Notation in logic paper

Question

I'm reading the paper "Elementary Intuitionistic Theory" by Craig Smorynski (https://doi.org/10.2307/2271732) and there is some notation I don't understand.

It looks like common logic symbols but with two (or one) bold dots on the sides, e.g. $\centerdot \equiv \centerdot$.

Here are formulae which use such symbols: \begin{align} &\begin{aligned} A \vee \neg A \supset [(A \wedge B) \vee (\neg A \wedge C) \centerdot \equiv \centerdot(A \supset B) \wedge (\neg A \supset C)], \end{aligned}\\ &\begin{aligned} A \vee \neg A \supset [(B \supset (A \wedge C &\centerdot\vee\centerdot \neg A \wedge D)) \\ &\centerdot \equiv \centerdot(A \supset (B \supset)) \wedge (\neg A \supset (B \supset D))], \end{aligned}\\ &\begin{aligned} A \vee \neg A \centerdot \wedge (A \supset B) \centerdot \equiv \centerdot A \wedge B \vee \neg A, \end{aligned}\\ &\begin{aligned} \bigwedge x (Ax \supset B) \centerdot \equiv \centerdot (\bigvee x Ax \supset B). \end{aligned} \end{align}

Does anyone know what do they mean?

P.S. As far as I know, $\bigvee$ and $\bigwedge$ are used here for universal and existential quantors respectively.

Answer 1

$\supset$ is an older symbol for implication ($\rightarrow$).

The dotted $\cdot \equiv \cdot$ is just equivalence (or bi-implication) $\leftrightarrow$ or $\Leftrightarrow$. The dots are an alternative system to parentheses.

You're almost right about the quantors:

$\bigwedge$ is $\forall$, while $\bigvee$ is $\exists$. (compare intersections $\bigcap$ and unions $\bigcup$).