TITLE: Notation Question: What does $\vdash$ mean in logic?
QUESTION [18 upvotes]: In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic.
I've been reading some of the logic questions on here to get used to notation, etc.  However, when I came across the question Visualizing Concepts in Mathematical Logic, I didn't understand what the $\vdash$ symbol means.
It's not in Discrete Math by Epp, nor is it in my mom's old logic book from when she went to college.
Wikipedia's Math Symbols page says it means "can be derived from" when used in a logic context.  However, that doesn't make any sense in the above question, as there is nothing on the left of the $\vdash$.
So, what does $\vdash$ mean, especially in the context of the question linked above?

REPLY [6 votes]: We should attend to a distinction between "$\vdash$" and "$\models$".  The notation $A\vdash B$ means $B$ can be deduced from $A$ in some reasonable system of deduction, and "reasonable" should mean at the very least

There is an algorithm for deciding which deductions are valid ("effectiveness"); and
If $B$ can be deduced from $A$ then $B$ is true in every structure in which $A$ is true (soundness).

One may also have

If $B$ is true in every structure in which $A$ is true, then $A\vdash B$ (completeness).

(The word "completeness" here should not be confused with the "completeness" referred to in Gödel's incompleteness theorem; that is different.)
The notation $A\models B$ means simply that $B$ is true in every structure in which $A$ is true.