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enter image description hereI would like to have access to a disjunction sign (and a conjunction sign), larger than $\vee$ and smaller than $\bigvee$ ($\wedge$ and $\bigwedge); its height should be the same as $\exists$ and $\forall$.

The British logician Geoffrey Hunter used such a $\mediumvee$ and $\mediumwedge$, in his book Metalogic: An Introduction to the Metatheory of Standard First Order Logic, for the existential and universal quantifiers. Sometimes one sees the use of much larger versions for the mentioned purpose, but I prefer the size I indicated.

How may the problem be resolved?

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I'd find it utterly confusing if the “Hunter-style” universal quantifier is used along the standard “and” symbol.

Here is a possibility, scaling \land to match the uppercase letters (I used T as the matching character).

\documentclass{article}
\usepackage{graphicx}

\DeclareRobustCommand{\hunterforall}{%
  % no need to make it scalable in sub/superscripts
  \mathord{\resizebox{!}{\fontcharht\textfont1 `T}{\raisebox{\depth}{$\land$}}}%
}
\RenewCommandCopy{\forall}{\hunterforall}

\begin{document}

$\forall x(F^*(x)\land F^{*\prime}(x))$

\end{document}

enter image description here

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