From what I understand α |= β means that if α comes out true, so does β. ie, α entails β.
This got me thinking, I know that there are truth tables for α → β, α ∨ β, etc. Is there a standard truth table for α |= β?
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$\begingroup$Your understanding is correct: $α \vDash β$ means that $α$ is a logical consequence of $β$.
There is no "standard" truth table for it, but of course you can use truth table: write a t-t for the formulas $α$ and $β$ and check if in every line of the t-t where $α$ is evaluated to TRUE also $β$ is TRUE.
If so, this will show that the second is consequence of the first.
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