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symbols that I want to have displayed as superscripts end up displayed right next to a formula when I use stmaryrd package.

Here's the code I used:

$\llbracket \lozenge \phi \rrbracket^{w,c,i}$

why do superscripts fail to work as usual in stmaryrd package and is there a way around this?

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    Welcome to TeX.SX! Please add a minimal but working example (MWE), that can be used to reproduce the problem, instead of the code snippet to your answer. We can use such an answer not only to see the problem but also to test out suggestions. So it would help us to help you. – Schweinebacke Aug 4 '17 at 8:57
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    Welcome to TeX.SX! If I try it, I get the expected superscript. Please, make a complete (but minimal) failing example. I guess you're loading some package for linguistics and it's essential to know which ones. – egreg Aug 4 '17 at 8:57
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If ^ does not work for some reason as token for superscript (maybe because of a different category code), then, the command \sp provides an alternative (the command name for subscript is \sb):

\documentclass{article}
\usepackage{amssymb}
\usepackage{stmaryrd}
\begin{document}
$\llbracket \lozenge \phi \rrbracket^{w,c,i}$

$\llbracket \lozenge \phi \rrbracket\sp{w,c,i}$
\end{document}

Result

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