The Semitic language family has two main branches: East and West. The West Semitic branch has two main branches: Central and South.
The simplest way to look at the Central Semitic languages is to think in terms of three branches to the family: Canaanite, Aramaic, and Arabic. There is scholarly debate on the details, but this is the easiest way. Canaanite includes Hebrew, Phoenician, Moabite, Punic, Ammonite, Edomite, Ugaritic, etc. Hebrew is the only surviving Canaanite language. Aramaic includes the ancient languages Neo-Babylonian, Biblical Aramaic, Syriac, Samaritan, and the modern languages that are often lumped together as Neo-Aramaic (including modern Assyrian and modern Chaldean). Arabic is not a single language, but is a close-knit group of languages that evolved from Classical Arabic.
Hebrew was the language of ancient Israel and Judah. But when the Babylonians conquered Judah in 586 BCE, the language went extinct. All the inhabitants of Judah started speaking Neo-Babylonian. The Jews who were deported to Babylon also started speaking Neo-Babylonian, but remembered Hebrew as a religious language and started compiling the Old Testament in Hebrew. In their daily lives, they all spoke Neo-Babylonian. Neo-Babylonian evolved into Aramaic.
When the exiled Jews returned to Palestine, everyone spoke a variety of Neo-Babylonian that is often called Biblical Aramaic. This was the situation when Jesus lived. Everyone in Palestine spoke Aramaic as their native language. The men learned Hebrew as children so that they could read the Old Testament.
The language Modern Hebrew was revived as an everyday language in 1948 by Jews returning to Palestine. It was based on the Hebrew that they learned as children in order to read the Talmud and the Old Testament. It is not identical to Biblical Hebrew, but is similar.
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