The term "establish Justice" in the preamble of the Constitution means to "begin fairness for all." There are reasons why this statement was added into the preamble of the Constitution. The preamble explains the reasons why the U.S. government was formed a certain way.
The preamble to the United States Constitution is an introductory statement outlining the Constitution's purposes both fundamentally and guiding. The revolution against England involved many important aspects, which were all important to the American citizens. The U.S. government has a legal system that is expected to make fair decisions regarding citizens' problems. The need developed for a judicial system that could handle any disputes arising between states or a dispute arising from people whom resided in different states. This ensured that all persons would be treated fairly and equally and guaranteed that their rights would be respected by others. It ensured that no individual nor entity had was superior to civil and criminal statutes. Adding this phrase into the preamble of the Constitution helped the American citizens feel secure in knowing that they would have justice under the new constitution. This phrase was meant to establish a fair court system. The court system referred to is the one which protects citizens' rights in the court of law. -- Source: http://Ask.com
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Establishing justice means that all people are to be ruled by the laws of their states or country that helps keep people safe and keeps certain rules about what people are supposed to do and not supposed to do protected.
Wikianswers Legal offered: In the Preamble Of the Constitution "Establish Justice" means that our government has a legal system which makes fair descisions to fix problems among its citizens. Establishing Justice means that all people are to be ruled by the laws of their states or country that helps keep people safe and keeps certain rules about what people are supposed to do and not supposed to do protected.

"... To establish Justice"
The essence of American democracy is contained in the Declaration of Independence, with its ringing phrase, "All men are created equal," and the follow-up statements "that they are endowed by the Creator with certain unalienable rights, that among these are life, liberty and the pursuit of happiness."
The Constitution makes no distinction as to the wealth or status of persons; all are equal before the law, and all are equally subject to judgment and punishment when they violate the law. The same holds true for civil disputes, involving property, legal agreements and business arrangements. Open access to the courts is one of the vital guarantees written into the Bill of Rights.
"... To Insure Domestic Tranquility"
The stormy birth of the United States and the unsettled conditions along the American western frontier convinced Americans of the need for internal stability to permit the new nation to grow and prosper. The federal government created by the Constitution had to be strong enough to protect the states against invasion from the outside, and from strife and violence at home. No part of the continental United States has been invaded by a foreign nation since 1815. The state governments have generally been strong enough to maintain order within their own borders. But behind them stands the awesome power of the federal government, which is constitutionally empowered to take the necessary steps to preserve the peace.
"... To Provide For The Common Defense"
Even with its independence secured, the new nation faced very real dangers on many sides. On the western frontier, there was the constant threat from hostile Indian tribes. To the north, the British still owned Canada, whose eastern provinces were jammed with vengeful American Tories. The French owned the vast Louisiana Territory in the continental midwest. To the south, the Spanish held Florida, Texas and Mexico. All three European powers had colonies in the Caribbean Sea, within striking distance of the American coast. Moreover, the nations of Europe were embroiled in a series of wars that spilled over into the New World.
In the early years, the constitutional objective of providing a "common defense" focused on opening up the territory immediately beyond the Appalachian Mountains and negotiating a peace with the Indians who inhabited the area. Within a short time, however, the importance of military strength was underscored by the outbreak of war with England in 1812, skirmishes with the Spanish in Florida and war with Mexico in 1846.
As America's economic and political power increased, its defensive strength grew. The Constitution divides the defense responsibility between the legislative and executive branches: Congress alone has the power to declare war and to appropriate funds for defense, while the president is commander-in-chief of the armed forces and bears primary responsibility for the defense of the country.
"... To Promote The General Welfare"
At the end of the Revolution, the United States was in a difficult economic position. Its resources were drained, its credit shaky and its paper money was all but worthless. Commerce and industry had come to a virtual halt, and the states and the government of the Confederation were deeply in debt. While the people were not in eminent danger of starving, the prospects for economic development were slim indeed.
One of the first tasks facing the new national government was to put the economy on a sound footing. The first article of the Constitution provided that:
The Congress shall have power to lay and collect taxes ... to pay the debts and provide for the ... general welfare of the United States ...
The tax power enabled the government to finance its war debts and to put the currency on a firmer basis. A secretary of the treasury was appointed to look after the fiscal affairs of the nation, and a secretary of state to handle relations with other nations. Also appointed were a secretary of war and an attorney general. Later, as the country expanded and the economy became more complex, the well-being of the people necessitated the creation of additional executive departments.
"... To Secure The Blessings Of Liberty To Ourselves And Our Posterity"
The emphasis on personal liberty was one of the salient features of the new American republic. Coming, as many of them had, from a background of political or religious suppression, Americans were determined to preserve freedom in the New World. The framers of the Constitution, in giving authority to the federal government, were careful to protect the rights of all persons by limiting the powers of both the national and state governments. As a result, Americans are free to move from place to place, make their own decisions about jobs, religion and political beliefs, and go to the courts for justice and protection when they feel these rights are being infringed upon.