I am doing
local w="" if [[ -v "$w" ]]; then echo printf '%s\n' "VARIABLE SET TEST" printf '%s\n' " w=\"$w\" | w: $w" printf '%s\n' \ " [[ -v \"\$w\" ]] TRUE, w is not set to a value by assignment" printf '%s\n' \ " [[ ! -v \"\$w\" ]] FALSE, w is not set to a value by assignment" fi [[ -v "$w" ]] should return True if the shell variable is set (has been assigned a value). But this is returning false.
As mentioned in comments, [[ -v var ]] takes the variable name, not the expanded value. With just the value, it wouldn't be able to tell the difference between an unset variable and a variable set to the empty string.
So,
[[ -v var ]] tests if var is set (but it can be empty)[[ -n "$var" ]] tests if var expands to a non-empty string, i.e. is set and not empty[[ "$var" ]] same as above[[ -z "$var" ]] tests if var expands to an empty string, i.e. is unset or set to the empty string.Though note that the latter two will croak if the variable is unset and set -u is in effect. You'd need to use e.g. the default value expansion "${var-default}" to avoid that. Also the -v test is not standard, look into DaG's answer for a method that should work in all POSIX-like shells.
To determine if variable is set
-z returns 0 (true) if length of string is zero.
${var+x} if variable (var) is set, expands to x
[ -z "${var+x}" ] && echo "variable unset" -n returns 0 (true) if length of string is non-zero
[ -n "${var+x}" ] && echo "variable set" To check variable and display error when unset use:
"${var?warn_unset}" To check if variables are unset or empty use : before +/? character.
[ -z "${var:+x}" ] && echo "variable unset or empty" "${var:?warn_unset_or_empty}" 
${var:+x}, you get x only if the variable is set and non-empty. Without the colon ${var+x}, you get x is it's set to any value. Same with ${var:?error}, it'd also trigger for a set but empty value, not just an unset one.
-v varnamenot its value-v "$varname"I think?$varnamethat I want.if test "${w+x}"(or equivalentlyif [ "${w+x}" ]). Cutely,if ${w+true} falseif [ "${w+false}" ].