Lets take, for example, the deduction theorem:
For any Well Formed Formulas group $\Sigma$ and for any 2 formula $\alpha, \beta$ ,
$$\Sigma \cup \{\alpha\} \vdash \beta\iff\Sigma \vdash (\alpha\to\beta)$$
Question:
What does $\vdash$ mean? and what does it mean in the specific example? (deduction theorem)
Please help.
The "$\vdash$" symbols reads as "proves". That is, $\Sigma \vdash \alpha$ means that the set of sentences $\Sigma$ proves $\alpha$. Usually, this is in some proof system that's already set out, or implicitly understood in the background.
In this case, the deduction theorem just says "$\Sigma$ together with the assumption $\alpha$ can prove $\beta$ if and only if $\Sigma$ (alone) can prove $\alpha \rightarrow \beta$."
The notation $\Sigma\vdash\varphi$ means that there exists a proof from $\Sigma$ to the statement $\varphi$.
A proof is a finite sequence of well formed formulas, where we are allowed to write in the sequence either formulas which appear in $\Sigma$, or formulas which are deduced from previously written formulas (in the sequence) by using inference rules.
For example, if we have modus ponens (that is $\alpha,\alpha\rightarrow\beta$ imply $\beta$) then given a proof where $\alpha$ and $\alpha\rightarrow\beta$ has been written, we are allowed to write $\beta$.
What are the inference rules? This depends on the exact logic.
