ELI5: How can a country bomb another country because they contain groups of ISIS without first declaring war?

And here is the text from the link for those who don't want to go off site:

Theories of International Relations

A theory of international relations is a set of ideas that explains how the international system works. Unlike an ideology, a theory of international relations is (at least in principle) backed up with concrete evidence. The two major theories of international relations are realism and liberalism.

National Interest

Most theories of international relations are based on the idea that states always act in accordance with their national interest, or the interests of that particular state. State interests often include self-preservation, military security, economic prosperity, and influence over other states. Sometimes two or more states have the same national interest. For example, two states might both want to foster peace and economic trade. And states with diametrically opposing national interests might try to resolve their differences through negotiation or even war.

Realism

According to realism, states work only to increase their own power relative to that of other states. Realism also claims the following:

  • The world is a harsh and dangerous place. The only certainty in the world is power. A powerful state will always be able to outdo—and outlast—weaker competitors. The most important and reliable form of power is military power.
  • A state’s primary interest is self-preservation. Therefore, the state must seek power and must always protect itself
  • There is no overarching power that can enforce global rules or punish bad behavior.
  • Moral behavior is very risky because it can undermine a state’s ability to protect itself.
  • The international system itself drives states to use military force and to war. Leaders may be moral, but they must not let moral concerns guide foreign policy.
  • International organizations and law have no power or force; they exist only as long as states accept them.

Machiavelli

One of the best-known realist thinkers is the notorious Niccolo Machiavelli. In his book The Prince (1513), he advised rulers to use deceit and violence as tools against other states. Moral goals are so dangerous, he wrote, that to act morally will bring about disaster. He also gave advice about how to deal with conflicts among neighboring states and how to defend one’s homeland. Machiavelli’s name has become synonymous with nasty and brutal politics.

Liberalism

Liberalism emphasizes that the broad ties among states have both made it difficult to define national interest and decreased the usefulness of military power. Liberalism developed in the 1970s as some scholars began arguing that realism was outdated. Increasing globalization, the rapid rise in communications technology, and the increase in international trade meant that states could no longer rely on simple power politics to decide matters. Liberal approaches to international relations are also called theories of complex interdependence. Liberalism claims the following:

  • The world is a harsh and dangerous place, but the consequences of using military power often outweigh the benefits. International cooperation is therefore in the interest of every state.
  • Military power is not the only form of power. Economic and social power matter a great deal too. Exercising economic power has proven more effective than exercising military power.
  • Different states often have different primary interests.
  • International rules and organizations can help foster cooperation, trust, and prosperity.

**Example:* Relations among the major Western powers fit a model of complex interdependence very well. The United States has significant disagreements with its European and Asian allies over trade and policy, but it is hard to imagine a circumstance in which the United States would use military power against any of these allies. Instead, the United States relies on economic pressure and incentives to achieve its policy aims.*

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