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I see the two used interchangeably. Is one the subset of the other or are they both the same thing?

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First-order logic and first-order predicate calculus are the same thing, which is exactly why the terms are used interchangeably. See, for example, Wikipedia:

First-order logic [...] is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic.

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