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To put it quite simply, a synthetic judgment is when you generalize (synthesize) existing knowledge, thereby producing new knowledge that you did not have before. For example: “By the brightest star of Ursa Minor, you can determine the direction to the north.” This judgment is synthetic, since there is nothing in the very properties of Alpha Minor Dipper (the North Star) that would connect it with a logical necessity with the direction to the north. You generalize your observations of the starry sky to come to this conclusion.
Analytical judgment is when you break down (analyze) existing knowledge about an object, without producing any new knowledge, but identifying characteristics that may have previously been hidden. For example, “a regular icosahedron has 20 faces.” In this case, you did not generate new knowledge, because the property of having twenty faces is contained in the very definition of a regular icosahedron. However, for a person who didn't know what an icosahedron is, this may be new information.
You can look at it this way. Synthetic propositions are propositions whose truth is random, not necessary, and depends on the state of affairs in nature (for example, Alpha Minor Dipper will stop pointing north due to precession after a while, and the sentence in the first paragraph will become false). Analytical judgments are necessary judgments, the truth of which does not depend on the state of affairs in nature, and the verification of which does not require an appeal to experience (an icosahedron will always have 20 faces, even if there is not a single icosahedron in the entire universe).
In the twentieth century, W. W. O. Quine, in his work “Two Dogmas of Empiricism”, by the way, quite convincingly showed that the distinction between “analytical” and “synthetic” judgments is conditional and there is really no hard border between them. In other words, purely synthetic judgments, as well as purely analytical ones, are impossible.
As an additional illustration: to the question ” what is happiness?” you can answer analytically and synthetically. Analytically, for example: happiness is a certain desired state of complete satisfaction, etc. Synthetically: “happiness is when you are understood” or “happiness is in cutting down the enemy”, etc. People often confuse analytical and synthetic meanings (judgments), passing one for the other. In the first case, we identify, so to speak, the form of the concept, and in the second, we fill it with one or another empirical content. Before you can understand what happiness is for someone, you need to find out what it's all about.