According to Paul's Gospel, all people, both Gentiles and Jews, will be judged by "law." However, as Romans shows, Paul does makes a distinction between "law" and "the law."
12 For all who sinned without-Law will also perish without-Law, and all who sinned under the Law will be judged by the Law. 13 For not the hearers of law[f] are righteous before God, but the doers of law will be declared-righteous. 14 For whenever Gentiles, the ones not having the Law, are doing by nature the things of the Law, these ones not having the Law are a law to themselves, 15 who are demonstrating the work of the Law written in their hearts— their conscience bearing co-witness and their thoughts accusing or even defending between one another 16 on the day when God judges the hidden things of people according to my good-news, through Christ Jesus. (Romans 2 DLNT)
f. This is lowercase because it applies to both groups in v 12 in the different senses Paul defines next.
Law is written with the article only twice in this passage.
Verse 14 speaks of Gentiles doing "the" law by nature. The likely meaning is the Mosaic Law.
1Verse 15 has τὸ ἔργον τοῦ νόμου, literally the work the law. Two meanings are possible: 1) "the" Mosaic Law doing "the" work on the heart or 2) "the" as in "any" law from God doing the work. With respect to being judged one could say the determining factor is that which is law for Gentiles and "the" law for Jews.
In his book, Greek Grammar, Beyond the Basics, Daniel B. Wallace provides an illustration
2showing how the article works. Next to this, I have added similar diagrams to depict all law, works and law, and the specific law of circumcision:
The force of the article is to "definitize." The most common use is anaphoric. When used as such the meaning is: that which I previously wrote about.
3With respect to the issue in Galatians, that type of use could be problematic because "law" for Paul has an inherent force without the article, as the second illustration shows. "The" Mosaic Law does not definitize by narrowing the focus. Rather, the Mosaic Law adds to any existing law. When writing to a mixed audience of Gentiles and Jews, the common application, to "definitize" law, that is to narrow the scope, would be taken to mean only those requirements which apply to Gentiles. However, all requirements for Gentiles are included in the Mosaic Law. Therefore, with regard to the scope of law, the article must be referring to the Mosaic Law. That is, according to Paul's Gospel Gentiles are judged according to those parts of the Mosaic Law they know.
With respect to circumcision, the article would definitize by narrowing the focus. However, since circumcision is required in three different settings for three completely different reasons (see below) what is made definite is only the act of circumcision.
Works of Law - Works of the Law
This issue is more clearly illustrated by the phrase ἐξ ἔργων νόμου, literally, by works of law. Regardless of which or any law Paul composed an answer based on salvation. Conceptually, Paul is arguing from the greater to the lesser. Like the Mosaic Law which encompasses all requirements, salvation by grace through faith encompasses all works, not just works of law:
This only I want to learn from you: Did you receive the Spirit by works of [article less] law, or by the hearing of faith? (Galatians 3:2)
8 For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, 9 not of [article less] works, lest anyone should boast. (Ephesians 2)
The statement in Ephesians, like Galatians omits an article with works, ἐξ ἔργων. Paul's Gospel is that one is saved by faith not by any work. Therefore, of law, without an article definitizes works. Not only is the article unnecessary to definitize, its use might imply some works might save, just not works of "the" law.
More importantly, if the force of the article is to definitize, then the force of omitting the article is to generalize when speaking of law and salvation:
Correct understanding: This only I want to learn from you: Did you receive the Spirit by any works of law, or by the hearing of faith?
Imprecise translation: This only I want to learn from you: Did you receive the Spirit by works of the law, or by the hearing of faith?
A translation of "the" law make sense but is imprecise. It shows a "definitization" which is contrary to the meaning of the actual text. By omitting the article the meaning remains as general as possible. Therefore, while it is true, the Spirit was not received by circumcision, or any other work of the law; the statement does not definitize law since all works are included in ἐξ ἔργων νόμου.
The primary reason for writing is to respond to the question of circumcision. Paul's answer is based on salvation, yet he must have the specific issue in view. As the illustration above shows, the article would be appropriate if one also had a definite law (circumcision) in mind. However, a comprehensive legal answer to the question of circumcision could address four specific laws:
Paul's use of Abraham's faith is most vulnerable on the question of circumcision. Didn't Abraham circumcise every male in his household as a sign of the covenant? In addition, only circumcised males are permitted to eat the Passover. Shouldn't Gentiles be circumcised before eating the Passover? If those of "the circumcision" are denying Gentiles the right to eat the Passover, they could argue theirs's is not a question of salvation. In this case, though it is likely an end run, those of the "circumcision" would be using eating the Passover to force circumcision.
Without ever citing a specific law, Paul knows the question of circumcision is answered at the end of the Mosaic Law. Deuteronomy only speaks of circumcision of the heart: one performed by the person and the other by God. Therefore, if one has made the choice of "heart" circumcision and God has performed this circumcision, isn't one (male and/or female) eternally circumcised? Furthermore, three individual laws of circumcision are qualified: foreskin is added in Genesis and Leviticus and the heart in Deuteronomy. The type of circumcision for the Passover is not specified. So with respect ot eating the Passover, either type of circumcision satisfies "the" law.
Galatians 3:10 and 3:21
Two verses have been composed using and omitting the article:
For as many as are of the works of law are under the curse; for it is written, “Cursed is everyone who does not continue in all things which are written in the book of the law, to do them.” (3:10)
Is the law then against the promises of God? Certainly not! For if there had been law given which could have given life, truly righteousness would have been by law. (3:21)
Verse 10 is reasonably straight forward. Works of law is definite without the article and "the law" is prefaced with "the book" which means the Mosaic Law. Paul immediately connects cursed with a specific law: Christ redeemed us from the curse of the law by becoming a curse for us—for it is written, “Cursed is everyone who is hanged on a tree” (verse 13). The article refers to the Mosaic Law, but the use with a specific law which does "curse" affirms the initial statement "any" works of "any" law are under a curse.
Verse 21 follows a similar pattern. "The Law" refers to the Mosaic Law and the other two mean "any law." The verse means, Is the Mosaic Law against the promises of God? Certainly not! For if there had been any law (including circumcision) which could have given life, truly righteousness would have been by law.
This second use sets the stage for Paul's response to a challenge based on Abraham circumcising every male in his household. An argument which continues to Chapter 4.
However, with respect to the OP's question about verse 3:21, it is worth noting in Romans Paul makes similar arguments regarding Abraham, and he also makes arguments making reference to the first man. If "law" means any instruction from God, then obviously the very first law could not bring life:
15 The LORD God took the man and put him in the garden of Eden to work it and keep it. 16 And the LORD God commanded the man, saying, “You may surely eat of every tree of the garden, 17 but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die.” (Genesis 2)
From the beginning, law brought death to that which God had given life.
When used with law, the article means a specific law or book of laws; in the context of Galatians, the article most likely is used to definitize the requirement of circumcision, regardless of which specific requirement.
In the specific case of verse 3:21, while adding the article does make sense, it also makes something specific which Paul composed to be general. That is, no law beginning with the first man in the Garden, could bring life. Quite the opposite, law brings death.
1. For example, a Gentile could voluntarily observe the Sabbath or follow a kosher diet.
2. Daniel B. Wallace, Greek Grammar: Beyond the Basics An Exegetical Syntax of the New Testament, Zondervan, 1996, p.
3. Referring to what was previously written in the document, not some other document.