# Meaning of $\dashv\vdash$

I was looking at ProofWiki's articles 'Definition:Equidistance' and 'Definition:Between (Geometry)'and came across the symbol '$\dashv\vdash$.' What does it mean?

• Basically "can prove and be proved by"; or "is syntactically equivalent with" (for a given formal system). – Graham Kemp Feb 25 '15 at 0:21

$\dashv\vdash$, means that two statements are interderivable.   It's an expression of logical equivalence, or syntactic equivalence, in formal proofs.

$$p\dashv\vdash q\qquad\text{ means that }\quad p\vdash q\quad\text{ and }\quad q\vdash p$$

.

PS: The turnstyle, $\vdash$, is the logical implication symbol in formal proofs; which may be read aloud as "therefore".

• I thought that '$A \vdash B$' meant 'if A, then B' and that '$\therefore$' meant 'therefore. – RandomDSdevel Feb 26 '15 at 0:47
• You can use "If $A$, then $B$", "Because $A$, therefore $B$", "From $A$, follows $B$", "$A$ proves $B$", or however you want to announce it. The tri-dot "therefore" held the same sense, but it is not often used as such any more, except to punctuate the end of a proof. – Graham Kemp Feb 26 '15 at 1:10
• Ah, I see. That makes sense, and I'd up-vote your comment if I had enough reputation on this Stack Exchange site. – RandomDSdevel Feb 26 '15 at 20:15
• @RandomDSdevel: Note that there are 3 distinct notions, "$A \vDash B$" (any interpretation that satisfies $A$ also satisfies $B$) and "$A \vdash B$" (from $A$ we can derive $B$) and "$A \Rightarrow B$" / "$A \rightarrow B$" (if $A$ then $B$). So technically what you said is not correct. In many formal systems, the 3 notions are in some sense equivalent, such as for first-order logic, where the first two because of the completeness theorem and the last two because of the deduction theorem. – user21820 Feb 24 '16 at 11:03