Porcelain was first made in China, so "porcelain" and "china" can be used interchangeably, especially when referring to dinnerware. However, there are nuances to take into consideration. There are actually two types of porcelain: hard-paste and soft-paste. Furthermore, in common usage, "china" has become the generic term for any ceramic dinnerware, including "bone china."
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"Porcelain," the term used when referring to high-quality ceramics--not just dinnerware--is made with kaolin, a white clay native to China. Porcelain is readily identifiable by its translucence and glassy sheen, obtained through an intense heat and fusion process called vitrification. If you hold porcelain up to the light, you should be able to see a shadow through it. Though often white, and sometimes heavy, porcelain always transmits light and is never opaque.
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Hard-paste porcelain is made from kaolin and fusible felspathic china stone that are heated at extremely high temperatures, causing them to fuse. This process is called vitrification and the results are a nonporous, very hard, white surface that has a light metallic ring when struck lightly. The finished product, often referred to as "true porcelain," has a translucent, glassy sheen, and is at once delicate and durable.
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Soft-paste porcelain is also made from kaolin but it is fused with a vitreous frit and heated at temperatures that are obviously high, but not as high as those used to make hard-paste porcelain. The results still bear the telltale translucent, glassy sheen, however, soft-paste porcelain is liable to break far easier than its counterpart. Nearly all 18th-century English porcelain is soft-paste.
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"China" is used to describe ceramic white dinnerware made with kaolin, originating in China, and copied ever since. "Bone china" is a specific type of porcelain that is made with the same kaolin mixture but which includes bone ash to make it a more sturdy and durable product.