How to write these symbols (double vee, wedge and bracket)

Question

I want to know how could I do these three things (in red):

For the last one, I'm curious about the brackets.

Probably they are not too dificult to "write" in this languege but I've no clue. I hope you can help me.

Answer 1

The double brackets are defined in stmaryrd as \llbracket and \rrbracket.

For the other symbols, provided they have to be used only in display mode, you can define

\DeclareMathOperator*{\bigdoublewedge}{\bigwedge\mkern-15mu\bigwedge}
\DeclareMathOperator*{\bigdoublevee}{\bigvee\mkern-15mu\bigvee}

MWE

\documentclass{article}
\usepackage{amsmath,stmaryrd}

\DeclareMathOperator*{\bigdoublewedge}{\bigwedge\mkern-15mu\bigwedge}
\DeclareMathOperator*{\bigdoublevee}{\bigvee\mkern-15mu\bigvee}

\begin{document}
\[\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \llbracket p_{1} \rrbracket_{v}\]
\end{document} 

---

EDIT

If \bigdoublewedge and \bigdoublevee have to be used not only in display mode, the following definitions are more suitable

\newcommand{\bigdoublewedge}{%
  \mathop{
    \mathchoice{\bigwedge\mkern-15mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-11mu\bigwedge}
    }
}

\newcommand{\bigdoublevee}{%
  \mathop{
    \mathchoice{\bigvee\mkern-15mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-11mu\bigvee}
    }
}

MWE:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\bigdoublewedge}{%
  \mathop{
    \mathchoice{\bigwedge\mkern-15mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-12.5mu\bigwedge}
               {\bigwedge\mkern-11mu\bigwedge}
    }
}

\newcommand{\bigdoublevee}{%
  \mathop{
    \mathchoice{\bigvee\mkern-15mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-12.5mu\bigvee}
               {\bigvee\mkern-11mu\bigvee}
    }
}

\begin{document}
\[\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]

\[\textstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]

\[\scriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]

\[\scriptscriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}\]
\end{document} 

Answer 2

A variant of karlkoeller's method that doesn't require guessing the amount of the back up. The relative position of the two symbols can be adjusted by changing the factor in

\makebox[1.35\wd\z@][s]{$\m@th#1#2\hss#2$}%

which in this example is set to 1.35.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\bigdoublevee}{\big@doubleop{\bigvee}}
\newcommand{\bigdoublewedge}{\big@doubleop{\bigwedge}}
\newcommand{\big@doubleop}[1]{%
  \DOTSB\mathop{\mathpalette\big@doubleop@aux{#1}}\slimits@
}

\newcommand\big@doubleop@aux[2]{%
  \sbox\z@{$\m@th#1#2$}%
  \makebox[1.35\wd\z@][s]{$\m@th#1#2\hss#2$}%
}
\makeatother

\begin{document}
\begin{gather*}
\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\\
\textstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\\
\scriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\\
\scriptscriptstyle\bigdoublewedge_{i\leq 0}\varphi_{i}=\varphi_{0}\qquad
  \bigdoublevee_{i\leq 0}\varphi_{i}=\varphi_{0}
\end{gather*}
\end{document}