declares God’s righteousness and reveals the good news that He has found a way to meet His holy claims against sin and thus be able to reach out in love to save sinners who believe. Thus, God is presented in the gospel as being “just and the Justifier of him which believeth in Jesus (chap. 3:26).
Many Christians have the idea that God's righteousness is something that is imparted to, or bestowed upon, or given to, the believer. However, Scripture does not present it in that way. Simply put, the righteousness of God refers to an act of God, not a commodity that He passes to men when they believe. If God gave His righteousness to us when we were saved, then He would no longer have it! W. Scott said, "It is God's righteousness, not man's. God cannot impute that which is essential to Himself in His dealings with men" (Unscriptural Phraseology, p. 10). It is true that God has given righteousness (Rom. 5:1717For if by one man's offence death reigned by one; much more they which receive abundance of grace and of the gift of righteousness shall reign in life by one, Jesus Christ.) (Romans 5:17)), but this is in the sense of having secured or provided it for mankind in Christ, the risen and glorified Man. Thus, Christ has been "made unto us wisdom and righteousness" (1 Cor. 1:3030But of him are ye in Christ Jesus, who of God is made unto us wisdom, and righteousness, and sanctification, and redemption: (1 Corinthians 1:30)), and we have been "made the righteousness of God in Him" in testimony to the world (2 Cor. 5:2121For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him. (2 Corinthians 5:21)). But the righteousness of God is not something that has been sent down from heaven and placed upon the believer, as some think.
W. Scott also said, “It is not putting a quantity of righteousness in, or on, a man" (Doctrinal Summaries, p. 15).
J. N. Darby remarked, “A man's being reckoned righteous is his standing in the sight of God, not a quantum of righteousness transferred to him” (Collected Writings, vol. 23, p. 254).
F. B. Hole said something similar: “We must not read those words [“the righteousness of God”] with a commercial idea in our minds, as though they meant that we come to God bringing so much faith for which we receive in exchange so much righteousness, just as a shopkeeper across the counter exchanges goods for cash” (Outlines of Truth, p. 5).
The righteousness of God points to what God has done in taking up the question of sin and settling it for His own glory and for the blessing of man. He sent His Son to be the Sin-bearer, and in His death, God judged sin according to His holiness. The Lord Jesus took the believer's place before God and bore his sins (the judgment of them) in His "own body on the tree" (1 Peter 2:2424Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed. (1 Peter 2:24)). His "finished" work on the cross (John 19:3030When Jesus therefore had received the vinegar, he said, It is finished: and he bowed his head, and gave up the ghost. (John 19:30)) has rendered a full satisfaction to the claims of divine justice and has paid the price for the believer's sins. Thus, God has not compromised what He is as a holy and just God in reaching out in blessing to man. God's love has come out to men with the good news that He can, on a righteous basis, redeem, forgive, justify, and reconcile the sinner who believes.

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