By the Law is the knowledge of Sin

Paul held a particular view on the law, as far as Romans 3 is concerned. He was addressing the Jewish Christians in the Church of Rome. Therefore, he meant the Torah, rather than laws in general. Let's rebuild the context.

Paul was criticizing the Jews who were entrusted God's revelation and God's Law. They should have been the instructors of the blind and the ignorant. Yet they themselves did the same evil as those Gentiles. So, Paul asked some rhetorical questions: what are the advantages of circumcision, of keeping the law and of being a Jew etc? Jews were the first to receive glory as well as the first to receive punishment (Romans 2:9-10). God gave them circumcision as a sign of covenant. Yet, their lack of faith rendered the covenant ineffective. The Jews had failed to make good use of the advantages. In the end, both Jews and Gentiles were under the control of sins (Romans 3:9). To support this conclusion, Paul pieced together passages from the Wisdom Literature (Psalms and Ecclesiastes) to show that nobody could be righteous before God (Romans 3:10b-18, See Appendix below).

Now, what are the purposes of the law and in particular, the Torah?
Modern people think of the law as a kind of social contract. Man cannot survive alone and must live in a group for protection/security purposes. Conflicts inevitably arise when men live in a community. Therefore, rules and regulations were agreed upon, firstly unwritten and later written, to guide the conducts of social transactions. Later, an impartial third-party was institutionalized to settle disputes. This is a pragmatic way of thinking about the law. But in ancient times, people did not think of the law in this manner.

For example, in ancient China during the Spring-Autumn Period, there was a debate over whether laws should be inscribed on bronze tripods so as to be publicly accessible 【鑄刑鼎】. This proposal was made out of pragmatic considerations. The Central Government was too weak to maintain orders among the warring states in the feudal system. Justice was arbitrary and favoured the nobility. Some political reformers saw that the productivity of the society was eroded by unrestrained personal greed of the rich and the nobility. Something needed to be done lest their country would be conquered by mightier warring states. Once the law was written down to be observed by all, commoners would be on equal footing with nobles. Hopefully, such reforms would rebuild the military strength of the country. This was a progressive political idea. Confucius opposed this idea. He believed that when the commoners knew more about the letters of the laws, they would raise more disputes to seek advantages instead of building up a good virtue which was the political ideal of Confucius who lived 500 years before Christ. His was an idealistic way of reasoning.

Paul was trained a Pharisee, an expert in the Torah, the five books of Moses. The Torah was more than regulations and legal codes. It was a book of guidelines, teaching the Israelites how to lead a life pleasing to God, to be a kingdom of priests, and an holy nation (Exodus 19:6). On a deeper theological level, the Torah was a sign of their covenant with God.
However, the way Paul put it, the Torah is rather negative.
that every mouth may be stopped, and all the world may become guilty before God
ἵνα πᾶν στόμα φραγῇ καὶ ὑπόδικος γένηται πᾶς ὁ κόσμος τῷ θεῷ (Romans 3:19b, KJV)
The KJV translation is well-known to be faithful. So, what did Paul had in mind when he said this? What was the purpose of the Torah? To stop people's mouth? To hold people guilty before God?

Earlier on, Paul explained that everybody should know God in the Creation and the law of God was written/hidden in the conscience of all men (Romans 2:15). This was the law of the unwritten kind. However, men, true descendants of Adam, always made excuses.
Oh! We know not the Law!
Oh! The Law is too idealistic!
Oh! The Law is too difficult etc.
Now, that the Torah was handed down from God, people could no longer make excuses that they did not know. With the law laid bare in front of them, what did they know?
In a negative way, they knew what they should not do. Hopefully, in a positive way, they knew what God intended them to do, to become: a kingdom of priests, a holy nation. The negative way was easier to grasp. Doing what should not be done (against God) is conveniently labelled a sin. This is how Paul continued.
by the law is the knowledge of sin.
διὰ γὰρ νόμου ἐπίγνωσις ἁμαρτίας. (Romans 3:20b, KJV)
Once written down, the law sets a standard against which people's actions could be measured. That is why Paul equates sins with the coming short of God's glory. God gave the Israelites the Torah to make them a kingdom of priests, a holy nation. The glory of God would be manifested by the lives of this people. When they failed to become a holy nation, God's glory was not achieved.
(For) all have sinned, and come short of the glory of God
πάντες γὰρ ἥμαρτον καὶ ὑστεροῦνται τῆς δόξης τοῦ θεοῦ (Romans 3:23).


Instead of being a source of wisdom and a guiding light for a righteous life (Psalm 119:104-105), the Torah has become a rod, a canon against which one's behaviour is measured. Paul was a bit too pragmatic and harsh here.

Dear Lord, Thy word is a lamp unto my feet, and a light unto my path. Strengthen my feet to walk along Your path. Amen.

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Appendix:

Οὐκ ἔστιν δίκαιος οὐδὲ εἷς, (Romans 3:10b)

ὅτι ἄνθρωπος οὐκ ἔστιν δίκαιος ἐν τῇ γῇ (Ecclesiastes 7:20a, LXX)

οὐκ ἔστιν ὁ συνίων, οὐκ ἔστιν ὁ ἐκζητῶν τὸν θεόν. πάντες ἐξέκλιναν ἅμα ἠχρεώθησαν· οὐκ ἔστιν ὁ ποιῶν χρηστότητα, [οὐκ ἔστιν] ἕως ἑνός. (Romans 3:11-12).
εἰ ἔστιν συνίων ἢ ἐκζητῶν τὸν θεόν. πάντες ἐξέκλιναν, ἅμα ἠχρεώθησαν, οὐκ ἔστιν ποιῶν χρηστότητα, οὐκ ἔστιν ἕως ἑνός. (Psalm 13:2b-3, LXX)

τάφος ἀνεῳγμένος ὁ λάρυγξ αὐτῶν, ταῖς γλώσσαις αὐτῶν ἐδολιοῦσαν, (Romans 3:13a, Psalm 5:10b)

ἰὸς ἀσπίδων ὑπὸ τὰ χείλη αὐτῶν· (Romans 3:13b, Psalm 139:4b, LXX)

ὧν τὸ στόμα ἀρᾶς καὶ πικρίας γέμει, ὀξεῖς οἱ πόδες αὐτῶν ἐκχέαι αἷμα, σύντριμμα καὶ ταλαιπωρία ἐν ταῖς ὁδοῖς αὐτῶν, καὶ ὁδὸν εἰρήνης οὐκ ἔγνωσαν, οὐκ ἔστιν φόβος θεοῦ ἀπέναντι τῶν ὀφθαλμῶν αὐτῶν (Romans 3:14-17, Psalm 13:3, LXX)