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10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!). | |
From: Wilfrid Hodges (Model Theory [2005], 3) | |
A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy! |